Christophe Guttin

Olivier Exshaw

Guillaume Brès

Jean- Luc Mocellin

Christophe Hoarau

Daniel

Sébastien Triqueneaux

Guillaume Donnier-Valentin

Anne Gérardin

Olivier Tissot En

Olivier Bourgeois

Olivier Arcizet

Benjamin Pigeau

Ben- Ii Jamin Besga

Clemens Winkelmann

Philippe Roche

Elsa Lhotel

Franck Balestro

Panayothis Spathis

Florence Lévy-Bertrand

Etienne Wolf

Nicolas Roch

Klaus Has- Selbach

Pierre Verlot

Jukka Pekola

Adrian Bachtold

Andrew

Eva Weig

Pierre-François Cohadon

Signe Seidelin

Jean-Christophe Toussaint

Julien Poëtte

Nicolas Ruty

Aurélien Kuhn

Elie Céline

Alexis Laura

Rémy

Victor

d'horizons divers : citons parmi d'autres l'ensemble de l'équipe TPS/pôle capteurs et tout particulièrement

Avant de céder aux sirènes d'un globish laxiste, je voudrais remercier un certain nombre de personnes qui ont contribué à l'excellent déroulement de ces quatre années.

Commençons donc par le plus facile : Eddy mérite les plus amples remerciements, lui qui a été moteur en toutes choses. J'ai tout particulièrement apprécié le fait de trouver un interlocuteur zen, patient et dépourvu de condescendance (une qualité plutôt rare par les temps qui courent), ainsi qu'un scientique de haut vol à l'ouverture d'esprit remarquable. Au-delà, j'ai beaucoup appris de nos looongues pauses café mêlant physique shadok (plus ce sera compliqué, plus cela aura de chances de marcher !), dettes abyssales envers la machine et destruction de feutres. On aura aussi massacré des zombies en surcharge pondérale (incarnations de referees obtus ?), bu quelques pintes aux quatre coins du globe, et déchiré nombre de codes mathematica "uc-tuv82" avec en toile de fond, "J'ai trouvé la solution en nageant hier soir !". Une façon idéale de faire de la recherche, en somme.

Il me faut ensuite mentionner les collègues de l'équipe UBT : du côté des permanents, Henri Godfrin, infatigable conteur, et Andrew Feerman, qui a été un interlocuteur de choix lorsque j'eectuais mes mesures de dissipation. Quant aux autres, eh bien...j'ai une pensée particulière pour Martial "Fokker-Bank" Defoort, qui m'a guidé pas à pas lorsque je n'étais qu'un bleu prétentieux et ignorant. En outre, je ne puis passer sous silence les concerts de salon avec des ballons et l'accueil californien, entre autres. Viennent ensuite Rasul Gazizulin et Xin "Go to the hell!" Zhou, avec qui j'ai plus ou moins directement travaillé, pour l'enrichissement culturel indéniable consistant à savoir jurer indiéremment en russe ou en chinois à une heure avancée de la nuit. Enn, merci et bon courage à ceux qui y sont encore, Annina, David, Dylan, Sumit, et ceux qui y étaient : Ketty, Ana, Yuriy, Ahmad mais aussi Kunal dont l'aide concentrée sur une semaine et un coup de téléphone en particulier m'auront été très

Fundamental experimental research, by denition, relies on the most sensitive detectors: this ranges from subatomic physics, where ongoing discoveries of new particles make use of state-of-the-art calorimeters, to general relativity, where one major milestone was the recent observation of gravitational waves predicted almost a century ago thanks to ultrasensitive interferometers [1], not to mention superconducting circuits for quantum-limited detection of extremely small electromagnetic signals [2], to name a few. These tremendous achievements go hand in hand with endless detector improvements, which further grow into research topics on their own, as in the case of optomechanics which expanded along with gravitational wave detectors [3,4]. Among these instruments, Micro-Electro-Mechanical Systems (MEMS) are probes of choice: they are transducers of electrical signals into mechanical ones (and vice-versa)

with very low masses, due to their small volume (at least one of their dimensions is in the µm range), which permits extreme sensitivities. They are now most conveniently used as biosensors or as accelerometers in cars or cellphones, but along these technological aspects they also present a fundamental interest for the physicist, as their small dimensions make them sensitive to forces that manifest only on small lengthscales, such as the Casimir force originating from quantum vacuum uctuations [5], or possible deviations to Newtonian gravity [6]. Even more appealing is the possibility oered by tremendous progresses in nanofabrication over the last decades, which have put Nano-Electro-Mechanical Systems (NEMS, having at least one sub-micronic dimension) within reach at the end of the 1990's [7,[START_REF] Cleland | A nanometre-scale mechanical electrometer[END_REF]. Since then, nanomechanics has extended its range to smaller and smaller systems that are now atomically thin, like 2 Chapter 1. General Introduction e.g. graphene and carbon nanotubes [START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF]. While industrial applications for such objects are still not widespread, they do push further sensitivity limits that allow to probe even smaller lengthscales previously out of reach for MEMS: an example is a 150 nm long carbon nanotube used as a mass spectrometer which enabled the detection of adsorption events of naphtalene molecules with a yoctogram resolution [START_REF] Chaste | A nanomechanical mass sensor with yoctogram resolution[END_REF], that is, at the proton level (see Fig. 1.1 upper left corner).

Not only NEMS are suitable sensors, they have also proven ideal, versatile model systems for fundamental studies. At the most basic level, a vibrating NEMS simply is a macroscopic mechanical degree of freedom, and possesses modes of vibrations that depend on the structure's geometry. Each one of these modes is at rst order a harmonic oscillator, but it can be pushed further to display a more subtle behaviour: a resonator undergoing a strong mechanical excitation can display a non-linear behaviour [START_REF] Lifshitz | Thermoelastic damping in micro-and nanomechanical systems[END_REF], adding a great deal of complexity. Thus, because of their relative simplicity of use, they are suitable to study in the non-linear regime e.g. bifurcation phenomena or chaotic dynamics [START_REF] Aldridge | Noise-Enabled Precision Measurements of a Dung Nanomechanical Resonator[END_REF][START_REF] Kozinsky | Basins of Attraction of a Nonlinear Nanomechanical Resonator[END_REF]. As mechanical modes are coupled to their environment, they experience also position uctuations, which are of interest when a non-linear interaction mixes them with a deterministic motion. As such, they are interesting testbeds for statistical physics, in and out of equilibrium [START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF][START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF]. The uctuations, that is, Brownian motion, are also representative of a mode's thermal occupancy, that can be for instance removed to cool this oscillator down to its ground state of motion. This quest was rst pursued by the close eld of opto-mechanics, where mechanical motion is detected and controlled by light elds, and where light conned in a cavity with a movable mirror can be used to cool down the mirror's mechanical motion (in the sense of having its thermal occupancy lowered) [3,[START_REF] Arcizet | Radiationpressure cooling and optomechanical instability of a micromirror[END_REF][START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. The experimental tour de force consisting in reaching the quantum regime, as far as electrical means are concerned, was demonstrated a few years ago in Ref. [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF] (see Fig. 1.1 upper right corner) with passive cooling of a GHz resonator in a dilution refrigerator, and in Ref. [START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF] with microwave sideband cooling of a thin membrane capacitively coupled to a quantum-limited detector (see Fig. 1.1 lower left corner). These impressive achievements are rst steps towards systematic studies of the elusive quantum coherence of macroscopic mechanical degrees of freedom, addressed theoretically for MEMS and NEMS in e.g. Refs. [START_REF] Armour | Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box[END_REF][START_REF] Sillanpää | Macroscopic quantum tunneling in nanoelectromechanical systems[END_REF], in relation with fundamental theories on decoherence of position states or wave function collapse associated with a quantum measurement.

Those very fundamental purposes force to ask ourselves what is truly quantum in a measurement [START_REF] Joos | Decoherence and the Appearance of a Classical World in Quantum Theory[END_REF]. In that regard, recent works highlight the formal analogies that can be made between coherently driven quantum two-level systems and classical coupled nanomechanical modes [START_REF] Faust | Nonadiabatic Dynamics of Two Strongly Coupled Nanomechanical Resonator Modes[END_REF][START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF], showing that the answer is less trivial than it may seem at rst glance (see Fig. 1.1 lower right corner).

The race towards ecient and reliable quantum opto-and electro-mechanical systems obviously requires a good mastering of the resonator's environment, hence a good understanding of it: these macroscopic model systems are coupled to micro-1.1. Nano-electro-mechanical systems: sensors and versatile platforms for basic research 3

Figure 1.1: upper left: suspended carbon nanotube used in Ref. [START_REF] Chaste | A nanomechanical mass sensor with yoctogram resolution[END_REF] where a yoctogram resolution is reached in mass sensing, through measurements of naphtalene molecules adsorption on the nanotube and xenon binding energy. Upper right: piezoelectric GHz device cooled down passively to a base temperature of 50 mK [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF]. A compression/dilatation mode is measured in its quantum ground state and coherently controlled at the single phonon level through capacitive coupling to a phase superconducting quantum bit circuit. Lower left:

MHz "drum" membrane capacitively coupled to a superconducting LC circuit [START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF] acting as a microwave cavity. One mode is selectively cooled down in its quantum ground state by driving the cavity with a tone detuned to lower frequencies (on the "red-detuned sideband"), where the detuning is equal to the mechanical frequency. Then energy is pumped out of the mechanical mode thanks to an equivalent opto-mechanical coupling: the circuit capacitance is modulated by thermomechanical motion. Lower right: silicon nitride nanobeam with two exural modes of dierent families coupled through capacitive actuation, forming an eective classical two-level system, where usual NMR and quantum information-like operations (e.g. Rabi pulses or Hahn echo) can be reproduced [START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF].

4

Chapter 1. General Introduction scopic degrees of freedom towards which mechanical energy ows in an irreversible way, limiting in the rst place mechanical coherence. But low temperatures allow to selectively turn o uncontrolled degrees of freedom, leaving only a few possibilities that can be addressed readily, be it the damping by a surrounding uid or intrinsic low energy excitations arising from defects in the constitutive materials of the NEMS.

In itself, it turns out to go beyond the focus on NEMS: the issue of two-level systems in amorphous materials is a long-standing problem [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF] that is still lacking of a compelling evidence as to whether they even exist or not in all glass-like materials. In that regard, a NEMS, with reduced dimensions, is a suitable condensed matter probe to bring new insights into the eld [START_REF] Hoehne | Damping in high-frequency metallic nanomechanical resonators[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF].

Context and structure of this thesis

This thesis deals with both model systems aspects and microscopic ones. Several subelds of physics are invoked, and yet, one common feature is noticeable: in all the results reported, uctuations or disorder play a role, which in the end aects mechanical damping or (classical) phase coherence. It can be a noisy signal in time, on the position or frequency, that aects the mechanical phase coherence, or it can be spatial disorder of e.g. the materials of which the NEMS is made of. To leave out possible additional diculty, we use what is arguably the most simple vibrating structures: nanometre-scale (in their cross-dimensions) beams, resonating in the MHz range, which have basic properties very well described by continuum mechanics and do not require heavy simulation tools. That allows us to reach a rather high degree of analytic, quantitative modelling of our results, making discussions sharper mathematically. As such, even though the present thesis is by no means a theoretical work, we shall put a strong emphasis on quantitative modelling when it is possible.

The nano-metre scale aspect is relevant for two reasons: rst, it makes the vibrating object very sensitive to non-linearities [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF], and second, it allows to study the role of reduced dimensionality on dissipation mechanisms, as it is still poorly understood up to now. Besides, the results reported in this manuscript are obtained at low temperatures, ranging from 30 K at most to 10 mK, with a good accuracy above typically 30 mK.

However they do not restrict to low temperature physics in their nature. Most of the time, a low temperature is convenient mainly because it guarantees a high level of environment control and reproducibility. Besides, some signatures, such as the Knudsen layer reported in Chapter 5 or the TLS-induced damping described in Chapter 6 would not manifest clearly at room or even liquid nitrogen temperatures. Finally, we point out that while some of our experiments can call for further developments in the quantum regime, and though the damping issue has a direct impact on the feasibility of quantum-limited experiments, our apparatus is classical, and our treatment of the various problems will mostly remain at the classical level, with the exception of 1.2. Context and structure of this thesis 5

Chapter 6, where we must on occasion invoke quantum tunneling of two-level systems to explain the microscopic origin of damping in our structures.

The manuscript is organized as follows: the rst Chapter aims at setting the experimental and theoretical grounds useful for the rest of the manuscript. It reports our calibration procedures in a high-impedance environment and some techniques, e.g. clean room fabrication, low temperature settings or the NEMS actuation and detection technique. As such, it is rather detailed for the interested reader but does not introduce new results. Therefore the experienced reader can skip it at rst reading.

The rst part of the results obtained in this thesis deals with the model system aspect, at a purely macroscopic scale. In Chapter 3, we provide an in-depth study of the frequency noise originating from position uctuations, that are translated as frequency noise through intrinsic geometric non-linearities. Though it has been reported qualitatively over the last few years [START_REF] Vinante | Thermal frequency noise in micromechanical resonators due to nonlinear mode coupling[END_REF][START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF], it had not yet been completely elucidated, and we show that its full explanation is rather involved, producing an analytic model based on Ref. [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF] which reproduces our data. Besides, it can be a major limitation as for the performance of some devices like e.g. carbon nanotubes [START_REF] Eichler | Symmetry breaking in a mechanical resonator made from a carbon nanotube[END_REF]. This is mainly what is reported in Ref. [START_REF] Maillet | Nonlinear frequency transduction of nanomechanical Brownian motion[END_REF], with a greater attention paid here to theoretical aspects based on the analysis of Ref. [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF]. We show also preliminary results obtained in the case of a strongly driven NEMS, where position uctuations themselves get altered and their dynamics is slowed down by the back-action of the strong drive.

The mechanism studied in Chapter 3 does not involve further energy exchanges, and yet it broadens our resonance lines. As such it can be understood in the framework of decoherence, here seen from a purely classical perspective. In Chapter 4, we build on the analogy by comparing time and frequency domain measurements of the mechanical response in the non-linear regime and under the inuence of external, gate-controlled frequency noise, extracting energy relaxation and phase coherence times in analogy with Nuclear Magnetic Resonance. It reports and goes a little beyond the work done in Ref. [START_REF] Maillet | Classical decoherence in a nanomechanical resonator[END_REF].

The second part of this thesis focuses more on the microscopic degrees of freedom to which our macroscopic mechanical mode couples. In Chapter 5, we investigate rst extrinsic sources of mechanical damping due to a surrounding cryogenic Helium gas, by deliberately injecting gas in the cell. The aim is twofold: while attempting to measure intrinsic damping, we must rst remove other external sources. But it is also a preamble to more involved studies of quantum uids such as superuid 3 He.

We report on the rst measurements by a nano-mechanical probe of the controversial Knudsen layer, which were published in Ref. [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF], and discuss the mechanisms behind its existence, as reported in Ref. [START_REF] Gazizulin | Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rareed Gas[END_REF].

Once this external source is well characterized, we focus on intrinsic sources of damping in Chapter 6, by measuring it down to millikelvin temperatures. We interpret our results within a framework adapted from the long existing so-called Standard Tunneling Model (STM) [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF][START_REF] Phillips | Tunneling states in amorphous solids[END_REF], to the case of a nanomechanical beam with reduced Chapter 2. Experimental and theoretical tools

Introduction

The rst Chapter of this thesis is essentially technical: it aims at providing the basic resources that are useful to understand the results presented further. In the rst section, nanofabrication techniques are described. Then, experimental techniques such as cryogenics, the magnetomotive scheme or lock-in amplier measurements, that are common to many experiments are reported. Finally, a comprehensive calibration of our devices is detailed: it involves common basic and simple theoretical tools such as beam mechanics, electronics and thermal diusion models. Some extensions to nonlinear theories are considered, as they are ubiquitous in our studies. These calibrations enable us to obtain quantities in real units, such as forces in newtons and displacements in meters, which is not a trivial task when it comes to vibrating nanostructures.

In the following, we put the emphasis on having measurements and theory as close as possible, with none or minimum free parameters when needed. We believe that it provides a solid basis to explain quantitatively more subtle results that are presented further. As this calibration procedure has been already described earlier in

Refs. [START_REF] Collin | In-situ comprehensive calibration of a tri-port nanoelectro-mechanical device[END_REF][START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF] for cantilever structures, some details are overlooked, especially when no dierence exists whether we use a cantilever or a doubly-clamped beam, or when they concern only cantilevers. As a proof of robustness of our calibration, we show measurements of parametrically amplied motion, which require that every calibration step be validated to be reproduced accurately by theory.

On a more technical note, while experimental frequencies f are in hertz (Hz) units, in the text we use angular frequencies ω = 2πf , in rad.s -1 , as they appear more naturally in theoretical descriptions, but values are given in Hz units.

Fabrication of suspended vibrating nanostructures

All devices presented in this work have been designed and developed in the nanofabrication facility (Nanofab) at Néel Institute. They only rely on mature and reproducible top-down fabrication techniques. We give here an overview of the general process, as further developments are only specic to single experiments.

The basic process only requires one lithography step on a 1 cm 2 silicon substrate covered with a 100 nm thick silicon nitride lm with a Young's modulus E SiN = 250 ± 150 GPa and a chosen stored stress value. The latter is obtained through dierential contraction between the silicon substrate and the grown nitride thin lm.

Typical internal tensile stress values for NEMS involved in this work are 100 MPa, 0.9 GPa, 1.1 GPa with a typical error bar ± 0.05MPa.

Sample design

While the nanomechanical strucure is itself small, one needs to design an environment that allows to address its motion with electrical means. Thus, as seen in Fig. 2.1, the NEMS is connected to electrodes that are large enough so that 1) their electrical resistance is negligble and 2) one can microbond electrical wires on it. Two electrodes are designed on each side of the NEMS (one side is connected to the ground), so one can perform four wire measurements. Clamps, as well as tracks connecting the NEMS to the electrodes, must be wide enough (at least 20 µm) so that they do not collapse during the etching step (see below). Each pad is annoted according to the bonding scheme. Right: zoom on the central part, with a Scanning Electron Microscope picture of the NEMS. The beam is here 10 µm long, with a gate electrode designed in its vicinity (see Section 2.8.3). Note the spongy silicon bottom, which is a consequence of XeF 2 etching.

Electron beam lithography

To allow selective etching of the structure a mask needs to be deposited. This mask will dene the structure around which the wafer will be etched, while functionalizing the device for electrical control.

To do so, after cleaning the wafer with acetone or isopropanol (IPA) we rst deposit a thin layer (about 100 nm) of polymethyl methacrilate (PMMA 4%) resist on top of the wafer. To obtain a uniform layer, the coated wafer is spinned at 6000 rows per minutes speed for 30 second, then baked at 180 • C for 5 minutes. Note that the resist thickness to be used depends on the metal thickness to be evaporated. It must be thick enough to clearly discriminate insulated parts from non-insulated, but not too thick so that e-beam insulation diuse over the whole resist depth. After resist deposition, the resist coated wafer is patterned through electron beam lithography performed with a Scanning Electron Microscope (SEM): the resist is selectively insulated with a beam of electrons accelerated at 20 keV, which then diuse into the resist. With this voltage value patterning can reach a resolution down to 30 nm for light electron doses and small SEM apertures. Typical doses used in this work are about 250 µC/cm 2 , which is enough to guarantee such a resolution. The insulated resist is then put in a mixture of methyl-isobutyl-ketone (MIBK) and IPA for 35 s, then in IPA for 1 min. This removes the insulated parts.

Si

Metallic layer evaporation and resist removal

The mask is obtained by evaporating a metal (in this work, aluminium) in a vacuum chamber. The evaporation is triggered by e-gun bombarding of a metallic crucible.

The evaporated metal sticking on the nitride lm is organized in crystalline grains of typical size 20 nm at a deposition speed of 0.1 nm/s in a typical vacuum of 10 -6 Torr, but this can be tuned with an appropriate choice of deposition speed: a faster evaporation will result in bigger grains, hence bigger grain boundaries. To be as sensitive as possible to the nitride properties, we evaporate a layer of typical thickness enough to ensure that the metallic layer is continuous (conducting) over the whole sample area. The result is displayed in Fig. 2.4. While the aluminium sticks on the SiN where lithography was performed, the other parts, sustained on resist, will be stripped o with the resist. To do so, the wafer is left in N-methyl-2-pyrrolidone (NMP) at 80 • C for at least an hour, and is occasionally shaked with weak ultrasound waves. In addition, after this, the wafer can be put in acetone to enhance the removing. After cleaning with ethanol and deionized water, the result is now a wafer covered by a mask on the parts where the nanostructures will be processed.

Si

Etching steps

Once the functional parts are delimited, they must be released from the silicon bulk to be able to vibrate. First the SiN layer is attacked throughout a 2 min 30 sec Reactive Ion Etching (RIE) step with a sulfure hexauoride plasma (SF 6 ). While being very selective (it does not attack aluminium), it is also very anisotropic, which allows to dig as vertically as possible. However, it is not very eective for silicon under-etching, which is limiting if we pattern long nanobeams or if we want huge gaps between the nanobeam and the bottom.

Our solution to etch the silicon part with better eciency is to use a Xenon uoride (XeF 2 ) gas. This allows us to selectively etch silicon over considerable thicknesses, as schematically depicted in Fig. 2.5 (right panel). The etching is very isotropic, thus releasing the beams. However, this also results in a so-called undercut, that is an etching of the silicun underneath the clamping electrodes. This limits our possibilities, as undercuts bigger than typically 20 µm might make the structure collapse. This is also the biggest gap between the beam and the bottom allowed by the straightforward use of this technique. Note that to etch deeper, more sophisticated processes must be developed, as in Chapter 5.

A puzzling feature observed on most of the fabricated samples is that the metallic layer gets sometimes "polluted" with uor, which corrupted the electrical and mechan- ical properties of the nanobeams when occuring. We therefore remove the metallic layer with melanine-formadehyde (MF-26) after etching, and re-evaporate the metal following the standard procedure, but without any mask (full eld evaporation). A typical NEMS obtained after the procedure described above is shown in Fig. 2.6. Note the distance between the suspended beam and the bottom. The signicant undercut as well as the spongy nature of the bottom are consequences of the XeF 2 etching. The whole process is very reproducible, with well-controlled dimensions: the typical error on the SiN lm dimensions is at most 5 nm, while the e-gun evaporation leads to at most a few nanometers error on the evaporated aluminum thickness monitored through the e-gun machine. • From a practical point of view, high magnetic elds can be generated with small coils made of conventional superconductors, which is helpful if we use the magnetomotive scheme (see Section 2.6.2).

• The electrical environment is less noisy, resulting in much cleaner measurements.

In addition, since measurements are performed well below the Debye temperature, we are not sensitive (or limited) by phonon-phonon scattering processes that are responsible for additional damping [START_REF] Cleland | Foundations of Nanomechanics[END_REF]. In particular, we resolve the low energy excitations which couple dissipatively to the mechanics (see Chapter 6).

• We have cryogenic vacuum for free, with residual pressures below 10 -6 mbar.

This greatly reduces the gas damping for the NEMS, while ensuring negligible outgasing.

• Thermal contraction/expansion is much smaller, almost inexistant. Therefore dimensions and internal stresses of our devices are well dened.

The sample cell is mounted on a stick, with its upper part xed to a room temperature ange, and its lower part xed to a solenoid coil, used as a magnet for our experiments. The room temperature ange integrates connectors for thermometers and coil wiring, two injection and two detection lines, and one gate electrode line.

The sample holder is a bulk copper piece, designed so that a plastic circuit board (PCB) can be screwed on it. The wafer is glued to this PCB with General Electric (GE) varnish, then electrically connected to it through microbonding wires from the PCB copper tracks to the designed pads (see Fig. 2.2). A carbon resistor is mounted at the back of the copper piece for thermometry in the Kelvin range, as well as a 100 Ω

resistor used as a sample holder heater. A picture of the actual sample holder is shown in Fig. 2.7.

The sample holder is joined together with the upper part of a stainless steel cell, on which connectors allow to pass currents from the top of the stick through coaxial cables. This upper part of the cell is soldered to a pumping line. The lower part is large enough to nest the sample holder while tting the inner diameter of the coil. Before cooling the experiment, the two cell parts are joined and sealed with indium wire, and the closed cell is pumped down to 10 -4 mbar. Lower pressures, below 10 -6 mbar, are reached through cryo-pumping during the cooling process, with the remaining gas being adsorbed on the cell's walls. The temperature is measured by a platinum resistor mounted on the stick down to 77 K by 4-wire measurements, while the carbon resistor mounted on the sample holder measures temperatures down to 1 K using a homemade (PID) balanced bridge technique. The same technique can be used to regulate the sample holder temperature up to 30 K.

The cryostat itself simply consists of two concentric stainless steel cans separated by a vacuum partition. The outer reservoir is lled with liquid nitrogen at 77 K, which precools the inner bath. The latter is then lled with liquid helium to cool down the experimental cell to 4 K. From then on, the liquid nitrogen bath protects the helium part from thermal radiation at 300 K, which reduces the helium evaporation rate.

The typical working temperature is that of liquid helium, 4.2 K. However, one can heat the sample to higher temperatures as mentioned above, or cool the whole bath by pumping on helium vapour in the bath. By doing so, one shifts the vapour-liquid equilibrium at the surface down to lower temperatures, following the transition curve in the P-T diagram. Temperatures down to 1.4 K can be obtained with this method, which allows one to go below the superuid transition occuring at 2.17 K and thus study dissipation or dispersion phenomena linked to 4 He superuidity (see Chapter 5).

However, due to the dramatic increase in helium heat capacity around the transition temperature, almost half of the helium bath is evaporated in the process, which limits the duration of a continuous experiment below 2.17 K to about a day.

Measurements at dilution temperatures A brief overview of dilution refrigeration

While obtaining cryogenic temperatures above 1 K is rather straightforward at Néel Institute, it is much more demanding to reach temperatures in the mK range. This can be achieved with a dilution refrigerator, which takes advantage of the mixture properties of 3 He and 4 He to reach temperatures down to 5 -10 mK. Detailed descriptions and discussions can be found in reference books, e.g. Refs. [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF] and [START_REF] Olli | Experimental principles and methods below 1K[END_REF].

Here we will simply give an overview, with a schematic drawing shown in Fig. 2.8, left panel, along with a picture of the cryostat we used (right panel).

While our laboratory has a long standing tradition in the development of dilution cryogenics, our experiments have been carried out on a commercial machine. The BlueFors cryostat used in this thesis uses a pulse tube cooler to cool and maintain the environment around the dilution stage at 3 K. Thus, as opposed to the usual, "wet" cryostat no liquid helium is required to keep the experiment cold. First, to limit heat leaks, the environment (called "calorimeter" or "vacuum jacket") is pumped down to below 10 -3 mbar, while the dilution unit is protected by a series of thermal screens screwed to the upper parts of the dilution cryostat to minimize heat radiation.

Once the cryostat is pre-cooled down to 3 K, the mixture is condensed by a compressor at 2 bar, then goes into the dilution circuit. Below 870 mK, the mixture separates into two phases: one is concentrated in 3 He, while the other is 3 He diluted in 4 He. Below 200 mK the concentrated phase is almost pure 3 He, while the dilute phase has a xed 3 He concentration of 6.4 %. This separation occurs in the mixing chamber (MxC), which is the coldest part of the dilution unit. As 4 He is heavier, the dilute phase lies beneath the concentrated phase. A pumping line connected to the distiller enables to circulate the mixture. The distiller itself is connected to the mixing chamber through a line that opens in the dilute phase of the mixing chamber.

With the still under pumping and at a higher temperature, an osmotic pressure gradient is created along the pumping line, which makes 3 He atoms go from the bottom of the mixing chamber up to the still, where mainly 3 He is evaporated, due to a higher saturated vapour pressure than 4 He. To maintain the concentration in the dilute phase at the mixing chamber level, 3 He has to move from the concentrated to the dilute phase.

This process relies on the dierent quantum properties of the two phases and is at the very core of the dilution refrigeration process. Indeed, liquid 3 He is a Fermi liquid, with each atom having an eective mass renormalized by its interactions with the other atoms. In the dilute phase, superuid 4 He acts as a vacuum background for diluted 3 He atoms, which are then much less interacting in this phase: their eective mass is closer to the bare mass. Therefore, 3 He somehow "prefers" to be in the dilute phase and the transfer from one phase to the other is endothermic, hence the cooling.

At the still level, 3 He will be pumped outside the still and will go back to the mixing chamber via an injection line. The cooling power Q is proportional to the 3 He Chapter 2. Experimental and theoretical tools still (~700 mK) Mixing chamber 3 He flow ow rate in the pumping line, ṅ3 , which is about a few hundreds of µmol/s, and to the enthalpy dierence between the two phases present in the MxC. It writes, as a function of the temperature at the phase interface T mix [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF]:

Q ≈ 83 ṅ3 T 2 mix .

(2.1)

It then depends on the heating power applied to the still, which regulates the ow through the evaporation temperature in the range 500 mK -1 K. To keep the dilution unit functional over time, the circuit is operating in closed cycle. Once pumped out of the still, the evaporated mixture goes outside the refrigerator, through the pumping system and a liquid nitrogen trap to prevent impurities to block the injection line, and is then re-injected in the cryostat. In the injection line, the mixture recondenses through an impedance while undergoing a Joule-Thompson expansion, and is thus cooled down to 1 K, this cycling playing the role of the usual 1 K pot used in wet cryostats. It is thermalized while going back to the mixing chamber through heat exchangers that thermally connect the condensing line to the pumping line all the way between the mixing chamber and the still.

With the dilution unit operating, our refrigerator can reach 7 mK within 48 hours.

Note that it is possible to heat the mixing chamber stage with a resistance, with heating powers in the mW range. Temperatures up to 1 K are reached, and higher temperatures are possible if one turns o the turbomolecular pump on the pumping line, as the ow through the pumping line increases a lot with the still temperature increasing.

Experimental setup

While having a 4 K environment for the NEMS is easily achievable, ensuring a good thermalization of the device especially below 100 mK is a dicult task. Moreover, since no exchange gas or cryogenic liquids can thermalize external leads, these must be well clamped to refrigerator parts, while also being either superconducting or very good electrical and heat conductors. This is particularly important as far as the coil used for the magnetomotive scheme is concerned: for experimental purposes, currents up to [START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF] A are needed in the coil, so it is vital to minimize the resulting heat loads.

Note also that ramping current in the coil will result in eddy currents heating the cold nger, hence heating the mixing chamber. A reasonable ramp speed chosen is 10 mA/ sec: for this range eddy currents do not heat much the mixing chamber stage, which goes quickly back to its base temperature. The magnet coil is made of niobium titanium and generates a magnetic eld of 58 mT/A, uniform at the cm scale. The actual coil setup, mounted on the still thermal screen, is shown in Fig. 2.9. Since this screen does not touch the mixing chamber plate, and since the PCB does not touch the coil, the mixing chamber stage is as thermally decoupled as possible from external heat loads. As seen in Fig. 2.9, the NEMS substrate is glued on a PCB with silver paste, which ensures better heat conduction than GE varnish. A carbon resistor is also screwed at Chapter 2. Experimental and theoretical tools the bottom of the PCB, which oers a thermometry point as close as possible to the NEMS. The PCB is then tightly screwed on a cold nger made of copper, which is itself tightly screwed to the mixing chamber plate for good thermal anchoring, and thin enough to t the coil inner diameter while leaving some space. As in the 4 K NEMS setup, SMA coaxial cables are going through the entire cryostat down to the cold nger, with thermal anchors at each stage of the refrigerator. These cables are superconducting below 8 K, which limits signicantly heat loads carried by our NEMS driving currents on the mixing chamber plate.

The current leads used to energize the magnet are made of copper until 50 K, where high T c lines are used down to the 4 K plate. The total wiring displays resistances of about 0.1 mΩ while the fridge is running. This residual resistance is most likely dominated by the leads resistance on the upper part of the cryostat, which is above 50 K. Thus, the heat load due to the coil current is negligible (in the most unfavorable case a few mW distributed over the still screen and mostly the 50 K zone), and does not aect directly the mixing chamber. While passing current through the coil, no signicant heating or voltage increase in the coil circuit was observed throughout experimental runs.

Thermometry

On each one of the upper stages (50 K plate, 4 K plate, still, and cold plate) a resistor measures the temperature. Since these thermometers are not primary (i.e. they do not have a universal and/or explicit dependence in temperature) they must be calibrated using another (already calibrated) thermometer. Currents as small as a few nA are used, so they do not constitute a signicant heat load. These resistors are measured with a home-made resistance bridge (TRMC2) through properly ltered lines. Temperatures as low as 10 mK are thus measurable.

On the mixing chamber, four resistances are connected, made either of Ruthenium Oxide or carbon. A carbon one is directly attached to the cold nger, and is well calibrated down to 10 mK where it saturates. This resistance is monitored by the TRMC2 to regulate the mixing chamber temperature through the heater. Note that it presents a small magnetoresistance, as reported in Ref. [START_REF] Valérie Goudon | Magnétisme nucléaire de l' 3 He liquide : nouvelle détermination du paramètre de Landau F a 0[END_REF], which can lead to discrepancies of at most 3 mK with another calibrated thermometer.

In addition to resistive thermometry, a noise thermometer, namely a SQUID MFFT setup from the company Magnicon, measures the temperature on the mixing chamber.

While it is not directly attached to the cold nger, it is almost a primary thermometer: it measures the current noise created by the electrical environment, and the integrated noise spectrum is proportional to the temperature (Nyqvist formula). The only nonuniversal parameter is the noise measurement bandwidth, which does not depend on temperature. Therefore, this thermometer only needs one calibration point, for instance at 1 K where other thermometers can display a reliable temperature. Once 2.4. Modal decomposition and vibration eigenmodes 19 this calibration is done, the SQUID thermometer can measure temperatures down to at least 300 µK [START_REF] Lusher | Current sensing noise thermometry using a low T c DC SQUID preamplier[END_REF].

Modal decomposition and vibration eigenmodes

Our patterned structures are rectangular beams, i.e. perhaps the most simple object described in the framework of continuum mechanics. An in-depth description can be found in reference books [START_REF] Timoshenko | Theory of Elasticity[END_REF][START_REF] Cleland | Foundations of Nanomechanics[END_REF]. Here we will focus on the main features that are relevant to our studies. Let us consider, a rectangular beam of length l, width w and thickness e with high aspect ratio, i.e. l e, w, as shown in Fig. 2.10. We introduce its cross-section A = w × e, mass density ρ (hence a total mass M = ρAl), bending rigidity (Young's modulus) E, second moment of area along the displacement axis I = we 3 /12 and in-built stress σ. Applying the least action principle to the displacement eld u(z, t)

of the beam, one can derive the well-known Euler-Bernoulli equation [START_REF] Timoshenko | Theory of Elasticity[END_REF]

EI ∂ 4 u(z, t) ∂z 4 + ρA ∂ 2 u(z, t) ∂t 2 -σA ∂ 2 u(z, t) ∂z 2 = 0. (2.2)
For the clamped-clamped geometry, it is clear that the displacement must be zero at each clamp. We are left with the following boundary conditions:

u z = ± l 2 , t = 0.
(

2.3)

There is no general analytic solution to Eq.(2. u(z, t) = Ψ(z)x(t).

(2.4)

The time-independent part Ψ(z) is the displacement prole, while x(t) represents the oscillation of the mechanical wave at a frequency ω: x(t) = x 0 e iωt . By making such a separation, we use the convention that the local maximum of the prole that is nearest to the center of the beam is scaled to 1. One can distinguish two limiting cases where an analytic solution exists, depending on whether tensile stress or bending rigidity dominates in the beam.

Bending limit

In the bending (or low-stress) limit, the potential energy term due to stress is small compared to the one arising from the bending moment. In the vanishing stress limit, Euler-Bernoulli equation can be simplied:

EI ∂ 4 u(z, t) ∂z 4 + ρA ∂ 2 u(z, t) ∂t 2 = 0. (2.5)
Moreover, an additional boundary condition to (2.3) must be introduced, that of an ideal clamp: for pure exure, the beam does not bend at the clamp level, which means that:

∂u(z, t) ∂z

z=± l 2 = 0. (2.6)
From the structure of this equation, one can introduce the following ansatz for the mode prole:

Ψ(z) = a 1 cosh(λz) + a 2 sinh(λz) + a 3 cos(λz) + a 4 sin(λz). (2.8)

This equation has an innite but discrete ensemble of solutions, hence the n index.

These solutions correspond to wave vectors for which the incoming and reected mechanical wave are dephased by 2π, i.e. constructively interfere, building a standing wave: the corresponding wave vectors dene the so-called vibration eigenmodes. While solutions to Eq. (2.8) are non analytic, a graphic representation as in Fig.

( .11) shows that in good approximation, the wave vectors are dened rather simply:

λ n l ≈ n + 1 2 π, (2.9)
when n 1, and λ n l = 4.73, 7.85, 11.00, 14.14 for n = 1 to 4 [START_REF] Cleland | Foundations of Nanomechanics[END_REF]. The next step is to nd the dispersion relation ω(λ) giving the eigenfrequencies of vibration. In principle, this can be obtained by injecting the ansatz (2.7) in Eq.(2.2):

ω n = EI ρA λ 2 n ≈ EI ρA n + 1 2 2 π l 2 .
(2.10)

String limit

In the high stress limit where the potential energy due to a tensile axial load is much bigger than the one due to the bending moment, the beam equation becomes that of a vibrating string:

ρA ∂ 2 u(z, t) ∂t 2 -σA ∂ 2 u(z, t) ∂z 2 = 0.
(2.11)

In this limit the sinh and cosh terms in the ansatz lead to imaginary, unphysical wave 22 Chapter 2. Experimental and theoretical tools vectors. We are left with the following prole:

Ψ(z) = a 1 cos(λz) + a 2 sin(λz).

(2.12)

Applying the boundary conditions (2.3), one nds the relation:

cos λ n l 2 sin λ n l 2 = 0. (2.13)
This leads to a family of eigenwave vectors:

λ n l = (n + 1) π.

(2.14)

Injecting the ansatz (2.12) in Eq.(2.11), one nds the natural frequencies of vibration:

ω n = σ ρ λ n = (n + 1)π l σ ρ .
(

The obtained frequencies are equally spaced, and do not depend on the resonator's transverse dimensions. Most of the NEMS used in our experiments are made out of highly pre-stressed silicon nitride, with a stored in-built stress of approximately 1 GPa. For these high-stress devices, the relation (2.15) is found to be always valid within ± 5 %, which justies the string limit approach in most cases. Other beams used are less pre-stressed, but they are still close to this limit within some correction involving their Young's modulus [START_REF] Lulla | Dissipation and nonlinear eects in nanomechanical resonators at low temperatures[END_REF]. An example of typical proles obtained for real high-stress beams is shown in Fig. 2.12. Of course one can never be perfectly in the string limit for a doubly clamped geometry, since it implies that there would be a discontinuity in the displacement eld derivative at the level of the clamp, which is unphysical. Rather, the prole is smoothed near the clamp, and one can show that the length of the zone where the string limit fails to apply near the boundaries vanishes as a power law of the parameter EI/(σAl 2 ).

f 0 f = 2 f 0 1 f = 3 f 0 2 Figure 2
.12: three rst exural modes of a doubly-clamped beam close to the string limit.

2.5. Time dependence: the 1D, linear, damped harmonic oscillator 23 Note that in this section, we did not take into account the bi-layer nature of the beam. However, aluminium and silicon nitride present similar mass densities, such that the bilayer eect only introduces small corrections owing to the fact that the stress is mostly present in the nitride layer and that the Young's modulus for aluminum is about a third of that of SiN. This correction is addressed in [START_REF] Collin | Metallic coatings of microelectromechanical structures at low temperatures: Stress, elasticity, and nonlinear dissipation[END_REF] and lead to an eective description with slightly renormalized mechanical parameters.

2.5 Time dependence: the 1D, linear, damped harmonic oscillator So far we have not explicited the time dependence of the displacement eld. In fact, once the spatial dependence is addressed and the mode's eigenfrequency is known, the dynamics of the abscissa x n (t) can be re-written as that of the simplest spring-mass system:

m n d 2 x n (t) dt 2 + m n ω 2 n x n (t) = F (t), (2.16)
with m n = M Ψ n (z) 2 dz the mode mass, where the integral is made over the whole beam length. Note that in the string limit, m n = M/2, and one also denes the mode spring constant: k n = m n ω 2 n . Compared to the initial displacement eld, we have ob- tained a simpler equation, for a collective variable x n (t) that describes the beam's motion with renormalized parameters (mass, frequency) that contain all the information on the excited mode, that is, its shape under deformation. This simplication is rather convenient if the actuation/detection technique is non-local, which is the case for the capacitive or magnetomotive techniques presented in the following.

Let us now assume that the exural mode n is excited by a sine wave. In this case, an excitation right at the resonance frequency causes the motion to diverge, which is unphysical. In fact, the modelling presented so far does not take into account the friction mechanisms that any real oscillator always experiences. These mechanisms act as damping forces, that counteract the motion set by the driving force. We will address some of these mechanisms later. Here, we simply mention that they can be all gathered in a single phenomenological constant which is called the damping rate of the mode Γ n . We assume in this section that the damping force is linear, that is, the damping rate does not depend on the displacement. The total damping force associated to the collective variable x n (t) writes then:

F d (t) = -m n Γ n dx n (t) dt .
(2.17)

This term accounts for the power dissipated by the mechanical motion into the sur-Chapter 2. Experimental and theoretical tools roundings. The actual dynamics equation is then:

m n d 2 x n (t) dt 2 + m n Γ n dx n (t) dt + m n ω 2 n x n (t) = F (t).
(2.18)

If initially, the mechanical amplitude x n (t = 0) is nonzero and no force is applied for t > 0, one can solve Eq. (2.18) and show that:

x n (t) = x n (t = 0)e -Γnt/2 cos(ω r t).

(2. [START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF] This means that the oscillator amplitude will decay to its rest position, releasing mechanical energy to its environment, over a typical timescale 2/Γ n . While relaxing to equilibrium, it will oscillate at a frequency ω r =

ω n 1 -1/(2Q n ) 2 , with Q n = ω n /Γ n
the quality factor of the mechanical oscillator. This Q factor is the number of oscillations performed during the typical decay time. For an underdamped oscillator, i.e. Q n 1, the pseudo oscillation frequency will be in very good approximation the natural eigenmode frequency.

Under harmonic forcing F (t) = F 0 cos(ωt + φ 0 ) with a phase reference φ 0 , a generic solution to Eq. (2.16) is:

x n (t) = x c n (ω) cos(ωt) + x s n (ω) sin(ωt), (2.20)
with:

           x c n (ω) = F 0 m n × (ω 2 n -ω 2 ) cos(φ 0 ) + Γ n ω sin(φ 0 ) (ω 2 n -ω 2 ) 2 + Γ 2 n ω 2 , x s n (ω) = F 0 m n × Γ n ω cos(φ 0 ) -(ω 2 n -ω 2 ) sin(φ 0 ) (ω 2 n -ω 2 ) 2 + Γ 2 n ω 2 .
(2.21)

x c n and x s n are the so-called quadratures of the mechanical motion. They are a linear combination of a component in phase with the excitation and its orthogonal component in quadrature, i.e. dephased by π/2 from the excitation. The quadrature component is nonzero, as the damping rate creates a nite time delay over which the response is fully established, hence a dephasing. Tuning the phase reference to φ 0 = +π/2, the two components, called X n (ω) (in-phase) and Y n (ω) (in quadrature), describing a point in the complex plane for a given excitation frequency ω, appear straightforwardly:

           X n (ω) = F 0 m n × Γ n ω (ω 2 n -ω 2 ) 2 + Γ 2 n ω 2 , Y n (ω) = F 0 m n × ω 2 -ω 2 n (ω 2 n -ω 2 ) 2 + Γ 2 n ω 2 .
(2.22)

The component X n (ω) is also called the absorption part of the mechanical response, while Y n (ω) is the dispersion part. Indeed, if the excitation frequency is at the resonance frequency, the applied force will be maximally converted into a displacement.

For excitations at higher frequencies, the system's response will be established with an increased delay, i.e. dephasing, until it cannot follow the excitation: it then becomes transparent to the excitation. At resonance:

       X n (ω n ) = F 0 Q n k n , Y n (ω n ) = 0, (2.23)
with Q n = ω n /Γ n the quality factor of the mechanical mode and k n = m n ω 2 n the eective mode spring constant.

A useful approximation can be considered when the resonator is in the high Q limit, i.e. Q n 1. The drive frequency is then such that (ω -ω n )/ω n 1, and ω + ω n ≈ 2ω n . In that limit Eq. (2.22) becomes:

           X n (ω) = F 0 2m n ω n × Γ n /2 (ω n -ω) 2 + Γ 2 n /4 , Y n (ω) = F 0 2m n ω n × ω -ω n (ω n -ω) 2 + Γ 2 n /4 . (2.24)
The absorption curve then has a Lorentzian line shape. As all our measurements are performed in the high Q regime, we use Lorentzian ts to extract relevant parameters.

In particular, the damping rate is directly the full width at half maximum (FWHM) of the X n curve.

Electromechanical transduction

Electrical setup

In absence of a magnetic eld, the nanomechanical beam simply acts as a resistance R N in series integrated in the experimental setup. This resistance is dominated by the aluminium layer on the beam itself, due to the submicronic connement on 2 dimensions. In practice, contacts also display very small resistances, on the Ω range, so we consider them as almost ideal. The typical resistances at 4 K range from about 50 Ω for small beams (5 µm long) to about 1 kΩ for the longest ones (300 µm). To prevent too huge currents from owing through the device, especially in the case of low impedance beams, a bias resistance R b = 1 kΩ is inserted in the injection line in series with the NEMS. It is also suitable for measurement purposes (see paragraph 2.6.3), since the NEMS is then kept in a high impedance environment on the injection side.

This arrangement is made at the expense of having a 50 Ω adaptation, thus leading to transmission losses in our setup. Nonetheless, these losses can be calibrated (see Section 2.8).

Magnetomotive actuation and detection of the motion

All the measurements presented in this manuscript have been made using the socalled magnetomotive transduction scheme. For fundamental research purposes, it has been widely used for decades in the superuid 3 He community to actuate and detect viscometers, thermometers, or bolometers [START_REF] Guénault | The mechanical behavior of a vibrating wire in superuid 3 He-B in the ballistic limit[END_REF][START_REF] Bäuerle | Temperature scale and heat capacity of superuid 3 He in the 100 µK range[END_REF], and has been introduced, for NEMS structures, in 1999 by A. N. Cleland and M. L. Roukes [START_REF] Cleland | External control of dissipation in a nanometerscale radiofrequency mechanical resonator[END_REF]. Since then, many progresses have been made in measuring nanometer scale motions, but this technique remains used mainly for its simplicity. While the current I(t) ows through the suspended beam (z direction), a static magnetic eld B = By, uniform at the scale of the sample, is applied perpendicularly to the device. This results in a Laplace force exciting the out-of-plane motion of the NEMS. Assuming that the current is sinusoidal at a frequency close to that of a mechanical mode n, such a force applied to an innitesimal oriented portion dl(z) at abscissa z of the beam writes:

dF L (z, t) = I(t)dl(z) × B. (2.25)
For all the structures measured in this thesis the maximum displacement of the beam from its rest position is at most 0.2 % of its length. It is then safe to simplify the above expression for a given mode n:

dF L (z, t) ≈ I(t)BΨ n (z)dz x. (2.26)
Integrating over the whole length l of the beam, we obtain the total Laplace force applied on the mode, in the x direction:

F L,n (t) = ξ n I(t)lB. (2.27)
Here ξ n is a so-called mode shape factor, which accounts for the fact that the force Chapter 2. Experimental and theoretical tools is not uniform along the z axis. This shape factor is then the only mode-dependent parameter in the problem, and an eective 1D force can be written, renormalized by ξ n :

ξ n = l/2 -l/2
Ψ n (z)dz.

(2.28)

For doubly-clamped beams in the string limit, Ψ n (z) = sin(λ n z), with λ n the eigenmode wave vectors in the string limit from Eq. (2.14), therefore ξ n = 2/ [(n + 1)π] for n odd with our notations. As this renormalization is done on the applied force, it should be done also for the displacement, which is distributed along the beam. Indeed, the local motion writes: u n (z, t) = Ψ n (z)x(t). Integrating over the whole beam, we can use a collective variable x(t) describing the displacement of the beam, and which is the maximum amplitude of the beam's deection. For instance, for the fundamental exural mode, x(t) will be the displacement of the central abscissa of the beam.

As induction creates the force applied on the beam, it is also used to detect the resulting motion. Indeed, as the beam is moving, an eective area is swept between the instantaneous position and the position at rest. For an oscillating excitation, this area will be varying in time. The magnetic ux crossing a dierential eective area d 2 A will be:

d 2 Φ(t) = B.d 2 A(t)y. (2.29)
Again, the double dierential area can be simplied in the small deection limit, for mode n:

d 2 A n (t) ≈ Ψ n (z)dzdx(t).
(2.30) After integration over the whole length, the one-time dierential magnetic ux will be:

dΦ n (t) = ξ n lBdx(t). (2.31)
As a time-dependent motion will respond to a time-dependent excitation, this magnetic ux is also time-dependent, and according to Faraday's law, this variation generates an electromotive force e(t) in addition to the Ohmic response of the NEMS:

e(t) = - dΦ n dt = -ξ n lB dx dt .
(2.32)

The experimental technique introduced here thus detects the velocity of the beam. Assuming that the drive excitation is sinusoidal, with angular frequency ω, the resulting velocity will be also an oscillation at ω: v(ω) = iωx(ω) in frequency domain. Then, the detected voltage drop e(ω) will be simply proportional to the complex amplitude x(ω) of the sine displacement, within a phase factor of π/2, which mathematically translates the fact that displacement and velocity are in quadrature:

e(ω) = iωξ n lBx(ω).

(2.33)

With a properly set phase reference for lock-in detection such that we look in the quadrature components X n , Y n directions dened in Eq. (2.23)one measures (in the high-Q limit):

V out (ω) ≈ ω n ξ n lB [X n (ω) -iY n (ω)] . (2.34)
This explains the choice of reference phase φ 0 made when producing Eq. 2.22. An example is shown in Fig. 2.15. From the t one extracts the mode's frequency, its height of vibration and its damping rate. Note that only the in-phase component is then needed. In particular, at the resonance frequency, the quadrature component must be zero. Note also that the cost of measuring velocity instead of the motion is a minus sign on the quadrature, with inverted denitions for in-phase and quadrature components. On a more practical level, the above description does not take into account attenuation in the measurement setup so far. This will be addressed later, in the calibration section.

This technique, on the whole, is very convenient mainly for two reasons:

• As opposed to e.g. a capacitive transduction scheme, the magnetomotive scheme is very linear, i.e. the applied force is independent of the displacement, and uniform at the NEMS scale, since magnetic eld gradients are negligible on the cm scale here.

• The actuation and detection prole match by construction since they are based on the two facets of the same principle and use the same element (a magnetic eld), and the applied force/detected displacement are both simple integrated quantities.

Besides, the scheme suers from a few drawbacks, which denitely rule it out of competition if very sensitive measurements are required:

• By construction, additional electrical losses are generated due to Lenz law of induction. This will be addressed in the next section.

• It is only possible to excite and detect modes with an even spatial prole, as those with odd proles will provide a zero net contribution to the integrated applied force or detected voltage: ξ n = 0 for n even (in our notations).

• Its sensitivity is rather poor, and decreases quadratically with the applied eld and beam length. A good lower limit in the driven regime is in the fraction of nm range for averaged signals, for all NEMS, and in practice, addressing the real Brownian motion at 4 K, of spatial extension k B T /k ∼ 10 pm at best (i.e. for long beams) is impossible: it lies well below the limits xed by the apparatus electrical noise, even at 4 K with a cryogenic amplier, including the bias resistance Johnson-Nyqvist noise. For such measurements optical techniques are by far more adapted [4,[START_REF] Mercier De Lépinay | A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2d force elds[END_REF].

As far as this manuscript is concerned, the required sensitivity is not tremendous.

When needed, Brownian motion is articially created by a white noise source (see Chapter 3). All the measurements realized during this thesis are performed through magnetomotive actuation and detection, and capacitive schemes are only used as an additional channel to achieve frequency tuning or parametric control of the mechanical resonator.

External loading by the electrical environment

We showed that in presence of a magnetic eld B, an electromotive force due to the NEMS motion, opposed to the Laplace force, is generated in addition to the usual Ohmic response described in Section 2.8.1. This e.m.f. has a parallel RLC circuit characteristic, so here we can model the measured nanomechanical mode from the electrical point of view as a parallel RLC circuit loaded by an external complex impedance (See Fig. 2.16). A direct correspondence, described in Ref. [START_REF] Cleland | External control of dissipation in a nanometerscale radiofrequency mechanical resonator[END_REF], is made between the equivalent electrical parameters (R m , C m , L m ) and the mechanical ones (intrinsic resonance frequency ω m and linewidth Γ m , mode mass m n and length ξ n l), together with the magnetic eld. The equivalent electrical parameters satisfy:

         ω m = 1 √ L m C m , Γ m = 1 R m C m .
(2.35)

The total impedance seen by the lock-in amplier, i.e. the one of the RLC circuit loaded in parallel with Z ext = R ext + iX ext can be thoroughly obtained: Note that in Fig. 2.16, C l may not be the total capacitance of the line, as Z ext is merely an impedance seen by the NEMS. Then, the capacitances located before the bias resistance do not signicantly contribute to the loading impedance in this frequency range. Besides, it also includes the geometric capacitance created by the gate-NEMS conguration in case we are using the gate. From the model presented in Fig. 2.16, the loading impedance is:

Z load (ω) = iω/C m iω Γ m + R ext |Z ext | 2 C m + ω 2 m -ω 2 + X ext |Z ext | 2 C m ω . ( 2
Z ext (ω) = R b + R N 1 + i(R b + R N )C l ω .
(2.37)

Chapter 2. Experimental and theoretical tools

Writing the equivalence between the observed e.m.f and the loaded electrical circuit, we obtain:

                         ξ 2 n l 2 B 2 m n = 1 C m , ω res = ω m 1 + X ext |Z ext | 2 ξ 2 n l 2 B 2 m n ω = ω m 1 - C l ξ 2 n l 2 B 2 m n , Γ = Γ m + R ext |Z ext | 2 ξ 2 n l 2 B 2 m n = Γ m + ξ 2 n l 2 B 2 (R b + R N )m n .
(2.38)

The measured properties are thus renormalized by the magnetic eld together with electrical and geometric properties. In the frequency ranges investigated, the measured relative change in frequency does not exceed 10 -5 . However, the correction to the damping rate is not necessarily negligible. In particular, longer beams are more aected by this loading correction while the extrapolated intrinsic damping rate decreases with length. For the longest measured beams in the present work, the loading contribution to the damping rate can reach 200 Hz for intrinsic rates of about 1Hz, as seen on Fig. 2.17. Meanwhile the capacitive contribution is often negligible, with loading capacitances on the 10 pF range at most. On the other hand for a given beam the mode dependence of the additional damping decreases as 1/n 2 , with n being the mode number. Higher modes of a beam are then much less aected by this loss mechanism.

Physically, this can be explained from two equivalent points of view. One is the illustration of the Lenz law of induction: the induced e.m.f. at the terminals of the conducting NEMS leads to a current proportional to the driving current and owing in the opposite direction. This translates into a reaction Laplace force opposed to the driving one and proportional to the velocity, hence the additional damping contribution scaling as B 2 . On the other hand, the measured resonance properties are that of a coupled system: the mechanical mode hybridizes with its electrical environment, through a coupling which strength is set by the magnetic eld. Thus, the more it is coupled to the electrical system, the more electro-mechanical energy is dissipated.

Meanwhile the coupling strength also depends on the impedance seen by the NEMS: if this loading external impedance is high, the NEMS is less coupled to the electrical circuit i.e. less easily detectable, but experiences less extra damping.

As a result, a trade-o situation must be found between having a high quality resonator (hence a high environment impedance) and good resolution (hence a NEMS long enough, which results in more electrical damping). On the detection side the lock-in amplier input impedance is set to 1 MΩ which can be regarded as an innite value for all NEMS investigated here. With a bias resistance of 1kΩ, added to that of the metallic layer of the NEMS, we ensure that at least short NEMS are very weakly loaded even for high elds.

Thermal properties of the system

In presence of electrical signals passing through resistive metals, or direct heating of the sample holder, the thermal environment of the NEMS will be aected, which in turn will modify the NEMS materials' properties. Since all the mechanical parameters (resonance, dissipation, amplitude) depend on the materials, it can be useful to estimate quantitatively thermal eects on the mechanics. Moreover, for calibration purposes (see Section 2.8.2), one needs at least a qualitative prediction of what happens to the mechanics for a given heating of the system.

Temperature dependence of measured mechanical properties

With a simple microscopic picture it is natural to think that materials properties will be aected by a temperature change: atoms' motion will grow bigger with temperature, which in turn will change the potential energy seen by each of them in the materials, hence the stiness. From a macroscopic perspective, materials will experience a change in Young's modulus as well as thermal expansion/contraction. While thermal expansions are small at cryogenic temperatures, there is nonetheless a non-negligible contribution in metal-coated doubly-clamped beams due to the expansion mismatch between the two layers. Thus, for a homogeneous temperature eld T a dierential strain ε(T, z) = δl(T, z)/l exists, and this in turn creates internal stress δσ according to Hooke's law for elastic media:

Chapter 2. Experimental and theoretical tools

δσ(T, z) = E(T )ε(T, z), (2.39)
if we restrict the discussion to 1D with the z axis. The potential energy associated to an oscillating motion writes:

E p = 1 2 x 2 n (t) l/2 -l/2 ∂ 2 Ψ n (z) ∂z 2 2 E(T )I - ∂ 2 Ψ n (z) ∂z 2 Ψ n (z)σ(T )A dz, (2.40)
where we emphasize the temperature dependence of Young's modulus and internal tensile stress σ(T ) = σ(T 0 ) + δσ(T ). Assuming the drive current is not heating much the device (which in normal operating conditions is reasonable), we can consider that the device is always at thermal equilibrium and forget for the moment the time and space dependences of the temperature eld. The integral in Eq. (2.40) is in fact a spring constant, and assuming small, low-order polynomial expansions in temperature for Young's modulus, in-built stress and dierential strain, the potential energy can be written:

E p = 1 2 [k n (T 0 ) + δk n (T -T 0 )] x 2 n (t), (2.41)
with T 0 a reference temperature (e.g. 4.2 K). This change in spring constant with temperature is measured through the resonance frequency. An example is displayed in Fig. (2.18), left panel. The shift in frequency is well captured with a polynomial dependence (see caption). The cubic dependence originates from the dierential strain and possibly the in-built stress, as it is absent for cantilevers measurements [START_REF] Collin | In-situ comprehensive calibration of a tri-port nanoelectro-mechanical device[END_REF], where dierential strain does not aect the neutral line. The weak, linear dependence, also observed for cantilever [START_REF] Collin | In-situ comprehensive calibration of a tri-port nanoelectro-mechanical device[END_REF] is then most likely due to a small, linear variation of the Young's modulus with temperature.

On the other hand, taking into account a small variation of the friction coecient with temperature in the considered range the instantaneous mechanical power dissipated writes:

P d = - 1 2 M ẋ2 n (t) l/2 -l/2 Γ n (T 0 ) + (T -T 0 ) dΓ n dT T =T 0 Ψ 2 n (z)dz.
(2.42)

A typical measurement of the linewidth up to 23 K is shown in Fig. 2.18, right panel. The dependence in temperature is linear, with a weak slope dΓ n /dT . Its value is rather reproducible from one sample to the other, between 1 % and 2 %. In addition, it has been observed that the extrapolated zero value scales with the metal thickness evaporated [START_REF] Olkhovets | Actuation and internal friction of torsional nanomechanical silicon resonators[END_REF][START_REF] Collin | Metallic coatings of microelectromechanical structures at low temperatures: Stress, elasticity, and nonlinear dissipation[END_REF][START_REF] Seitner | Damping of metallized bilayer nanomechanical resonators at room temperature[END_REF] for the same sample, which seems to show that metallic coatings are the dominant source of damping for temperatures above 1 K in our structures. In Ref. [START_REF] Quirin | Damping of Nanomechanical Resonators[END_REF], a macroscopic model describing losses in nanomechanical resonators based on continuum mechanics showed that the bending energy is responsible for losses.

This explains, at least partially, why the Q factor of stressed SiN resonators have rather high values [START_REF] Defoort | Stressed Silicon Nitride Nanomechanical Resonators at Helium Temperatures[END_REF]: the added in-built stress does not contribute to the dissipated energy while increasing the stored energy (since it is the dominant contribution to the stiness). However, the actual microscopic friction mechanisms remain elusive.

In our case, thermoelastic damping [START_REF] Lifshitz | Thermoelastic damping in micro-and nanomechanical systems[END_REF] and clamping losses [START_REF] Wilson-Rae | Intrinsic dissipation in nanomechanical resonators due to phonon tunneling[END_REF] should be negligible for such materials and frequencies, at such temperatures. Then, one candidate could be friction occuring at grain boundaries, with a temperature dependence arising from thermal contraction or expansion. Below typically 1 K, new mechanisms start to be dominant, which will be addressed in Chapter 6.

Thermal balance in presence of a driving current

In this section we shall derive a simple thermal model describing the eect of external Joule heating on the thermal prole T (z, t) of the beam, in the range 1 K-30 K. This will be useful for calibration purposes (see Section 2.8.2). The major contribution for thermal conductivity in the aluminium layer will arise from conduction electrons. In the Kelvin range the Wiedemann-Franz law linking the electrical resistivity ρ e and the thermal conductivity κ e is accurate. Thus κ e = π 2 k 2 B T /(3e 2 ρ e ), and we can neglect the contribution from phonons in the aluminium. Conduction electrons will be also the dominant contribution to specic heat: C V,e = γ e T up to 10 K.

In the nitride layer, the phonon bath is responsible for a specic heat C V,ph ∝ T n , where n depends both on materials disorder and sample dimensionality. The latter is to be compared with the dominant phonon wavelength λ dom = hc/(2.82k B T ), with c the sound speed. Meanwhile, thermal conductivity cannot be properly dened, as it is proportional to the phonon mean free path, which is much bigger (around 1 cm) than the beam's cross-dimensions at cryogenic temperatures. Rather, we use the thermal conductance K ph , which depends on the layer's cross-dimensions, dened as:

K ph = β cas Λ ph T 3 .
(2.43)

Here Λ ph = 1.12 √ e SiN w is the eective mean free path in the diusive (Casimir) regime of thermal phonon transport: in this limit thermal phonons are inelastically scattered at the NEMS boundaries. Meanwhile, β cas is a constant extracted from kinetic theory, which depends on dimensions, and is found experimentally in similar devices [START_REF] Tavakoli | Universality of thermal transport in amorphous nanowires at low temperatures[END_REF] to be:

β cas = 3.2 × 10 3 2π 2 k 4 B 5 3 c 3 2/3 e SiN w l .
(2.44)

Below 15 K thermal conductance due to phonons in the nitride layer does not exceed a fraction of the one due to electrons in the metallic layer. However, as the ratio between the two K ph S e /(κ e l) ∝ T 2 , this quickly becomes wrong above approximately 10 -15 K. Therefore, for simplicity we rst assume in our modelling that heating will not bring the device to too high temperatures so that we can keep only the electronic contribution in the model.

We assume that intially, the beam is thermalized at a base temperature T 0 , be it the cryogenic temperature 4.21 K or the one obtained by heating the sample holder.

Due to the huge aspect ratio, we can reduce the situation to a 1D heat diusion model in the aluminium layer. After switching on a heating electrical current, contacts are assumed to be big enough so as to remain thermalized at the base temperature, but due to the small section S e of the resonator's metallic layer, any electrical current I(t) owing through the latter will result by Joule eect in a heat load per unit volume q(t) = ρ e I 2 (t)/S 2 e . S e denotes here the cross-section of the aluminium layer, which is the only conducting part of the beam. The evolution of the temperature prole along the beam (z axis) is determined through a heat diusion equation:

µ e C V [T (z, t)] ∂T (z, t) ∂t = ∂ ∂z κ [T (z, t)] ∂T (z, t) ∂z + q(t), (2.45)
where µ e = 2.7 g/cm 3 is the mass density of the aluminium layer. The boundary conditions express the symmetry of the heat load distribution and the thermalization of the device, at the clamps, to the reservoir formed by the contacts, which are at temperature T 0 :

       T (z = ±l/2, t) = T 0 , κ [T (z = 0, t)] ∂T (z = 0, t) ∂z = 0.
(2.46)

Temperature prole along the Joule-heated nanobeam

We write the heating current as I(t) = I h cos(ω h t), so the heating power per unit volume will be:

q(t) = ρ e I 2 h [1 + cos(2ω h t)] 2S 2
e .

(2.47)

Eq. (2.45) is solved in details for a dierent NEMS geometry in Ref. [START_REF] Collin | In-situ comprehensive calibration of a tri-port nanoelectro-mechanical device[END_REF], we give here a solution adapting the procedure for our geometry. The AC part of the solution oscillating at 2ω h scales in amplitude with a thermal skin depth vanishing as 1/ √ ω h . As the lock-in detection averages over a time ω -1 h , We can safely neglect it in the problem, which amounts to saying that the NEMS is only sensitive to the average temperature at a given abscissa z. We are left with the DC temperature prole:

T DC (z, I h ) = 3e 2 ρ 2 e I 2 h 2π 2 k 2 B S 2 e l 2 4 -z 2 + T 2 0 .
(2.48)

The prole is maximum at the center of the beam, which is consistent with the fact that it is the most isolated part from the thermal reservoirs, i.e. the contacts, while the current is uniform along the beam. Thus the steady-state temperature will be further from equilibrium at the center, while the clamps will more easily leak to the contacts and therefore thermalize to base temperature.

Let us dene ∆T DC (z, I h ) = T DC (z, I h ) -T 0 the dierence between the heated beam temperature at an abscissa z and the base temperature (i.e. temperature at the clamps), and ∆T max = ∆T DC (0, I h ) the temperature at the centre of the beam, i.e. the maximum temperature:

∆T max (I h ) = 3e 2 ρ 2 e I 2 h l 2 8π 2 k 2 B S 2 e + T 2 0 -T 0 .
(2.49)

This heating results in a shift of the spring constant which is an average over the temperature gradient along the beam:

Chapter 2. Experimental and theoretical tools

ω n (I h ) = ω n (0) + a∆T max (I h ) ∂ 2 Ψn(z) ∂z 2 2 ∆T DC (z,I h ) ∆Tmax dz ∂ 2 Ψn(z) ∂z 2 2 dz + b∆T 3 max (I h ) ∂ 2 Ψn(z) ∂z 2 Ψ n (z) ∆T DC (z,I h ) ∆Tmax 3 dz ∂ 2 Ψn(z) ∂z 2 Ψ n (z)dz . (2.50)
This expression shows that the frequency shift due to a heating current depends only on the geometry (through the temperature prole calculated with the thermal model and the mode shape) and on the coecients found for a uniform heating a and b.

The fractions in Eq.(2.50) are not analytic but can be evaluated numerically. Over the range of heating currents considered, they yield constant values of respectively 0.92 for the bending contribution and 0.79 for the stress contribution in the doublyclamped geometry. In addition, the NEMS resistance R N = ρ e l/S e is found to be independent of temperature between 1 and 30 K, within 1 %. the center of the beam, a discrepancy appears. This corresponds to the appearance of another heat conduction channel, namely phonons in the nitride part of the beam, similar to silicon based devices [START_REF] Meschke | Electron Thermalization in Metallic Islands Probed by Coulomb Blockade Thermometry[END_REF]. It explains why the frequency is less strongly shifted, as the real temperature is lower than the one predicted by our model: indeed, the additional phonon conduction enhances the beam thermalization to the reservoirs, therefore reducing the temperature increase due to Joule heating.

In-situ calibration of the system

The magnetomotive technique introduced in paragraph 2.6.2 should in principle allow to determine the applied forces and the displacements in real units, since all parameters are known. However, this accuracy is strongly limited by the apparatus. The injection and detection cables, which are about a few meters long each, are responsible for low-pass ltering due to their capacitance and can exhibit parasitic resonances for high frequencies 10 MHz. The NEMS resistance itself is not necessarily 50 Ω, its environment is high impedance for measurement purposes (see Section 2.6.3), while the coaxial cables impedance is 50 Ω, leading to impedance mismatches for an AC drive.

A thorough denition of each imperfection is unrealistic, but it is possible nonetheless to obtain in-situ calibration factors for both injection and detection lines, called G I (ω) and G D (ω) in the following. The technique is described extensively in Ref. [START_REF] Collin | In-situ comprehensive calibration of a tri-port nanoelectro-mechanical device[END_REF], and here we will focus on the essential features.

Characterization of the global transmission in frequency

The transmission is simply measured at the lock-in detection through the Ohmic response of the NEMS to a current owing in its metallic layer. The injection and detection lines are coaxial cables with a capacitance dristributed all over their length (a few meters for both). The estimated lineic capacitance is 100 pF/m.

G(ω) = V out (ω) V in (ω) = 1 1 + R b /R N + jR b Cω . ( 2
G(ω → 0) = R N R b + R N .
(2.52)

An example of transmission line characterization is shown in Fig. 2.20, with a NEMS 1 0 0 0 0 0 1 0 0 0 0 0 0 0 , 1 0 , 2 0 , 3

T r a n s m i s s i o n G a i n F r e q u e n c y ( H z )

Characterization of the injection and detection lines

To obtain the injection coecient |G I (ω)| independently from the global transmission, we use the following technique: while the usual drive current is applied around the resonance frequency, a strong heating current I h will be added at a frequency ω h oresonance. For this we use a homemade additioner, of bandwidth 100 MHz and gain 1. By Joule eect, a heat load will ow through the beam, resulting in a temperature increase. The thermal model at stake is analyzed in Section 2.7. Here we will simply use the fact that heating the structure will shift its resonance frequency towards low frequencies.

• For ω h → 0 (DC heating), we assume that there is no loss: G I (ω h → 0) = 1.

The actual current owing through the beam is then the calculated current • For ω h = 0, |G I (ω h )| = 1 as a consequence of losses and/or reections in the line, so the actual current owing through the device is not the calculated one. This current will nonetheless heat the device, so we can have access to its actual value, by simply rescaling the calculated heating current I h (ω h ). The scaling factor will be by construction the injection factor |G I (ω h )| such that the resonance shift curve δω 0 [|G I (ω h )|I h (ω h )] will collapse on the reference curve δω 0 [I h (0)]. For each heating current frequency, a dierent rescaling factor is applied, so that all the data collapse on the DC heating curve. The line is the same as in Fig. 2.19.

I h = V in,h /(R b + R N ).
An example of this procedure is shown in Fig. 2.21. To each dataset corresponding to one heating current frequency, a scaling factor is applied on the current injected, to obtain the actual current owing through the beam.

Chapter 2. Experimental and theoretical tools

The main advantage of this technique is to rely only on genuine (and local) properties of the device, that unequivocally depend on the amount of heating current passing in the metallic layer. Moreover, while we derived a thermal model that accounts fairly well for the observations in presence of a moderate heating current, it is not crucial, as the calibration procedure is relative to the DC transmission, which is 1 by denition.

Thus, the accuracy on the measured injection loss coecient value is very good, within ±5 %.

However this technique has got one drawback for doubly-clamped beams. This class of structure experiences a rather strong nonlinear behaviour (see Section 2.9 below), so possible nonlinear mixing eects (such as the self-coupling scheme described in [START_REF] Defoort | Modal self-coupling as a sensitive probe for nanomechanical detection[END_REF]) can arise if ω h is too close to the resonance frequency. To prevent this situation to happen the characterization around resonance frequency is made with detunings from ω 0 of about ±10 kHz at least, while G I (ω 0 ) is deduced by interpolation. In addition, since thermal properties are eld-independent, we repeat the procedure for several magnetic elds to rule out non-linear mixing eects that would depend on the magnetic eld, since they are of mechanical origin. The actual force exerted on the beam then writes: (2.54)

F L,n (ω) = ξ n lB |G I (ω)|V 0 R b + R N .
Knowing the detection coecient, we can now obtain the displacement of the beam in real units: Nonetheless, we have another way to determine the detection coecient accurately, assuming that all parameters are known otherwise. At resonance, the relation (2.23) links the applied force (calibrated, with known injection coecient) and the displacement (which should be obtained through the detection coecient), through the resonance linewidth, which is measured. Then, the linewidth can be recalculated with a detection coecient left as a single free parameter:

x(ω) = V out (ω) iωξ n lB|G D (ω)| . ( 2
Γ calc = F 0 m n ω n X n (ω n ) = F 0 m n ξ n lBG D (ω) V out (ω) .
(2.56)

Then, the actual detection loss coecient is the one such that the recalculated linewidth matches the measured one. This self-consistent check enables to quantifying potential cross-talks in the detection line.

Capacitive tuning with a gate electrode

With our fabrication technique it is straightforward to design a sample with a metalcoated electrode placed in the vicinity of the resonator, with a gap between them that Points at high amplitudes (≥ 50 nm) are taken in the nonlinear regime (see Section 2.9).

can reach values as low as 100 nm routinely. This so-called gate electrode forms with the beam a geometric capacitance with the vacuum gap in between acting as a dielectric medium. It can be used to tune properties of the mechanical resonator such as its resonance frequency or its nonlinear coecients [START_REF] Kozinsky | Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators[END_REF]. It can also be used as a transduction scheme based on dielectric forces [START_REF] Quirin | Universal transduction scheme for nanomechanical systems based on dielectric forces[END_REF], while one can think of it as a way to parametrically pump a mode [START_REF] Collin | Nonlinear parametric amplication in a triport nanoelectromechanical device[END_REF] or couple two modes [START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF]. Here we present some of these aspects which will be useful for our work. The same calibration technique used before for the injection line applies to the capacitive line, so we do not describe it here. All further references to gate voltages concern the actual ones at the level of the electrode.

We name C g the geometric capacitance created by the NEMS-gate electrode conguration. Applying a DC bias voltage V 0 on the gate electrode, the charging potential energy for the capacitance is:

E C = 1 2 C g V 2 0 .
(2.57)

As the beam moves, electrostatic eld lines are modied, in a non-uniform way, such that it creates a capacitance gradient. As a result, an electrostatic force is exerted on the beam:

F C = 1 2 V 2 0 ∇C g .
(2.58)

2.9. Basic nonlinear behaviour of nanomechanical systems 45

One can show that for exural out-of-plane motion, in our geometry, the gradient in y and z direction is small [START_REF] Collin | Nonlinear parametric amplication in a triport nanoelectromechanical device[END_REF], hence leading to very small forces applied to the beam in these directions. In the limit of small displacements, we can expand the capacitance gradient in the x direction at rst order in displacement, which yields for the electrostatic force:

F x C = 1 2 V 2 0 ∂C g (0) ∂x + ∂ 2 C g (0) ∂x 2 x + O(x 2 ) (2.59)
The rst term is a static force which magnitude is in practice very small, and it does not aect substantially the NEMS dynamical behaviour. The high order terms lead to renormalizations of the NEMS nonlinearities, as they can be recasted as eective nonlinear terms in the dynamics equation [START_REF] Kozinsky | Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators[END_REF]. Here we drop them in the modelling, but they can be used for instance with cantilevers, where they oer the possibility to tune with great precision e.g. the non-linear behaviour (as the intrinsic one is small in this case) [START_REF] Collin | A Tunable Hybrid Electro-magnetomotive NEMS Device for Low Temperature Physics[END_REF].

The rst order term can be recasted in the dynamics equation as a static modication δk(V 0 ) of the bare spring constant k 0 :

δk(V 0 ) = - 1 2 ∂ 2 C g (0) ∂x 2 V 2 0 . (2.60)
For small changes of the spring constant, the resonance frequency is then:

ω res (V 0 ) = ω 0 1 + δk(V 0 ) k 0 ≈ ω 0 - 1 4m 0 ω 0 ∂ 2 C g (0)
∂x 2 V 2 0 .

(2.61)

The parameter ∂ 2 C g (0)/∂x 2 therefore characterizes the coupling strength between the gate electrode and the mechanical motion, and we can measure it through the resonance shift as a function of the gate voltage. One must take into account that trapped charges can exist on the gate electrode, leading to small oset voltages (see Fig. 2.24).

2.9 Basic nonlinear behaviour of nanomechanical systems 2.9.1 Geometric nonlinearity When oscillating, any real beam undergoes a deformation. As such, because of the nite length of the beam and the boundary conditions for the oscillating motion (the displacement eld must be zero at the clamps), some additional stretching will appear, hence an additional axial load (stress). This can be quantied from the curvature radius of the deformation u(z, t), which is to be compared to the beam's length l. x direction [START_REF] Cleland | Foundations of Nanomechanics[END_REF]. As the clamps impose a non-uniform displacement to the neutral axis, the latter stretches on each small portion, and for not too large deformations (compared to the portion length) its length is easily calculated at rst order (see Fig.

2.25)

. If the deformation is small compared to the beam's length, the total length writes [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF]: Using the decomposition of the displacement eld u(z, t) = Ψ(z)x(t), we can obtain the dynamics equation for the displacement, which is that of an anharmonic oscillator with a non-linear restoring force acting on the beam:

l + δl = l 1 + ∂u(z,
F nl = mγx 3 , (2.64)
where γ is the non-linear interaction constant [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF][START_REF] Nayfeh | Nonlinear Oscillations[END_REF]:

γ = E 2l 2 ρ l ∂Ψ(z) ∂z 2 dz 2 .
(2.65)

Note that here the nonlinear constant is positive, meaning that the geometric nonlinearity is stiening the resonator. For cantilevers the same eect would be softening, due to the contraction of the cantilever during its deection [START_REF] Matheny | Nonlinear Mode-Coupling in Nanomechanical Systems[END_REF]. However, in this case the situation is more subtle than for doubly clamped beams, and to date experimental results are not satisfactorily explained by the simplest 1D theories [START_REF] Villanueva | Nonlinearity in nanomechanical cantilevers[END_REF].

The nonlinear coecient expression (2.65) can be well simplied in the case of a doubly-clamped beam in the high-stress limit, where Ψ n (z) = sin[(n + 1)πz/l]: γ = (n + 1) 4 Eπ 4 8l 4 ρ .

(2.66)

The Dung oscillator

In good approximation, the main eect of geometric nonlinearities, from a single mode perspective, can be encompassed in the so-called Dung model. A more exhaustive analysis is proposed in Ref. [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Dung model[END_REF], that takes into account non-linearities for damping Chapter 2. Experimental and theoretical tools and inertial terms. Here we focus on the main geometric eect, which appears through a non-linear restoring force cubic in displacement:

mẍ + mΓ ẋ + mω 2 0 x + mγx 3 = F (t).

(2.67)

Here γ is the non-linear coecient. Solving directly Eq. (2.67) is no easy task. The cubic term will generate, from a monochromatic forcing F (t) = F 0 cos(ωt), a multifrequency response: a sum of a static term, the ω response, and harmonics at ± 3ω...to capture this, we write a generic displacement ansatz in the Fourier basis, following Landau and Lifshitz [70]:

x(t) = +∞ n=0 [x c n (ω) cos(nωt) -x s n (ω) sin(nωt)] .
(2.68)

Injecting this ansatz in Eq.(2.67) and projecting the result onto the Fourier basis components, one can in principle fully solve the equation. This task is tedious, and one can show that in the high-Q limit, at leading order, only the term n = 1 is relevant, the other amplitudes decreasing as 1/Q n . This is an adiabatic approximation, meaning that the Dung non-linearity essentially does not allow at rst order any energy ow from the fundamental response to higher-order harmonics. Within this approximation, in the frequency domain, the oscillation amplitude can be written in a self-consistent way:

x(ω) = F 0 2mω 0 ω 0 -ω + β|x(ω)| 2 + iΓ/2 .

(2.69)

In other words, the Dung non-linearity introduces a correction to the mechanical linear susceptibility, which scales as |x| 2 . Experimentally, the Dung term will result in distorted lineshapes, as shown in Fig. 2.26.

We choose to name β = 3γ/(8ω 0 ) the Dung coecient, as it is immediately accessible experimentally. Indeed, in the high-Q limit (always valid in the present work), the renormalized resonance frequency writes:

ω res = ω 0 + β|x max | 2 .
(2.70)

It is then easy to extract a non-linear coecient β by measuring the resonance shift with an increasing driving force, hence an increasing resonance amplitude. An example is given in Fig. 2 in the calibration procedure, as well as the uncertainty on materials properties such as Young modulus or internal tensile pre-stress. As far as experimental results are concerned, we will use the measured value to quantitatively address nonlinear phenomena in the following.

Knowing the β coecient, we can now t the distorted lines without free parameter. Indeed, for maximum amplitudes below a critical threshold x c , Eq. (2.67) possesses only one analytic solution in real space. A typical application is shown in Fig. 2.26.

Moreover, one cannot directly extract a damping coecient through the linewidth.

However, Eq. (2.23) is still valid if one substitutes ω n with the Dung-shifted resonance ω res . Then, one can extract the actual damping even in the non-linear regime, through the recalculated nonlinear linewidth.

Above a certain maximum threshold amplitude x c corresponding to an applied force

F c such that x 2 c = √ 3Γ 2β
, the self-consistent equation (2.69) possesses 3 solutions in real space. We will always refer to them in an implicit way, as their expressions are tedious to manipulate even though they are analytic. Physically, this threshold corresponds to a Dung frequency shift that becomes comparable with the damping rate. This multivalued zone adds a great deal of physical complexity, and can be useful as a model system to study non-trivial nonlinear dynamics, such as the bifurcation phenomenon [START_REF] Kozinsky | Basins of Attraction of a Nonlinear Nanomechanical Resonator[END_REF][START_REF] Aldridge | Noise-Enabled Precision Measurements of a Dung Nanomechanical Resonator[END_REF][START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF]. Some of these aspects will be addressed in Chapters 3 and 6.

If we study in detail the stability of these three possible states, we see that two of them correspond to stable orbits in the phase space (the so-called attractors), while the third one is an unstable state. In the coexistence zone, for a given driving frequency, the system can jump by thermal activation from one stable state to the other. The critical point F c beyond which a coexistence region appears is called the spinodal point. If one measures the response at a given force F 0 beyond the spinodal point, sweeping the frequency upwards or downwards will not give the same results, as the system is initially prepared in the most energetically favorable state, which is not the same whether one sweeps upwards or downwards. Thus, experimentally, a hysteretic behaviour is observed, as shown in Fig. 2.28.

This hysteresis is delimited by two frequencies ω bif,↑ and ω bif,↓ . At the hysteresis opening (F 0 = F c ), we have ω bif,↑ = ω bif,↓ , but ω bif = ω res . The underlying phenomenology around these peculiar points will be detailed in Chapters 3 and 6. Here, we just mention that at these frequencies sharp jumps are observed in the response. This corresponds to one metastable state merging with the unstable one, thus becoming unstable, as shown in Fig. 2.29. Hence only one state becomes possible at these frequencies and outside the hysteresis zone: the merging maps the transition from three motional solutions to only one, and can be viewed as an example of saddlenode bifurcation, where two stable states correspond to potential wells, separated by a potential barrier (the saddle).

The properties of the bifurcation process and the spinodal point will be exploited further in Chapters 3 and 6.

γ m,n = E l 4 ρ I n,n I m,m 2 + I 2 m,n , (2.72)
where I m,n = Ψ n (z)Ψ m (z)dz. The most natural conguration to witness this coupling will be that a mode m will undergo a strong excitation (a pump tone) at its resonance frequency ω m while the mode n will be weakly excited, acting as a probe.

Indeed, the resulting non-linear restoring force F n,m exerted on the probe mode can be derived from the coupling potential energy (2.71):

F m,n = - ∂V m,n ∂x n = - 3 2 γ m,n x 2 m x n .
(2.73)

With an adiabatic treatment similar to that of Section 2.9.2, one can discard fast oscillating terms appearing in the dynamics equation with this non-linear force. We are then left with the following mode n mechanical response:

x n (ω) = F 0 2m n ω n (ω n -ω + β m,n |x m,res | 2 ) 2 + iΓ n /2 , (2.74)
with β m,n = 3γ m,n /(4ω n ). Note that while γ m,n = γ n,m , this does not hold for the β factors, which depend also on the probe mode's frequency. Assuming that the pump amplitude stays constant, the eect on the probe mode is just a shift of its resonance frequency depending quadratically on the pump resonant amplitude [START_REF] Westra | Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators[END_REF][START_REF] Lulla | Nonlinear modal coupling in a high-stress doubly-clamped nanomechanical resonator[END_REF][START_REF] Matheny | Nonlinear Mode-Coupling in Nanomechanical Systems[END_REF] :

ω n,res = ω n + β m,n |x m,res | 2 .
(2.75)

One can then quite easily measure the coupling strength given by the factor β m,n .

An example is shown on Fig. 2.30. As for the Dung coecient, our measured mode coupling factors are usually in good agreement with theoretical expressions, again within 20 % for the same reasons as in paragraph 2.9.2. While being another proof of robustness of our calibration procedure, these mode coupling constants measurements are also useful when it comes to quantitatively address the phenomena described in Chapter 3.

The parametric amplication scheme

We have seen in section 2.8.3 that a gate electrode enables additional control of the nanomechanical beam through e.g. resonance frequency tuning. If now the ap- plied voltage V g on the gate is modulated in amplitude at a frequency ω , such that V g = V 0 + V m cos(ω t + φ g ), the result will be a periodic modulation of the NEMS spring constant i.e. of the linear restoring force. Indeed Eq. (2.60) becomes:

δk(t) = - 1 2 ∂ 2 C(0) ∂x 2 V 2 0 + V 2 m 2 + 2V 0 V m cos(ω t + φ g ) + V 2 m 2 cos(2ω t + 2φ g ) . (2.76)
If one chooses to modulate at ω = 2ω where ω is the magnetomotive excitation frequency, the last term in Eq. (2.76) will yield fast oscillating terms in the dynamics and can be dropped. The rst two DC terms correspond to the situation described in the previous paragraph, where the AC amplitude is a correction to the usual DC frequency shift. Of utmost interest is the middle term. It corresponds to a modulation around twice the resonance frequency, analogous to a child's swing motion which is sustained by the child's legs moving at half the swing period. It is then possible, for certain conditions in phase, to amplify or reduce the motion in certain directions in the phase space. This is the so-called parametric amplication scheme, since a parameter (mass, spring, damping) is modulated to enhance the dynamics. Note that the scheme can be implemented for modulation frequencies ω = 2ω/n, yet the 2ω modulation avoids additional resonant forces that would be superposed with the magnetomotive scheme. Dropping the non-resonant term, with k

= k 0 -1/2 × ∂ 2 C/∂x 2 (V 2 0 + V 2 m /2),
the dynamics equation becomes:

mẍ + mΓ ẋ + [k + δk 0 cos(2ωt + φ g )] x = F 0 cos(ωt + φ 0 ), (2.77)
where δk 0 = -∂ 2 C/∂x 2 V m V 0 . To solve this non-linear equation in x one can use again the ansatz (2.68). An exhaustive solving is proposed in [START_REF] Collin | Nonlinear parametric amplication in a triport nanoelectromechanical device[END_REF]. Here we focus on the resonant terms, which are dominant in the dynamics. At resonance, the amplitude is renormalized by a factor which depends on the phase of the modulation signal φ g and its amplitude, which we write in a normalized form h = δk 0 2k Q:

x(ω res ) = F 0 Q k sin 2 π 4 + 2φ 0 -φg 2 1 + h 2 + cos 2 π 4 + 2φ 0 -φg 2 1 -h 2 .
(2.78)

We can see that the appropriate choice of phase parameter φ = φ 0 -φ g /2 allows one to either amplify or squeeze the resonant amplitude. Maximum squeezing (+) or amplication (-) occur for φ = ±π/4, with the two quadratures writing:

           X(ω) = F 0 m • ωΓ (1 ∓ h) (ω 2 res -ω 2 ) 2 + ω 2 Γ 2 (1 -h 2 ) , Y (ω) = F 0 m • ω 2 res -ω 2 (ω 2 res -ω 2 ) 2 + ω 2 Γ 2 (1 -h 2 )
.

(2.79) The two quadratures are then asymmetrized, as seen in Fig. 2.31. Note that whether motion is amplied or squeezed, in both cases, the in-phase response is still a Lorentzian in the high-Q approximation, with renormalized amplitude and linewidth:

2.9. Basic nonlinear behaviour of nanomechanical systems 55

       X(ω res ) = F 0 Q k • 1 1 ± h , Γ r = Γ √ 1 -h 2 .
(2.80)

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0 1 1 0 1 0 0 G a i n h ( d i m .

l e s s )

A m p l i f i c a t i o n S q u e e z i n g 0 , 9 6 0 , 9 8 1 , 0 0 5 0 In addition, we see that in both cases, for both quadratures, the eective linewidth goes to zero the same way as h → 1. That, and the divergence for the same limit in the amplication case, indicate that there is an instability appearing at h = 1. This is the threshold for the transition to the parametric oscillation regime, where oscillations are self-sustained.

1 0 0 1 5 0 2 0 0 G a i n h 0 , 0 0 ,
Close to the threshold, the resonant amplitude, as well as the eective Q factor, are supposed to diverge in the parametric amplication regime. However, saturations are observed, which are due to nonlinearities, either geometric [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF] or coming from the anelasticity of the constitutive materials. Here, gains up to 100 were measured in a perfectly linear regime, as shown in Fig. 2.32 (left panel). A saturation seems to appear beyond a pump factor h = 0.99 (Fig. 2.32, left panel inset). However, one must be cautious in analyzing its origin: voltage uctuations on the gate at the mV scale as well as errors of even a few percent in the pump phase setting can lead to increasingly dramatic errors in h close to the threshold, as discussed in Ref. [START_REF] Collin | Nonlinear parametric amplication in a triport nanoelectromechanical device[END_REF]. This is captured in the error bars, which are big even for large DC bias voltages on the gate

(5 V at best).

56

Chapter 2. Experimental and theoretical tools

Conclusion and perspectives

In this chapter the basics of our experimental techniques along with some necessary theoretical grounds have been described. Their purpose is to set safe grounds for more sophisticated model studies which will be detailed in the chapters to come. Indeed, they rely on careful, comprehensive calibrations, and our procedure can allow one to obtain unprecedented agreements between experimental data and theoretical models without need of free parameters. Therefore, in the following, measurements will be presented calibrated as much as possible, that is, in real units. Future works on other actuation and detection schemes such as the capacitive ones [START_REF] Quirin | Universal transduction scheme for nanomechanical systems based on dielectric forces[END_REF] can nonetheless be considered. Even with the magnetomotive scheme, tilting the sample holder at 45 • should allow to address mode coupling between two exural modes of dierent families of nearby frequencies, in the same fashion as Refs. [START_REF] Faust | Nonadiabatic Dynamics of Two Strongly Coupled Nanomechanical Resonator Modes[END_REF][START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF] where energy can be exchanged between the two modes, acting as a classical two-level system that can be driven coherently.

Despite the characterization of external electrical damping, the issue of the fundamental origin of damping mechanisms was eluded in this chapter. The dissipation measured between 1 K and 30 K is almost constant, which is a characteristic feature of dissipation mediated by two-level systems in the framework of the Standard Tunneling Model. In the third part of this manuscript these mechanisms are investigated with more details, especially below 1 K. In addition, unusual features observed for gas damping will also be reported, that are attributed to the interaction with a gas in conned geometries.

Last but not least, these characterizations provide us with a solid basis to embed beam-based NEMS structures in more sophisticated detectors that can operate at the quantum limit, to eventually reach the quantum ground state of motion [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF]. For this purpose, dedicated actuation/detection schemes are needed, by far more sensitive than the magnetomotive scheme and also enabling quantum-limited operations [START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF].

Résumé en français

Introduction

The present chapter, together with the next one, attempts to propose an in-depth, quantitative study of a phenomenon which is ubiquitous in vibrating nanostructures: frequency uctuations. Identifying intrinsic frequency noise sources in vibrating resonators has drawn a lot of attention recently for essentially two reasons. The rst one is obviously of practical nature: a huge eort in the two last decades has been made on using nanomechanical systems as mass sensors [START_REF] Hanay | Single-protein nanomechanical mass spectrometry in real time[END_REF], with increasingly good accuracy, to a tremendous mass sensitivity at the proton level [START_REF] Chaste | A nanomechanical mass sensor with yoctogram resolution[END_REF]. This last point relies crucially on the frequency resolution of a mechanical oscillator: a change in the resonator's mass due to e.g. a single protein sticking on it provokes a small change of its eigenfrequency. Thus, damping must be minimized, but also dephasing of the mechanical signal, as it also leads to spectral broadening [START_REF] Ben | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF]. In particular, while being promising candidates due to their extremely low mass, graphene and carbon nanotubes mechanical resonators suer a lot from excess spectral broadening. Several papers in the last few years have addressed the issue, e.g. Refs. [START_REF] Eichler | Symmetry breaking in a mechanical resonator made from a carbon nanotube[END_REF][START_REF] Barnard | Fluctuation broadening in carbon nanotube resonators[END_REF][START_REF] Miao | Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators[END_REF], while signatures of noise which does not fall into known sources and whose origin remain unclear [START_REF] Sansa | Frequency uctuations in silicon nanoresonators[END_REF] have been reported. But on a more fundamental basis, these can lead to elegant, in-depth experimental studies. The stochastic behaviour of (nano-)mechanical systems is in itself a teeming topic: recent theoretical and experimental studies have shown that their dynamics when coupled to various noises sources is highly non-trivial and requires subtle descriptions [START_REF] Dykman | Spectrum of an Oscillator with Jumping Frequency and the Interference of Partial Susceptibilities[END_REF][START_REF] Atalaya | Diusion-Induced Bistability of Driven Nanomechanical Resonators[END_REF][START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF][START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF][START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF], which go far beyond the mere technical issue of stability.

In the following Chapter we focus on one mechanism which is ubiquitous in micro/nanomechanical systems: thermal position uctuations experienced by the resonator are transduced through geometric, dispersive non-linear interactions (Dung and mode-coupling) into frequency uctuations. This interplay between position uctuations and non-linearities has already been addressed experimentally in a few papers [START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF][START_REF] Vinante | Thermal frequency noise in micromechanical resonators due to nonlinear mode coupling[END_REF], but always in a limited range, which always prevented previous studies to obtain a full description of the problem. Here, we take advantage of our experimental schemes to address the physics over an unprecedented wide range. While the temperature is articially tuned, the mechanism is deeply intrinsic, as all nanomechanical structures present some geometric non-linearity. Our experimental ndings are captured by a theoretical description largely based on Ref. [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF], which we try to explain in simple terms here. An eort is made on drawing parallels with Nuclear Magnetic Resonance (NMR), with whom the studied phenomenon shares some common features.

We also propose a derivation of an ultimate limit to frequency resolution imposed by 3.2. Fluctuation-dissipation theorem 63 this mechanism, which appears to be signicant for low-mass, low-stress structures such as carbon nanotubes. Finally, we extend experimentally the study to the bistable regime of the NEMS response, where position uctuations are themselves modied by the Dung nonlinearity, revealing a great richness especially close to bifurcation and spinodal points [START_REF] Dykman | Theory of uctuational transitions between stable states of a nonlinear oscillator[END_REF]. Our discussion on this last point will nonetheless stay at a phenomenological level and ignore the frequency uctuation aspect.

Fluctuation-dissipation theorem

As discussed previously, the NEMS is not isolated from its environment: phonons of the resonant mode are coupled to dierent thermal baths, among which bulk traveling phonons (coupling through the clamps), electrons in the metallic layer and external impedances, defects, two-level systems, etc. These are all couplings to uncontrolled degrees of freedom, towards which the resonator dissipates kinetic energy in an irreversible way while moving. This irreversibility appears through the rst derivative in the dynamics equation, which breaks time-reversal symmetry. The coecient Γ associated to the rst derivative term is a phenomenological quantity, the damping rate, i.e. the rate at which the system will return to equilibrium after a small perturbation by losing energy towards these degrees of freedom. From a macroscopic perspective, an analogous Mathiessen's rule makes this electromechanical damping rate the sum of the damping rates assigned to each bath:

Γ = {µ} Γ µ , (3.1)
where the sum is formally made over all dissipative channels. Here we assume that all baths degrees of freedom are statistically independent. A more detailed discussion on microscopic mechanisms from which damping channels emerge will be given in the next part. As mechanical energy is dissipated, it is converted into heat to its surroundings, which in turn favours scattering between bath constituents and mode phonons: such processes thus turn heat (which is incoherent) into kinetic energy by creating a random motion, the well-known Brownian motion.

In the canonical picture of Brownian motion, the position of a potential-free motion variable follows standard diusion laws and its average spatial extension grows with time as √ t. In our case, the resonator's collective mode variable x(t) also experiences Brownian motion, however its spatial extension remains conned within the restoring potential, as the latter counteracts the diusion process. On timescales larger than Γ -1 , the spatial extension ∆x 2 = x 2 keeps a nite, constant value. Note that here brackets indicate statistical averaging over realizations of the Hamiltonian; yet the ergodicity principle ensures that for long measurement times Γ -1 , time averaging is equivalent to statistical averaging. The spatial extension is easily obtained through Chapter 3. Interplay between non-linear dynamics and position uctuations the equipartition theorem: the mean potential energy U = mω 2 0 x 2 /2 associated to the motional degree of freedom x is k B T /2, and therefore:

∆x 2 = k B T mω 2 0 . (3.2)
We see that the same mechanism links Brownian motion (position uctuations) and damping (energy dissipation). This link has been mathematically demonstrated in a more general context by several authors in the 20 th century [START_REF] Callen | Irreversibility and Generalized Noise[END_REF][START_REF] Kubo | The uctuation-dissipation theorem[END_REF], and constitutes the uctuation-dissipation theorem, referred to as FDT in the following. It states that the autocorrelation function, which quanties the resonator's position uctuations at thermodynamic equilibrium, is proportional to the retarded (i.e., dissipative) response of the system to an external perturbation. Assuming we are in the linear response regime, in the classical limit ω k B T , the FDT writes in the frequency domain [START_REF] Kubo | The uctuation-dissipation theorem[END_REF] for a degree of freedom x with a susceptibility χ:

S x (ω) = 2k B T ω Im [χ(ω)] , (3.3)
where S x is the power spectral density of position uctuations, i.e. the spread in frequency of the position noise power. Wiener-Khinchin theorem establishes that it is merely the Fourier transform of the autocorrelation function x(t)x(t + τ ) , which is characteristic of the uctuations, acting a "memory":

S x (ω) = +∞ -∞
x(t)x(t + τ ) e iωτ dτ.

S x (ω) = |χ(ω)| 2 S F (ω), (3.5)
where S F is the spectrum of the Langevin force F th responsible for position uctuations. We assume here that this force is zero on average, and is δ-correlated, which in fact is a requirement of the FDT: in the so-called Markovian approximation, the cor-

S F (ω) = 2mΓ k B T. (3.6)
Note that the amplitude of the Langevin force, as well as the position uctuations (which are merely a linear lter to the force noise) are assumed to follow a Gaussian distribution: we have made the approximation that τ b Γ -1 , and therefore statistics is made on an innite number of independent collision events. This leads to Gaussian uctuations through the central limit theorem.

3.3 Dispersive coupling to Brownian motion through geometric nonlinearities

General framework

The previous section concerns an ideal, linear system, e.g. a perfectly linear NEMS with uncoupled eigenmodes. However, this picture will partially break if one goes beyond linear response. In particular, the FDT as written in Eq. (3.3) is not valid anymore.

Let us now consider the case of a real structure, where geometric nonlinearities will impose de facto coupling between the dierent eigenmodes of the structures. The typical situation during our experiments will correspond to the case where we focus on the dynamics of a peculiar driven mode with index n. In the simplest case, the Hamiltonian describing the driven mode and encompassing the situation writes:

H = p 2 n 2m n + 1 2 m n ω 2 n x 2 n + 1 4 m n γ n x 4 n + 3 4 m n m =n γ m,n x 2 m x 2 n -F n x n cos(ωt), (3.7)
where p n = m n ẋn is the momentum, F n is the driving force amplitude, γ n is the term describing the Dung interaction, while the bi-quadratic terms in the sum together with their coecients γ m,n account for the dispersive modal coupling to all the other eigenmodes by construction, as dened in section 2.9.3. One can obtain the global equation of motion of a mode n from a generalized Hamilton-Jacobi equation:

m n ẍn + ∂H ∂x n = -m n Γ n ẋn + F th,n (t).
(3.8)

The right hand side term describes phenomenologically the energy decay of the n-th resonator mode at a rate Γ n into the thermal bath, which acts in return on the resonator's dynamics through the Langevin force F th,n . A set of equations of motion for 66 Chapter 3. Interplay between non-linear dynamics and position uctuations each mode n is obtained through Eq. (3.8):

ẍn + Γ n ẋn + ω 2 n x n + γ n,n x 2 n + 3 2 m =n γ n,m x 2 m x n = F th,n (t) m n + F n cos(ωt) m n . (3.9)
The last term on the left hand side acts as a global non-linear restoring force which directly translates the energy stored in the involved modes into frequency pulling through geometric nonlinearities as an anharmonic correction. As this sum is proportional to the displacement, nonlinear terms act, within this model, as a correction to the harmonic potential.

If now the energy stored in the modes is only of thermal origin (i.e. arising from the modes coupling to a thermal bath, see section 3.2), the displacement variables of all the undriven modes are noises. As a result, the thermal displacement noise on a mode is transduced as a frequency noise in another through nonlinear mode coupling and intra-mode coupling, i.e. Dung nonlinearity.

Frequency noise correlator and spectral density

Before going deeper into calculations, it is useful to qualtitatively describe the frequency noise introduced in the previous section. In what follows, for the sake of simplicity we restrict ourselves to the case where one mode n is driven, with a certain response x n , and coupled to only one another undriven mode m of amplitude δx m .

The response can also be coupled to its own uctuations with the Dung interaction term, in the same fashion as the intermodal coupling case, with simply a numerical renormalization factor. In fact, we will see later that our experimental scheme can allow us to address these ideal situations.

From above, we know that the mode m will create a frequency uctuation δω n,m ∝ δx 2 m on the probed mode. The autocorrelation function for this frequency noise can be obtained through Wick's theorem, separating a 4-point correlator into a sum involving only 2-point correlators:

δx 2 m (0)δx 2 m (τ ) = ∆x 4 m + 2 δx m (0)δx m (τ ) 2 .
(3.10)

The corresponding noise spectral density is then readily obtained:

S x 2 m (ω) = ∆x 4 m δ(ω) + 1 π +∞ -∞ S xm (ω -ω )S xm (ω )dω . (3.11)
The frequency domain expression gives an intuitive picture of the noise contributions.

A sketch of the situation is represented in Fig. 3.1:

3.3. Dispersive coupling to Brownian motion through geometric nonlinearities 67

• The "δ-peak" term is a DC contribution which is simply the average value (rst order moment) of the frequency uctuations. Indeed, the position uctuations of every mode have a Gaussian distribution centered in zero. Then, by construction, the frequency uctuations are highly non-Gaussian and asymmetric, having a non-zero average value.

• The convolution term can be split into two parts. The high-frequency one, centered around 2ω m , represents a variation on a timescale (2ω n ) -1 too short to be followed by the resonator whose dynamics is set on a timescale Γ -1

m . As such, within the high-Q mode approximation, this fast varying term is ltered out, which is the philosophy of the rotating wave approximation. Meanwhile, the low-frequency part of the noise spectral density is spread over a width ∼ Γ m and will be responsible for slow variations of the modes resonance frequencies, which are the key aspect of the work presented here: the driven mode will have its frequency adiabatically following the frequency change.

0

Γ m Γ m β 2 m ∆x 4 m δ( ω)
F r e q u e n c y P o w e r S p e c t r a l D e n s i t y 2 ω m h i g h f r e q u e n c y n o i s e , f i l t e r e d i n R W A (3.12)

The obtained correlation decay rate Γ m of the frequency uctuations is a crucial parameter that will set the physics of the problem. Equivalently, a typical correlation time τ c = 1/Γ m can be dened for the uctuations. It represents the "memory" time of the uctuations, and its value will set the crossover between the dierent regimes experimentally observed. We see that even though the process described does not allow additional energy transfer, it is limited by the energy relaxation rate of the uctuating mode creating frequency noise.

Langevin equations in the rotating wave approximation

The problem at hand is tedious to solve in its raw form, yet the approximations stated above make a considerable simplication possible. We write the motion variables in terms of slow motion amplitudes, in the so-called rotating frame:

x n (t) = (A n (t)e iωt + A * n (t)e -iωt
) /2 with ω the driving eld frequency and A n the slow complex amplitude of motion. This decomposition can be re-injected in Eq. (3.9). Here, since we are in the high Q limit, we can make the following approximations:

• Terms which are not resonant with the driving frequency, i.e. proportional to e -iωt , e ±2iωt , e ±3iωt , are dropped.

• Since the amplitude term is varying on timescales comparable to Γ -1 n , we can neglect most of the derivative terms: Än , Γ n Ȧn

ω 2 A n , ωΓ n A n .
The same transformation can be applied to mode m, with its slow amplitude δA m . Such a notation is a reminder of the uctuating nature of mode m. As the latter is undriven, so its rotating frame frequency will be taken as ω m . In addition, we consider rst the case where the driving force F n is small enough not to signicantly distort the response, which in fact amounts to neglecting the terms β n,m |A n | 2 δA m . Note that if this approximation cannot be made, new phenomena emerge, which are described in the last section of this Chapter.

This leads to a linearized version of the problem. The equations of motion write for modes m and n, respectively:

                 δ A m + Γ m 2 δA m = F th,m (t)e -iωmt 2m m ω m , Ȧn + Γ n 2 A n + i    ω -ω n -β n,m (1 + δ n,m )|δA m | 2 δωn,m    A n =
F n e iφ 4m n ω n .

(3.13)

Dispersive coupling to Brownian motion through geometric nonlinearities 69

The probe mode n is then aected by noise on the mode m, through an eective frequency noise δω n,m (t

) = (1 + δ n,m ) β n,m |δA m (t)| 2 , where δ n,m is Kronecker's symbol.
The latter allows a unied description of the inter-mode coupling and intra-mode coupling ("self-coupling" of the response to its own position noise) cases. To allow such a unied description, here we have assumed that the uctuating mode is linear as far as its uctuations are concerned. The case where the non-linearity comes into play will be addressed in section 3.5. For the time being we are only concerned with the uctuations statistics, and one can show that the deviation to Gaussian statistics in the steady-state is signicant only if the Q factor is of order 1 [START_REF] Thomas | Derivation and Application of the Fokker-Planck Equation to Discrete Nonlinear Dynamic Systems Subjected to White Random Excitation[END_REF], hence negligible in our high Q devices. We can decompose the slow complex amplitude into its quadratures δA m = δX m + iδY m . These two quadratures are uncorrelated (within our linearization), and are both so-called Ornstein-Uhlenbeck noises, i.e. are Gaussian processes with a nite steadystate variance due to the restoring potential. They satisfy the following Langevin equations:

           δ X m + Γ m 2 δX m = F th,m (t) cos(ω m t) 2m m ω m , δ Y m + Γ m 2 δY m = - F th,m (t) sin(ω m t) 2m m ω m , (3.14)
with autcorrelation functions δX m (0)δX m (τ ) = δY m (0)δY m (τ ) = ∆x 2 m exp(-Γ m τ /2), and δX m (0)δY m (τ ) = 0. While δX m , δY m are Gaussian and centered, it is obvious that the eective frequency noise δω n,m is neither Gaussian nor centered. Moreover, the spectrum of the noise is similar to the position spectrum, thus providing a good example of a highly colored noise.

Averaging over the frequency noise: two interesting limit behaviours

Our measurements always yield quantities that are averaged. To model this theoretically in the situation at hand is not a simple task for two reasons. First, as said above, the frequency uctuation at a time t is not distributed according to a Gaussian law:

since position uctuations are Gaussian, frequency jumps take only positive values and are distributed according to an exponential law. The stationary joint probability density for the displacement noise of the uctuating mode is:

p(δX m , δY m ) = 1 2π∆x 2 m e - δX 2 m + δY 2 m 2∆x 2 m . (3.15)
Chapter 3. Interplay between non-linear dynamics and position uctuations

The mode-coupling induced frequency noise variable writes δω n,m = (1+δ n,m )β n,m (δX

2 m + δY 2 m ), with its magnitude Σ n,m = 2(1 + δ n,m )β n,m ∆x 2 m .
The frequency noise stationary probability density is readily obtained through a Jacobian transform:

dp(δω n,m ) = Θ(δω n,m ) Σ n,m e - δω n,m Σ n,m dδω n,m dθ 2π , (3.16)
with Θ the Heaviside function. However, due to the nite decay rate of the uctuations, one cannot always simply average the susceptibility over time-independent distributions. A simple picture can clarify the situation: if we assume that the resonator experiences at a time t a frequency change of typical magnitude Σ n,m , the minimum time needed by the resonator to synchronize with the jump and "sample" this change of frequency accurately will be ∼ Σ -1 n,m . This is the characteristic time to be compared with the typical decay time τ c = Γ -1 m , or "memory", of the frequency noise correlations. Two limits exist, which are detailed below.

Inhomogeneous broadening limit

The rst limit situation occurs when Σ n,m τ c 1 (see Fig. 3.2): in this regime, the uctuations are essentially correlated over an innitely long time, which means that every change in frequency will be well resolved and adiabatically followed by the resonator: the latter can sample the change with increasingly good accuracy. Provided that the averaging measurement is done over a innitely long time, say

Γ -1 m and Γ -1 n ,
it is equivalent to say, by ergodicity, that the average susceptibility in the frequency domain is the simple sum of partial stochastic susceptibilities χ ( ω -δω n,m ) over all realizations of δω n,m with the exponential distribution derived above:

χ(ω) = +∞ -∞ χ(ω -δω n,m )Θ(δω n,m ) e -δωn,m/Σn,m Σ n,m dδω n,m . (3.17)
The distribution involed is none other than the Boltzmann weight, as noticed in

Ref. [START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF]: frequency uctuations are directly proportional to the uctuating mode's stored energy. One notes the treatment similar to Nuclear Magnetic Resonance [START_REF] Slichter | Principles of Magnetic Resonance[END_REF],

where the eective experienced eld is averaged over spatial inhomogeneities. This analogy must be taken carefully: of course, here the system experiences time-controlled disorder instead of spatial. In the experiment described here, this analogous regime can be reached for frequency noise magnitudes high enough. The physics is then reduced to the interplay of two parameters: the frequency noise magnitude and the probe mode's decay time. This will be addressed in greater detail in Chapter 4.

Motional narrowing limit

In the opposite limit where Σ n,m τ c 1 (see Fig. 3.3) correlations decay too fast for the NEMS to resolve a small frequency change: the memory of this change is erased faster than the time required to sample it. Then, the eective distribution "seen" by the probe's mode is not obtained from the full reservoir's dynamics but only from a truncated part of it, as only a fraction of the realizations on the time disorder are accessible.

This is somehow analoguous to NMR again: by replacing a spatial disorder on the surrounding magnetic eld by a time disorder on the frequency, we observe a crossover from the previously described inhomogeneous broadening (abbreviated IB) to a motional narrowing regime (abbreviated MN), where essentially the NEMS will only have its parameters renormalized by the rst (resonance shift) and second (linewidth broadening) moments of the uctuating frequency distribution. This transition was observed in mechanical devices in the case of articially created telegraph frequency noise in Ref. [START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF], where the authors varied the noise correlation time rather than its magnitude. From a well resolved two-peak structure in the inhomogeneous broadening regime due to the two only possible values of the frequency noise, they observed the transition towards the MN regime where the two peaks merged into a single one which gets closer to the original resonance as the correlations decay time is reduced. The originality in the case described in this manuscript is that it relies on really intrinsic properties of the resonator: its thermal uctuations and its geometric nonlinearities. In the context of NMR, motional narrowing arises from fast diusion of the inhomogeneous eld sources [START_REF] Slichter | Principles of Magnetic Resonance[END_REF], and therefore it increases with temperature. Here the phenomenon is smeared out as the eective temperature T eff ∝ ∆x 2 m increases. This has led authors who observed a similar phenomenon in the optical emission spectrum of quantum dots to name it an "unconventional" motional narrowing [START_REF] Berthelot | Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot[END_REF], "unconventional" referring to the unusual temperature dependence. However, in the present context temperature does not play an analogous role as in NMR. In the original context, it speeds up (spatial) inhomogeneities sources, whereas here it somehow slows the (time) sources down with respect to the memory time.

Joining the two limits with a stochastic path integral approach

The previous section provides rather intuitive ways to consider the phenomenon. However, our experimental results often fall in between the two marginal cases, and a simple quantitative formulation is not easy to nd in that situation. Nonetheless, there is an elegant way to obtain the susceptibility averaged over frequency uctuations in the whole dynamic range through a stochastic path integral formulation [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF].

The idea is that in a given time interval, between known initial and nal positions in the phase space, the Brownian motion of the mode creating frequency noise can take many possible paths in the phase space, and that the statistical averaging procedure is merely taking into account all these paths with their weight. Therefore, the accu-3.3. Dispersive coupling to Brownian motion through geometric nonlinearities 73

mulated phase can also take many paths, and in fact, experiences diusion, as shown in Fig. 3.4. Each path's weight can be obtained from the Langevin dynamics of the uctuating mode in the Markovian bath approximation. From what we have described above, we know that the average susceptibility will be a function of the scaling param-

eter called "motional narrowing parameter" α n,m = Σ n,m τ c = 2β n,m (1 + δ n,m )∆x 2 /Γ m .
0 , 0 0 0 0 , 0 0 2 0 , 0 0 4 0 , 0 0 6 0 , 0 0 8 0 , 0 1 0 0 , 0

2 , 0 x 1 0 -1 1 4 , 0 x 1 0 -1 1 6 , 0 x 1 0 -1 1 8 , 0 x 1 0 -1 1 1 , 0 x 1 0 -1 0 1 , 2 x 1 0 -1 0 1 , 4 x 1 0 -1 0 A c c u m u l a t e d p h a s e ( V 2 )
T i m e ( s ) δY δX 0 , 0 0 0 0 , 0 0 2 0 , 0 0 4 0 , 0 0 6 0 , 0 0 8 0 , 0 1 0

0 , 0 4 , 0 x 1 0 -1 2 8 , 0 x 1 0 -1 2 1 , 2 x 1 0 -1 1 1 , 6 x 1 0 -1 1 2 , 0 x 1 0 -1 1 A c c . p h a s e s t a n d a r d d e v i a t i o n ( V 2 )
T i m e ( s )

A n (t) = t -∞ dt F n e iφ 4m n ω n exp - Γ n 2 + i(ω -ω n ) (t -t ) + i t t δω n,m (t )dt . (3.18)
This solution is simple to interpret: indeed, the exponential term is merely a slow time domain susceptibility χ sl (t, t ), relating the response at a time t after a perturbation at a time t , modulated by a stochastic accumulated phase term, such that:

A n (t) = t -∞ F n e iφ 4m n ω n χ * sl (t, t )dt . (3.19)
By denition, the lock-in amplier measures an averaged response at a given frequency.

Within the applied force factor and calibration coecients, this is just the susceptibility in frequency domain, which is by construction:

Chapter 3. Interplay between non-linear dynamics and position uctuations

χ n (ω) = i 2m n ω n t 0 χ sl (t , 0) dt . (3.20)
We can now write an averaged slow susceptibility, which amounts to the usual certain part times the averaged stochastic modulation:

χ sl (t, 0) = exp - Γ n 2 + i(ω -ω n ) t e i∆φn,m(t) , (3.21)
where ∆φ n,m (t) = t 0 δω n,m (t )dt is the accumulated phase during one Brownian trajectory of the mode m between time 0 and t. From the integral form in the exponential argument one can see that the averaging has to take into account all realizations of a full trajectory in the phase space of the Langevin process (δX m , δY m ) between time 0 and time t. It is useful to notice that averaging over the frequency noise is equivalent to average over the two quadratures which are uncorrelated in the limit of small driving elds: thus each averaging can be done separately. Since the bath is Markovian, one can convert Langevin equations (3.14) on quadratures into Fokker-Planck equations, which govern the evolution of transition probabilities. We note

W X i,f ≡ W (δX m,f , t f |δX m,i , t i
) the probability to be at the position δX m,f at a time t f starting from position δX m,i at time t i , and equivalently for δY m . The Fokker-Planck equation writes:

∂W X i,f ∂t f = Γ m 2 ∂ δX m,f W X i,f ∂δX m,f + Γ m ∆x 2 m 2 ∂ 2 W X i,f ∂δX 2 m,f . (3.22)
The same equation applies also to

W Y i,f ≡ W (δY m,f , t f |δY m,i , t i ).
The general solution to this equation writes:

W X i,f = 1 2π∆x 2 m (1 -e -Γm(t f -t i ) ) exp - δX m,f -δX m,i e -Γm(t f -t i )/2 2 2∆x 2 m 1 -e -Γm(t f -t i ) . (3.23)
For long times the memory of the initial position (δX m,i , δY m,i ) is erased in the phase space, and we recover the steady-state, Gaussian probability distribution (3.15) on δX m,f . More interestingly, in the limit of short time intervals

δt = t f -t i τ c ∼ Γ -1 m ,
one obtains the propagator:

W X i,f ≈ δt Γ -1 m 1 2πΓ m ∆x 2 m δt exp - [δX m,f -δX m,i (1 -Γ m δt/2)] 2 2Γ m ∆x 2 m δt . (3.24)
It is now possible to reconstruct, from this propagator, the probability associated to the realization of one path. Let us cut the time interval [0, t] into N small equal intervals, such that δt = t/N . Since the process is assumed to be Markovian, one can 3.3. Dispersive coupling to Brownian motion through geometric nonlinearities 75

derive the probability density associated to one path using conditional probabilities:

W[δX m ] [0,t] = N -1 i=1 W X i,i+1 = 1 2πΓ m ∆x 2 m δt N/2 exp    - N -1 i=1 δX m,i+1 -δX m,i δt -Γm 2 δX m,i 2 δt 2Γ m ∆x 2 m    . (3.25)
Taking the limit N → ∞, i.e. δt → 0, the probability density functional on the path δX m between 0 and t is:

P[δX m (t)] = exp - 1 2Γ m ∆x 2 m t 0 δX m (t ) + Γ m 2 δX m (t ) 2 dt . (3.26)
The expectation value for the accumulated phase can then be expressed through the functional integral:

e i∆φn,m(t) = DδX m DδY m e i t 0 (1+δn,m)βn,m(δX 2 m (t )+δY 2 m (t ))dt P[δX m ]P[δY m ], (3.27)
with the formal dierential term in the trajectories space:

DδX m (t) = N →∞ N i=1 dδX m,i (2πΓ m ∆x 2 m δt) 1/2 .
(3.28)

Calculating the average dephasing term from Eq. (3.27) is no easy task. Basically, one has to discretize once more the integral, which decomposes over sums on Gaussian variables. This tedious task is performed in Ref. [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF]. After calculation, one gets explicitly the expectation value of the accumulated phase term e i∆φn,m(t) ≡ G(t):

G(t) = e Γmt/2 cosh(a n,m t) + Γ m (1 + 2iα n,m ) 2a n,m sinh(a n,m t) . (3.29)
where a n,m = Γm 2 1 + 4iα n,m . Note that this path integral approach provides us with another level of interpretation. At the most basic level, a mechanically resonant mode is merely the constructive interference of an incoming and a reected sound wave in the beam, which acts as an acoustic cavity. As the nonlinear interaction creates an additional stochastic phase, this interference is blurred, because each wave of the sum is partially incoherent: the stochastic phase term competes with the background phase coherent term of the standard response.

Chapter 3. Interplay between non-linear dynamics and position uctuations

In the motional narrowing limit, the initial phase coherence is more protected since noise correlations, decaying too fast, become localized in time. Therefore the eect of phase diusion on resonance properties is limited, as the most noisy paths have a reduced impact.

In the opposite, inhomogeneous broadening regime, the impact of phase diusion is not attenuated anymore by time localization eects and therefore blurring masks the background phase coherence. To paraphrase Ref. [START_REF] Dykman | Spectrum of an Oscillator with Jumping Frequency and the Interference of Partial Susceptibilities[END_REF], the average (3.17) can be interpreted as an incoherent sum over partial susceptibilities χ(ω n -δω n,m ), as described in paragraph 3.3.4.

Even though the expression of G is quite unpractical, one can expand it in the motional narrowing limit, nding again the results obtained in Section 3.3.4:

G(t) ≈ e iαn,mΓmt-Γmα 2 n,m t/2 .
(3.30)

The complex term corresponds to a frequency shift, the real one is an additional decay term. In frequency domain, it leads to a complex Lorentzian averaged response G(ω) with renormalized resonance frequency ω res and linewidth ∆ω:

ω res ≈ ω 0 + Γ m α n,m = ω 0 + 2β n,m (1 + δ n,m )∆x 2 m , (3.31)
∆ω ≈ Γ n + Γ m α 2 n,m 2 = Γ n + 2 [2β n,m (1 + δ n,m )∆x 2 m ] 2 Γ m .
(3.32)

3.4 Results in the driven case

Experimental setup

The experimental setup is shown in Fig. 3.5. As the fundamental (noted 1) and third (noted 3) mode are rather easy to actuate and detect, we will focus our study on them.

Thermal uctuations at 4 K are way below our sensitivity limit and too small to reach the inhomogeneous broadening limit, in particular for a doubly-clamped structure which does not move too much. Thus we deliberately feed the rst mode with a current noise. The latter goes through injection line with the additioner used for the calibration procedure (see paragraph 2.8.2) and is translated by the magnetomotive scheme into a noise force.

In principle, this puts the noise-fed mode highly out of equilibrium. However, the force noise is strictly equivalent to an eective temperature dened through the of the resonator, the noise power spectral density S F (ω) is at. Thus, the eective temperature mentioned above is dened such that S F (ω

) = 2m 1 Γ 1 k B T eff .
We carefully characterized the delivered noise: from spectrum analyzer measurements we know that there is only a 5% decreasing slope between 0 and 10 MHz, irrelevant at the NEMS scale, and made sure the noise is Gaussian, centered, as shown in With such a scheme, we are able to tune one mode's eective temperature and study its coupling to the driven mode while leaving the other modes utterly irrelevant to the study, as thermomechanical noise at 4 K is completely negligible in comparison.

In addition, we carefully characterize both modes, following the procedure described in Chapter 2: applied forces as well as displacements are known in real units, which enable to know nonlinear coecients β i,j , {i, j} = {1, 3}. These calibrations put together give us a 20 % margin. The experiments have been performed on 4 dierent devices, whose characteristics are gathered in the following table.

Starting from a voltage noise δV in shown in peak-to-peak (V pp ) units, we can recalculate the current noise spectrum S I (ω) (force S F ) owing through the NEMS: we have Chapter 3. Interplay between non-linear dynamics and position uctuations

-0 . 4 -0 . 2 0 . 0 0 . 2 0 . 4 0 5 0 0 0 1 0 0 0 0 C o u n t s V o l t a g e ( V ) 5 0 0 m V p p o n 1 M o h m σ = 0 . 1 2 5 V 0 . 0 1 0 . 1 1 1 0 1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 F i l t e r t r a n s m i s s i o n
F r e q u e n c y ( M H z ) spring checked that the root mean square (rms) value delivered by the generator is dened such that the peak-to-peak value displayed is 8 times the rms value, as shown in Fig.

k n freq. ω n /2π Γ n /2π non-lin. β n,m /2π n = 1 300 µm-n • 1 0.40 N/m 0.59 MHz 100 Hz β 1,1 = 1.35 15 Hz/m 2 n = 1 300 µm-n • 2 0.45 N/m 0.66 MHz 140 Hz β 1,1 = 1.70 15 Hz/m 2 n = 1 15 µm-n • 1 1.2 N/m 6.9 MHz 550 Hz β 1,1 = 5.0 10 19 Hz/m 2 n = 3 300 µm-n • 1 3.4 N/m 1.74 MHz 40 Hz β 3,1 = 2.

left panel.

Besides, this noise is spread over the output bandwidth (9.5 MHz) of the AFG, hence a value given in V rms / √ Hz. The resulting voltage noise goes through the bandpass lter (described in the above paragraph), and the injection line, which sets an integrated value for the voltage noise applied to the NEMS+bias resistance ensemble. Within a numerical shape factor, the current noise spectrum in A 2 /Hz units that ows through the NEMS is then: 3.1), in the motional narrowing (left panel) and inhomogeneous broadening (right panel) regimes in the case of intra-mode, Dung mediated frequency uctuations, at a constant driving excitation. Solid lines are direct applications of Eq. (3.34). The grey line is a measurement of the standard response at the same drive tone for comparison, with complex Lorentzian ts (dashed lines).

S I (ω) = 0.95 √ πG I (ω 1 ) 8(R N + R b ) δV in √ 9.5 MHz
χ n (ω) = +∞ -∞ χ n (ω -δω n,m )Θ(δω n,m )G(δω n,m )dδω n,m .
The susceptibility is probed through our standard response measurements, with long enough ( Γ -1 m , Γ -1 n ) lock-in time constants so as to integrate all realizations of the slow frequency uctuations. One important point is to keep coherent driving excitations at a low enough level, in order not to enhance other, unwanted non-linear interaction terms that go beyond the linearized model presented in section 3.3.3. These additional terms can indeed result in correlations between the two quadratures of the uctuating mode, and extensions of the model would have to be considered (see section 3.7). In our case, a good criterion is to apply drive forces such that the unperturbed response (grey dots in Fig. 3.7) still appears symmetric (or anti-symmetric, for the inphase response). Two resonance lines for the "self-coupling" case, i.e. the driven mode coupled to its own Brownian motion are shown in Fig. 3.7. Each one is representative 80 Chapter 3. Interplay between non-linear dynamics and position uctuations of an asymptotic regime, for the same sinusoidal excitation. One can see that for low frequency noise (left panel), i.e. in the motional narrowing regime, the lineshape remains essentially Lorentzian. This is not surprising, as the calculated lines are simply renormalizations of the noiseless lines in this regime (see previous sections).

This is equivalent to say that the pseudo-distribution is simply a Lorentzian which carries only the rst and second moments of the "full" exponential distribution. The resonance shift, being proportional to α 11 is signicant, while linewidth broadening, being proportional to α 2 11 remains marginal. Thus, phase diusion is rather limited, while its drift, seen through the resonance frequency shift, is consequent.

On the contrary, the shift does not increase much more in the high amplitude noise (inhomogeneous broadening, right panel) limit, because the distribution gets closer to the real one, which is real-valued, exponential. Note that as it gets closer, its asymmetric shape leaves an imprint on the response, which also becomes highly asymmetric as a result of the convolution. Indeed, the uctuations take only positive values, which explains the long-tail feature in the frequency region above the resonance frequency.

As a matter of fact, the line gets closer to an exponential shape, i.e. the stationary frequency noise distribution. We use the most simple convention: the chosen inhomogeneous linewidth parameter is the full width at half height (FWHH, or FWHM).

It allows to easily extract numerically the parameter from the theoretical expression (3.34), which is neither explicit nor easy to manipulate in its most general form. Note also that the signal-to-noise ratio worsens as the noise becomes larger, because a much longer averaging time is required to obtain cleaner lines: yet it is unrealistic to measure for too long, as uncontrolled drifts due e.g. to two-level uctuators (see Chapter 6) can perturb signicantly the measured lines on long timescales. The theoretical lines reproduce rather well the measured lines, with no free parameters.

The evolution of the two relevant parameters (shift, broadening) as a function of the Brownian motion amplitude, i.e. the magnitude of the frequency noise (or equivalently, the eective temperature) is shown in Fig. 3.8, with broadening obtained from FWHH measurements. The agreement between experimental data and theory is rather good, and there is no free parameter, as our noise source is calibrated within ±15 %.

The case of inter-modal coupling is also addressed, with again representative examples in Fig. 3.9. Note that the line with the highest level of noise applied is still close to the motional narrowing limit. This is due to values for inter-mode nonlinear coecients smaller than for Dung ones, as mode overlaps are signicantly smaller.

Thus, to witness an eect, very high noise excitations are required, which explains why the most noisy lines are still rather close to a Lorentzian. Note, however, that the agreement between theoretical expressions and experimental data is not as good as in the Dung case, and higher noise would result in even poorer resolution: thus the inhomogeneous broadening regime could not be properly observed. This can also explain why the work presented in Ref. [START_REF] Vinante | Thermal frequency noise in micromechanical resonators due to nonlinear mode coupling[END_REF] was limited to the low noise, motional narrowing limit: although temperature could also be enhanced by a few orders of F r e q . s h i f t ( H z )

∆x 2 1 ( m 2 ) 4 β 1 , 1 ∆x 2 1 I B M N 1 0 -1 5 1 0 -1 4 1 0 1 0 0 F r e q . s h i f t ( H z ) ∆x 2 1 ( m 2 )
0 , 0 5 , 0 x 1 0 magnitude, mode-coupling geometric nonlinear coecients were too small to observe a transition, a situation expected for a cantilever. F r e q u e n c y ( H z )

Figure 3.9: third mode in phase (blue) and quadrature (orange) resonance lines measured for a 300 µm long device, for low (left) and intermediate (right) noise levels on the rst mode, at a constant driving excitation. Solid lines are the direct application of Eq. (3.34). Grey lines are measurements of the standard response at the same drive tone for comparison, with complex Lorentzian ts (dashed lines).

The resonance shift and the broadening over the experimentally studied noise range are shown in Fig. 3.10. We see rst that data taken remain close to the motional narrowing limit even for high noises, as mentioned above. Besides, the agreement in the case of the shift is not as good, again, as in the Dung case, with discrepancies Chapter 3. Interplay between non-linear dynamics and position uctuations up to 20 %. This is likely due to a competition between the mode coupling eect, hardening the resonator, and the fact that the required noise starts to signicantly overheat the structure physically, leading to softening of the resonator (see Section 2.7): this competition would thus result in a smaller shift than expected. F r e q . s h i f t ( H z ) From these measurements one thing is clear: in the linear regime of driving force, the observed eect only depends on the dynamics of the noisy mode, while the probe's dynamics has little inuence, whether the probe is another mode or the noisy mode response itself. This shows that the two eects dier only quantitatively. Only in the case where Γ n Γ m one might observe a departure from the model, because then a fraction of the eective frequency uctuations would be too fast to be followed by the probe mode, which in fact means that this fraction would fall outside of the rotating wave approximation, i.e. would be ltered by the probe's dynamics.

∆x 2 1 ( m 2 ) 2 β 3 , 1 ∆x
In addition, measurements for very small position uctuations unambiguously demonstrate the motional narrowing eect. While we have not used noises large enough to depart from the motional narrowing regime on the 250 µm sample, the resolution is excellent: in Fig. 3.11 we see that for an integrated position noise on the fundamental mode ∆x 2 1 = 6.1 × 10 -16 m 2 , theoretical curves obtained through full averaging with Eq. (3.34) and through averaging with Eq. (3.17) are signicantly dierent: while the application of the full theory matches well the resonance lines, the IB-type averaging clearly fails to reproduce the data.

An interesting feature is that this discrepancy appears only on the right side of the response: this is expected, as the motional narrowing process prevents large excursions in frequency, and those happen by construction towards higher frequencies since δx 2 m ≥ 0. If one considers the unperturbed driven response as the result of a frequency noise magnitude going to zero, the motional narrowing is merely the onset of the constructive F r e q u e n c y ( H z )

Figure 3.11: X (left) and Y (right) response curves taken for a small position noise, in the motional narrowing limit. Two ts are presented: one (light green) is from the analytic result (3.34), while the other (dark green) is obtained by averaging with the static exponential distribution on frequency noise, with a manually added shift in the latter case to allow a visual comparison.

interference at the origin of the mechanical resonance. As the noise magnitude, i.e. the disorder, goes to zero, there is less diusion around the most probable accumulated phase path (the black line in Fig. 3.4 left panel), which enhances phase coherence and results in more Lorentzian lines.

Observation of non-linear position noise spectra

As seen in section 3.2, the response and the position noise spectrum are linked through the uctuation-dissipation theorem in the linear regime. Yet, in presence of a nonlinear interaction, even though the phenomenological explanation should still hold, a direct relation such as Eq. (3.3) is not necessarily valid anymore.

Nonetheless, the results obtained in the previous sections for a driven mode obviously apply qualitatively when looking at the position spectrum of the uctuating mode, without any driving sine force. Under high force noises, the nonlinear restoring force term becomes increasingly relevant. This impacts the position spectrum, which undergoes frequency noise and line shape distortion as the driven responses. Including now a non-linear term γ m,m δx 3 m (t) on the dynamics of the noisy mode in Eq. 3.9 and performing a RWA, its Langevin equation writes now:

δ A m + Γ m 2 δA m + β m,m |δA m | 2 δA m = F th,m (t)e -iωmt 2m m ω m . (3.35)
From rules on derivation of correlation functions, and using δA m = δX m + iδY m , one can write coupled Langevin equations on both quadratures. The procedure is detailed in [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF] and follows the same idea as what has been exposed above. In the end, the Chapter 3. Interplay between non-linear dynamics and position uctuations average accumulated phase term is rather similar to the driving case:

G s (t) = e Γmt cosh(a m,m t) + Γ m (1 + 2iα m,m ) 2a m,m sinh(a m,m t) . (3.36)
The non-linear spectrum thus writes:

S x (ω) = +∞ -∞ S x (ω -δω n,m )Θ(δω n,m )G s (δω n,m )dδω n,m , (3.37)
where G s is the Fourier transform of G s . This section presents our technique to measure the spectrum of position uctuations, as well as the results obtained under large uctuations.

Acquisition of noise spectra

The most conventional way to measure noise is to use a spectrum analyzer, which measures the autocorrelation function of the noise signal in a given bandwidth and display its Fast Fourier Transform (FFT). While it is a straightforward measurement procedure, having the measured quantities in real units can be tedious, as the displayed spectra are only relative to a certain reference level and go through a complex chain of processing. In the present experiment, having real units is crucial to quantitatively address the phenomenon. Meanwhile, lock-in amplier measurements are already calibrated so that forces and displacements are known in real units. Thus, we developed an alternative way to measure position spectra using only the lock-in amplier. Another advantage of this technique is the possibility to obtain phase-resolved information on the uctuations, which proves useful when nonlinearities break the symmetry of the 2D Gaussian statistics in the phase space (see section 3.7 below).

By denition, a lock-in amplier demodulates in amplitude a given input signal by mixing it with a reference signal. The resulting voltage is averaged during a time set by the lock-in time constant and long enough in principle, so that only the signal present at the reference frequency remains with minimum noise. In the frequency domain, this procedure is the convolution of the input signal with a narrow lter centered around the swept reference frequency, of bandwidth inversely proportional to the time constant. Therefore, the narrower the lter, the better the resolution on the measured quantity, at the expense of a fast measurement.

However, it is implied that the wanted signal, in such a procedure, is a deterministic one, obtained through a response to a drive signal synchronized in frequency with the reference signal. What we want to measure here is a noise, which by denition is spread over a large range of frequencies. Therefore, one can use the lock-in amplier in a reverse way by opening the bandwidth centered around the reference frequency, to acquire as much noise as possible coming from the system.

Within this bandwidth BW, a signal is acquired and digitized with a DAQ card operating much faster than the lock-in time constant. The acquired signal length is proportional to the bandwidth, but we loop the measurement to obtain a trace with a number of samples N statistically signicant. A computer routine then calculates numerical correlators

C XX [j] = δX i δX i+j i , C Y Y [j] = δY i δY i+j i , C XY [j] = δX i δY i+j i ,

Pulling Brownian motion out of the measured signal

The procedure explained above does not take into account the fact that the NEMS is actuated in an invasive way through current feeding and is inserted in a disadapted transmission line. These factors will be responsible for additional parasitic noise sources, which will result in dramatically limited performances of our setup compared to, say, optical means [4]. Let us now focus more on these dierent contributions. The noise current injected writes in the rotating frame δI(t) = δI c (t) cos(ωt) -δI s (t) sin(ωt). δV (t), the detected signal at the level of the lock-in, is the sum of:

• the electromotive force noise corresponding to the Brownian motion itself:

δV B (t) = -(ξlB) 2 d dt (χ * δI)(t), (3.38)
• the Ohmic response of the conducting layer having a resistance R N :

δV ohm = R N δI(t), (3.39)
• a capacitive response due to the nite transmission line capacitance C eq :

δV capa (t) = δI(t ) C eq dt , (3.40)
• all the other noise contributions that do not arise from the electrical noise current, e.g. the lock-in amplier input noise. These are gathered in a quantity δV oth (t).

Chapter 3. Interplay between non-linear dynamics and position uctuations

The voltage noise spectrum is the Fourier transform of the total voltage noise autocorrelation function δV (t)δV (t + τ ) . Within the rotating wave approximation, the terms δI c , δI s can be neglected. Moreover, the noise current is ltered with a 0.7 MHz bandwidth lter, and thus has a typical correlation time ∼ 1 µs, which is reasonable when compared to the NEMS decay time in the ms range. Therefore the background noise can be safely approximated as δ-correlated. In addition, noise arising from uncontrolled sources is uncorrelated with the current noise. The main drawback of this technique is a cross correlation appearing between the Brownian motion term and the ohmic/capacitive contributions. Especially, the Brownian-capacitive cross-correlation will introduce, in the frequency domain, a term proportional to Re [χ(ω)], which is anti-symmetric, as opposed to the Brownian spectrum and the Ohmic contribution cross-terms. This will result in a distorted total spectrum lineshape, as shown in Fig. 3.12 left panel. The total spectrum writes:

S V (ω) = S th (ω) + S cross (ω) + S back (ω), (3.41)
with each contribution:

S th (ω) = |χ(ω)| 2 (ξlB) 4 ω 2 S I (ω), (3.42) S cross (ω) = -2R N Im [χ(ω)] + 2 C eq ω Re [χ(ω)] (ξlB) 2 ωS I (ω), (3.43)
S back (ω) = S oth (ω) + R 2 N + 1 C eq ω 2 S I (ω). (3.44)
Knowing the metallic layer resistance R N , only the transmission line capacitance C eq and the noise level S oth (ω) (supposedly white in the frequency range considered) are left as free parameters to t the measured spectra. An example of application of this procedure is shown in Fig. 3.12.

By subtracting the cross correlation term and the background term to the raw, measured spectrum, one is left only with the actual Brownian motion spectrum. Since this rectication procedure involves subtraction of unwanted contributions, it introduces negative values in the spectrum, but these are mere artefacts. We measure spectra for dierent applied (current) noises, checking that the equipartition relation (3.2) holds, with mode parameters (spring, mass) measured independently, e.g. through standard driven response procedures (see Fig. F r e q u e n c y ( H z ) It is useful to notice that each contribution has a dierent eld dependence. Therefore, to ensure that the procedure is valid in the whole parameter range for the chosen t parameter C eq , we measure spectra at dierent elds and report the tted magnitudes of each contribution at the NEMS resonance as a function of the magnetic eld, as shown in Fig. 3.14. The agreement between the tted contribution and Eqs. Last but not least, one must nd an optimal bandwidth BW with respect to the Brownian spectrum width. In the ideal case, taking a working point at the highest possible eld with the smallest bandwidth would be the best option. However, doing so is time consuming, so the bandwidth chosen is generally not the smallest one. Yet, as the useful noise has a spectral spreading equal to the system linewidth, an upper limit on the measurement bandwidth exists. Indeed, if this bandwidth is too large, the eective thermomechanical noise will be articially broadened and its resonance height decreased, since the peak will be "diluted" over the measurement bandwidth due to the convolution process. Therefore, a trade-o bandwidth must be found so that the mechanical spectrum has its true linewidth, while the signal is measured over reasonable times. We have measured spectra for dierent electromechanical linewidths tuned with the magnetic eld, as reported in Fig. 3.15. We have checked that all background contributions scale as BW as expected for a white noise. These peak height and width plots, normalized to mechanical parameters (true height, true damping rate at a given eld) show that the data taken at dierent damping rates all collapse on the same curve when the measurement bandwidth is normalized to the damping rate. An optimum value in this case is a lock-in bandwidth equal to 30 % of the damping rate.

0 . 0 1 0 . 1 1 1 0 0 . 1 1 0 . 3 N o r m . H e i g h t N o r m . B W 1 E -3 0 . 0 1 0 . 1 1 1 0 1 1 0 N o r m . F W H H N o r m . B W 0 . 3
One important point is that in all these measurements, the intercorrelation spectrum S XY has always been 0, and spectra measured on both quadratures were identical S XX = S Y Y . This does not hold anymore if a strong sine wave driving excitation is used (see section 3.7), as the linearization used to obtain Eq. (3.13) is not valid anymore in the self-coupling case: the term |A n | 2 δA n cannot be discarded in this situation. [START_REF] Landau | Statistical Physics, Third Edition[END_REF] Chapter 3. Interplay between non-linear dynamics and position uctuations 3.5.3 Distortion of position spectra at high eective temperatures

In the limit of high eective temperatures, the nonlinearly-induced shift and broadening observed for response measurements becomes also visible for the position spectrum, with its two limit behaviors (MN and IB). However, one has to take into account overheating of the structure due to high injected powers, which becomes relevant in the very high eective temperature limit ( 109 -10 10 K). ). The grey line is a spectrum obtained in the linear regime, for low enough eective temperatures so that the eect of Dung non-linearities can be neglected, while the dashed line is a Lorentzian t with the linewidth corresponding to that of the driven response.

An interesting point is that numerical integration of the spectra, i.e., the spatial extension of the Brownian motion ∆x 2 1 , matches well the expected value insofar as

∆x 2 1 = Q 2 1 S F /k 2
1 , as shown in Fig. 3.13. Thus, the equipartition hypothesis is not aected by the reported phenomenon. If it were the case, the features observed in driven response measurements in section 3.4 would anyway be incorrectly reproduced by theory: indeed, these features are dependent on the integrated position noise, which in the case of driven measurements was inferred through the equipartition hypothesis.

Consequences on fundamental limitations to frequency resolution 91

The fact that energy is conserved is another evidence that we genuinely witness a distortion due only to frequency noise, a purely dispersive phenomenon, mathematically described by the convolution. Measured shifts and FWHH as a function of position noise magnitude are plotted in Fig. 3.17. The agreement between data and theoretical lines is unprecedented: to our knowledge, only one study reported thermally non-linear spectra [START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF] in the case of levitated nanoparticles in an anharmonic trap, and no quantitative agreement could be found, while the discussion only considered the case where the partial susceptibilities were weighted by Boltzmann's distribution, i.e. the inhomogeneous broadening regime.

Interestingly, one sees that the lines obtained for the spectrum and the imaginary part of the response are not the same. Therefore, even though the equipartition hypothesis is still valid, the uctuation-dissipation relation (3.3) does not hold anymore. This is not surprising, however, since the present study actually goes beyond the simple linear response, which is a prerequisite for the validity of the relation. F r e q . s h i f t ( H z )

∆x 2 1 ( m 2 ) 0 , 0 5 , 0 x 1 0 -1 4 1 , 0 x 1 0 -1 3 1 , 5 x 1 0 -1 3 2 , 0 x 1 0 -1 3 2 , 5 x 1 0 -1 3 0 5 0 0 1 0 0 0 1 5 0 0 4 ( 2 β 1 , 1 ∆x 2 1 ) 2 / Γ 1 ∆x 2 1 ( m 2 ) B r o a d e n i n g ( H z ) I B M N Figure 3
.17: frequency shift (left) and linewidth broadening (right) of the fundamental mode position power spectrum as a function of its intensity. Solid lines are the theoretical parameters extracted numerically from Eq. (3.37), while dashed lines show the laws obtained in the motional narrowing regime.

Consequences on fundamental limitations to frequency resolution

Previous sections focused on the subtleties of the Dung-mediated frequency noise.

However, it also presents some important consequences for applications. Indeed this eect is ubiquitous, relying only on really intrinsic and genuine properties such as the dissipative coupling to a thermal bath and geometric nonlinearities. In addition, while we focused on the eect due to one mode for the sake of demonstration, any real vibrating structure will have a collection of modes which will contribute to the 92 Chapter 3. Interplay between non-linear dynamics and position uctuations observed frequency noise. In particular, while geometric nonlinearity-mediated mode coupling was long observed within a given family of modes, recent works have shown that geometric mode coupling was also possible between two exural modes of dierent families in nanomechanical structures [START_REF] Cadeddu | Time-Resolved Nonlinear Coupling between Orthogonal Flexural Modes of a Pristine GaAs Nanowire[END_REF]. Coupling to longitudinal or torsional modes is however overlooked here, as it is in practice very small and falls outside standard beam theory.

In essentially all situations where real nonlinearity-induced frequency noise is observable, the temperature is such that the phenomenon occurs in the motional narrowing limit. Therefore, the resonance will be a dressed Lorentzian, with a frequency shift ∝ T analogous to a "thermal" mechanical Lamb shift and a thermal decoherence-like broadening ∝ T 2 . Including another family of exural modes with motion variables {δy m }, the renormalized parameters for a driven mode n write:

ω n = ω 0 n + 4β n,n ∆x 2 n + 2 m =n β n,m ∆x 2 m + 2 m β n,m ∆y 2 m , (3.45
)

∆ω n = Γ n + 2(4β n,n ∆x 2 n ) 2 Γ n + 2 m =n (2β n,m ∆x 2 m ) 2 Γ m + 2 m (2β n,m ∆y 2 m ) 2 Γ m . (3.46)
These expressions are generic. In the case of doubly-clamped beams, one can use expressions for βs derived from the nonlinear description of beams (see Chapter 2, section 2.9). Within numerical sum factors, and after application of the equipartition hypothesis (3.2) to ∆x m and ∆y m , the dependencies on the in-built parameters appear as follows:

ω n -ω 0 n ω 0 n = A n × E beam A 2L 3 k B T 2k 2 n , (3.47)
∆ω n -Γ n Γ n = B n × E beam A 2L 3 2 k B T 2k 2 n 2 Q 2 n . (3.48)
Numerical prefactors A n and B n include sums over all the coupled modes, introducing the integrals dened with the mode function overlaps (as dened in Chapter 2):

A n = m I n,n + I m,m + 2I 2 n,m k m /k n + m I n,n + I m ,m + 2I 2 n,m k m /k n , (3.49)
B n = m Q m Q n I n,n + I m,m + 2I 2 n,m 2 (k m /k n ) 2 (ω 0 m /ω 0 n ) + m Q m Q n I n,n + I m ,m + 2I 2 n,m 2 (k m /k n ) 2 (ω 0 m /ω 0 n ) , (3.50)
where we deliberately made a distinct sum over the other family of eigenmodes, as their modal functions are slightly dierent in the case of a rectangular beam, where moments of area dier. Note that in both limiting cases (high and low stress), overlap 3.6. Consequences on fundamental limitations to frequency resolution 93 integrals, resonance frequencies and spring constants ratios can be simplied, yielding only dependencies in mode numbers in the sums [START_REF] Maillet | Nonlinear frequency transduction of nanomechanical Brownian motion[END_REF].

However, A n and B n do depend on the type of dispersion relation ω m (λ m ), and no analytic expression can join the high and low stress limits. Left alone is the dependence of the Q factor with the mechanical mode number for the thermal dephasing prefactor B n . Although there is currently no universal description of loss mechanisms in nanomechanical resonators, observations [START_REF] Quirin | Damping of Nanomechanical Resonators[END_REF][START_REF] Suhel | Dissipation mechanisms in thermomechanically driven silicon nitride nanostrings[END_REF] suggest that for SiN beams the Q factor depends on the bending energy as: Q m = (U bend + U tension )/∆U bend , with U bend + U tension the total potential energy dened by Eq. (2.40). The phenomenological model does not propose a microscopic analysis of the damping, but assumes that losses ∆U bend are proportional to the bending energy and introduces a complex Young's modulus to account for it which is left as the only free parameter in the model.

Previous studies of our SiN beams showed rather good agreement with this approach, and a mode dependent expression can be found in Ref. [START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF]. The approach is phenomenological, with the mode dependence t to the model with one free parameter ∆U bend . E beam I . Solid lines are the result of numerical sum calculations, while dashed lines show both limit cases (high and low stress), calculated for a single family of modes.

The results of the computation for the fundamental exure are summarized in Fig. 3.18, for a single family of modes. It essentially doubles for a square beam, taking into account the other family. Note that the frequency dressing term is simply a shift, but this term would lead to extra dephasing if temperature were not stable. Furthermore, the limitation imposed by thermal decoherence crucially depends on the aspect ratio L/e. In particular, we see that the eect quickly decreases with added stress, but for low stress structures with high aspect ratio, the fundamental limit imposed by this mechanism is not negligible, e.g. for carbon nanotubes. This could be an explanation for recent observations [START_REF] Barnard | Fluctuation broadening in carbon nanotube resonators[END_REF][START_REF] Eichler | Symmetry breaking in a mechanical resonator made from a carbon nanotube[END_REF], where the mechanism was qualitatively mentioned. The 94 Chapter 3. Interplay between non-linear dynamics and position uctuations authors from Ref. [START_REF] Barnard | Fluctuation broadening in carbon nanotube resonators[END_REF] reached the same conclusion, although with a very dierent, microscopic approach.

3.7 Anomalous uctuations for non-linear driving elds

Beyond linearization

The results exposed in the sections above are obtained and well described in a linearization approximation: in order to obtain the set of equations (3.13), we have neglected the term |A n | 2 δA m , which corresponds experimentally to a drive low enough so the response remains linear in the driving force. In other words, the interplay between the uctuations and the drive was unilateral, and modifying the driving force did not aect the uctuations. However, as shown in Fig. 3.19, in the "self-coupling" situation, the linear relation x n ∝ F n does not hold anymore when the NEMS is strongly driven: its peak value is lower than expected, as if there were an excess position noise transduced into frequency noise, which over-deteriorates the response. Moreover, this eect is asymmetric: while the X quadrature is over-blurred, the Y one is less aected (see Fig. 3.19 inset).

In fact, keeping the full non-linear term in the RWA introduces a new phenomenology that has been overlooked so far in this Chapter. It arises from the possibility for the driven motion to act back on its own uctuations, if the nonlinearity is strong enough, i.e. βx 2 max ∼ Γ . In that limit where the system is driven far from its equilibrium, it is not possible to obtain separate equations such as the system (3.13). While a complete resolution is tedious, we shall expose qualitative theoretical and experimental features that we believe can explain the excess frequency noise observed. The interested reader may nd a more detailed derivation of the uctuation problem for a strongly driven nonlinear resonator in Refs. [START_REF] Dykman | Theory of uctuational transitions between stable states of a nonlinear oscillator[END_REF][START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF]. On the experimental side, let us mention that at this stage the apparatus was not developed enough to enable systematic and quantitative measurements. Therefore, the results obtained in this section are only preliminary and used as a qualitative demonstration.

It is convenient to start from the Dung equation (2.67), rst without considering uctuations. Within the rotating wave approximation, and introducing reduced pa-

rameters Z(θ) = 3γ 16ωΓ A 2θ Γ , g 0 = √ 3γF 0 8(ωΓ ) 3/2 m , g th = √ 3γF th 4(ωΓ ) 3/2 m e -iωt
, Ω = 2(ω -ω 0 )/Γ with a dimensionless slow time θ = Γ t/2, one obtains:

Ż(θ) = -(iΩ + 1)Z(θ) + i|Z(θ)| 2 Z(θ) -ig 0 -ig th (θ), (3.51)
where the dot notation refers to the derivative with respect to θ, A is the slow, deterministic complex amplitude in the rotating frame and all indexes are dropped since we are only interested in one mode. First we consider a situation without noise, i.e. g th = 0. The steady state solution writes:

Z 0 = g 0 |Z 0 | 2 -Ω + i = |Z 0 (Ω)|e iφ 0 (Ω) . ( 3
δZ R δZ I = -1 Ω -Z2 0 3 Z2 0 -Ω -1 δZ R δZ I .
(3.53)

Chapter 3. Interplay between non-linear dynamics and position uctuations

This equation is the core of the issue: without the Dung interaction there would not be a dependence in the stationary solution and the eigenvalues of the system would give the usual relaxation time 2/Γ in both directions. Here we can expect the Dung nonlinearity to break this symmetry, as the eigenvalues yield:

v 1,2 = -1 ± 3 Z2 0 -Ω Ω -Z2 0 .
(3.54)

We see that as soon as the oscillation is large enough to distort a resonance line, there is a separation of the two quadratures decay rates: indeed, in real units, the decay rates The separation, at a constant force, is triggered as soon as Ω ≥ Ω res , as shown in Fig. 3.20. For forces close to the critical point, there is a clear separation of timescales on quadratures, and when the critical force is applied, the relaxation rate of the slow quadrature reaches zero, at the spinodal point, while the other is doubled compared to the bare relaxation rate. This behaviour remains beyond the critical force, when the driving force is brought close to both bifurcation points: for instance, at the upper branch bifurcation where the high amplitude metastable state and the unstable state merge, we have v 1 = 0 and v 2 = -2, that is, system-independent eigenvalues, which do not depend on the amplitude and frequency anymore. The quadratures in the eigenbasis (δZ 1 , δZ 2 ) behave according to: to study quantitatively escape dynamics from the metastable state [START_REF] Defoort | Scaling laws for the bifurcation escape rate in a nanomechanical resonator[END_REF]. This phenomenon should be visible directly if one measures the correlation functions on both quadratures with dierent phases, to extract these decay times. We do so instead of measuring the spectrum: indeed, since this time we drive the NEMS while we measure the uctuations, we obtain an integrated r.m.s. information for each drive frequency,

write ∆ω 1,2 = -Γ Re[v 1,
   δZ 1 = 0, δZ 2 = -2δZ 2 .
but not the full spectral information. Since the latter is supposed to depend on the driving eld, we can only have a partial information if we use our spectrum reconstruction technique. However, by measuring the correlation function at each drive setting with a bandwidth wide enough (i.e. a good resolution in time), we have access to the relaxation time at each drive and we can observe its evolution along the frequency sweep. The correlators computed from both quadratures are obtained from a noise sample digitized by the DAQ card (see section 3.5.1), and averaged over many samples.

To obtain a clean signal on the correlation functions and remove the backgroundmechanical cross component (3.43) that will alter the exponential decay, we have designed a simple opposition line based on a LC circuit with a tunable capacitance that is mounted in parallel with the NEMS circuit (see Fig. 3.21): the incoming voltage noise is splitted between the two lines, and the LC circuit acts as a dephaser with an adjustable gain, so that the voltage noise obtained at the end is subtracted to the background voltage noise obtained at the cryostat output. The Ohmic component of the cryostat output is, within a numerical factor, the same as the opposition line signal, plus a constant dephasing due to lineic capacitances. This holds true as long as the bandwidth of the RLC circuit is large compared to the mechanical bandwidth, which is the case in practice with our setup. With the right tuning of the capacitance (between 0 and 200 pF) and the appropriate cable length for the opposition line, it is possible in theory to catch up the dephasing, match the level of Ohmic noise and signicantly cancel the background noise. However, due to the noisy nature of the signal, the task is tedious. Nevertheless, we manage to suppress the background by a factor 5 in power, which is enough to signicantly suppress the cross component (3.43) as well.

Slowing down the decay of position noise correlations near bifurcation points

We have measured correlation functions C XX (t), C YY (t), C XY (t) for a force near the critical point where the hysteresis opens (see Chapter 2 paragraph 2.9.2). We see while sweeping the frequency Ω that the dynamics is slowed down as we run across the spinodal point x s where ω bif,↑ = ω bif,↓ , with a decay time for correlations that is multiplied by a factor up to 20, as shown in Fig. 3.22, left panel.

The increase in relaxation times is concentrated for driving frequencies near the spinodal point in the (ω, x) plane (which is not the resonance position, i.e. the maximum in amplitude: ω bif,↑,↓ = ω res ). As seen in Fig. The cable length is adjusted to optimize the noise cancellation. V 0 and δV in generate the driving force and noise respectively, while V is the detected voltage fed to the lock-in. Note that the measurements presented here have been realized at xed driving phase. Thus this (X, Y ) quadrature basis is not the one adapted to the present problem: indeed, the eigenbasis associated to the eigenvalues is obtained by rotation from the initial basis (the one in which we have the in-phase and quadrature response) with an angle that depends on the frequency and amplitude of the oscillation. Therefore, the reference phase must be adapted with an appropriate compensation scheme for each driving frequency to follow the directions where the asymmetry is maximum, which is not the case here. The signature of this is that the correlation function C XY is nonzero for some drive frequencies, as shown in Fig. 3.22 left panel inset. In any case, qualitatively, we observe a slowing down of the uctuations dynamics, which is a rst new feature compared to previous sections.

For applied forces larger than the critical force (F 0 > F c ), asymptotically close to ω bif,↑ , we also observe the predicted separation of timescales for the two quadratures, as shown in Fig. 3.23. Note that getting close to the bifurcation point is a delicate operation, as the frequency and the damping of the resonator are uctuating, thus aecting the hysteresis (see Chapter 6). Moreover, being close to the bifurcation point makes the resonator likely to jump between the two states, and this likelihood exponentially increases with the force noise injected since the latter acts as an eective temperature. Thus, we cannot get too close to the bifurcation frequency, which might explain why the measured relaxation times in the privileged direction are not tremendously high.

We see that one quadrature has correlations that decay very fast while the other has a correlation time multiplied by 10, i.e. more than one order of magnitude higher than the other, thus reducing the physics to the very convenient true 1D problem through timescale separation as explained theoretically above. Note that it is unclear whether the sharp decay on the X quadrature is due to the background or the NEMS, since the signal on X is very weak.

Asymmetric amplication of uctuations in the phase space

So far we have shown preliminary evidence that the dynamics of the NEMS is slowed down near peculiar points when driven in the nonlinear regime. But the magnitude of its uctuations is also aected: for each driving frequency we have monitored the trajectory of the Brownian motion in the quadrature space in the RWA. An example near the bifurcation frequency is shown in Fig. 3.24.

Two features are noteworthy: rst the Brownian motion is squeezed along one quadrature while amplied along the other (see the oval in the gure replacing the Chapter 3. Interplay between non-linear dynamics and position uctuations perfect circle of standard Brownian motion). It was observed in Ref. [START_REF] Almog | Noise Squeezing in a Nanomechanical Dung Resonator[END_REF] in a somewhat dierent way, by measuring r.m.s. noise values near a bifurcation point at dierent reference phases. It is indeed expected, as a general consequence in a system whose frequency is modulated at twice its resonance frequency. This is the case here, because of the term |A n | 2 δA n that was neglected in the previous sections: the uctuations are parametrically amplied, in a similar fashion to the gate modulation scheme detailed in Chapter 2, and used for noise reduction schemes [START_REF] Rugar | Mechanical parametric amplication and thermomechanical noise squeezing[END_REF]. Note that the squeezing achieved cannot be accurately evaluated, because of the residual Ohmic background, which is not aected by nonlinearities. The background eventually masks the squeezed component for strong squeezing. On the other hand, we observe a slight asymmetry in the amplied direction. This appears to be specic to the bistable regime. Indeed, one can show (see e.g. Ref. [START_REF] Dykman | Theory of uctuational transitions between stable states of a nonlinear oscillator[END_REF]) by keeping the second order in the uctuations in Eq. (3.51) that close to the bifurcation frequency, the 1D problem is equivalent to that of a ctive particle moving in a cubic potential. This is schematically pictured in the inset of Fig. 3.24: the uctuations are asymmetric because of the potential asymmetry. Such features are similar to what is observed for bistable optical [START_REF] Reynaud | Photon noise reduction by passive optical bistable systems[END_REF] and optomechanical systems [START_REF] Fabre | Quantum-noise reduction using a cavity with a movable mirror[END_REF].

Even though the measurements presented here are only qualitative and preliminary, they comfort us in the idea that the framework used to describe the nonlinear frequency transduction of Brownian motion in the previous sections needs to be adapted when the Brownian motion itself is aected by the driving eld: indeed the magnitude of uctuations as well as their correlation time is deeply changed, which in return aects the nonlinearity-mediated frequency uctuations of the NEMS.

Conclusion and prospects

Summary of the results

In this chapter, we have reported the experimental demonstration of non-linear transduction of Brownian motion into frequency noise over an unprecedented parameter range. We showed that this transduction followed a non-trivial evolution, scaling with the ratio between the magnitude of the created frequency noise (Σ) and its decay rate (τ -1 c ). This led us to observe two asymptotic regimes that are identied by analogy with NMR: inhomogeneous broadening (Στ c

1) where the noise is "static", innitely correlated with respect to the uctuating mode's dynamics, and motional narrowing on the spectrum, where the usual Lorentzian peak was undergoing a transition from a weak "dressing" (Lorentzian with renormalized parameters) to an asymmetric, inhomogeneous shape due to full averaging over partial susceptibilities. While thermal non-linearities for position uctuations spectra had been reported in Ref. [START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF], the description stayed mainly phenomenological, and limited to the inhomogeneous broadening case. We demonstrate here the phenomenon from low to high noise limit, with an excellent theoretical agreement.

To further develop on the transduction phenomenon with a more applied approach, we derived as quantitatively as possible the ultimate limit on frequency resolution imposed by this mechanism. In-built stress is found to dramatically improve the performance of the device. Yet, for low-stress devices, the strong dependence on the devices' aspect ratio can lead to non-negligible limitations, which are crucial as far as their potential for sensing is concerned.

Finally, we have shown the limitations of our model when the NEMS driven motion is in the nonlinear range: here the motion back-acts on its uctuations, creating an anomalous position and frequency noise. This anomaly is explained qualitatively with squeezing/amplication of position uctuations and slowing down of timescales, even though the obtained results are still preliminary and cannot be yet described quantitatively.

Prospects

Regarding the frequency noise transduction experiment itself in its current form, a rather quick investigation might be done by keeping a constant force noise while tuning the damping rate of the uctuating mode with the magnetic eld (loading eect, see 2.6.3), i.e. the frequency noise correlation time. This is the other way of tuning the motional narrowing parameter α, and it is in fact the approach adopted by authors in

Ref. [START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF] in the case of telegraph frequency noise.

Another in-depth investigation would be to access the position spectrum while a driving force is applied. It is predicted in Ref. [START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF] that unusual spectral features would appear: in particular, one expects a pedestal structure to form around the sine drive δ peak and an excess spectrum due to the non-linear interaction between the probe driving and the uctuating mode. Such features were reported in Ref. [START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF] for a carbon nanotube resonator, although through a dierent mechanism involving a gate electrode. In the current state of our experiment, however, it is rather dicult to probe this eect, as the method used for measuring spectra is not well adapted: close to the driving frequency, the drive amplitude will necessarily pollute the spectrum due to bandwidth opening. Therefore, a spectrum analyzer would be more adapted, but it does not avoid the issue of additional spectral contributions described in section 3.5.

Besides, while some further eorts are required to explain quantitatively the anomaly reported in the last section, we see also that a subtle phenomenology is at stake: on 104 Chapter 3. Interplay between non-linear dynamics and position uctuations some peculiar points of the mechanical hysteresis, position uctuations are diverging, both in magnitude and in correlation times. This could be further developed quantitatively in the framework of critical phenomena and phase transitions.

Other schemes could be foreseen: engineering more sophisticated structures with a more sensitive read-out technique, one could couple modes (not necessarily of the same family) with clever schemes that allow one to observe e.g. anomalous phase diusion [START_REF] Sun | Correlated anomalous phase diusion of coupled phononic modes in a sideband-driven resonator[END_REF]. Moreover, if a quantum-limited read-out of a mechanical resonator is made possible, the extension of the presented results to the quantum regime [START_REF] Katz | Signatures for a Classical to Quantum Transition of a Driven Nonlinear Nanomechanical Resonator[END_REF] is naturally appealing. An educated guess is that in the limit where all modes are in the quantum ground state, i.e.

ω m k B T , the mechanism survives only in the form of a mechanical Lamb shift [START_REF] Rentrop | Observation of the Phononic Lamb Shift with a Synthetic Vacuum[END_REF], where a mode eigenfrequency is dressed by the interaction with quantum vacuum uctuations of all the other modes. It could also be implemented for quantum non-demolition (QND) measurements of another mode Fock states, since the interaction Hamiltonian is bi-quadratic. To witness these eects, however, is far beyond the current reach of state-of-the-art quantum nanomechanics. between the two has been overlooked, since dephasing eects were assumed to be small enough to attribute spectral broadening only to dissipative processes. Recently, the rise of graphene and carbon nanotubes as mechanical resonators has led to reconsider non-dissipative processes, which also limit spectral resolution to a non negligible extent [START_REF] Moser | Ultrasensitive force detection with a nanotube mechanical resonator[END_REF][START_REF] Miao | Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators[END_REF][START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF], as shown in Chapter 3. In particular, recent experiments have investigated non-linear damping signatures in graphene and carbon nanotube mechanical resonators [START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF][START_REF] Güttinger | Energy-dependent path of dissipation in nanomechanical resonators[END_REF]. Another study showed a non-linear dephasing signature in such devices, where non-linear damping was ruled out thanks to time-resolved measurements [START_REF] Ben | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF]: a signicant discrepancy was observed between the spectral Q factor and the one obtained in time domain. Let us mention that such signatures were not observed for atomically thin vibrating MoS 2 membranes [100], which strongly suggests a carbon- based mechanism. Generally speaking, this dephasing is due to dispersive coupling to a noisy signal, which could be e. g. a gate electrode voltage due to uctuating oset charges.

Résumé en français

In the present chapter we detail a comprehensive approach which enables to thoroughly separate dissipative from non-dissipative processes in a nanomechanical measurement. We rst present a generic way to calculate analytically the broadening induced by frequency uctuations in the case where the latter are Gaussian, i.e. the reservoir quantity is linearly coupled to the motion. We present the corresponding measurements in frequency domain, which bear some similarities with the previous Chapter's results. We then extend our measurements to the case of quadratic coupling to the reservoir, which leads to asymmetric noise. In addition, we perform time domain measurements which enable to separate damping from dephasing.

Analogy with Nuclear Magnetic Resonance

The separation between damping and dephasing processes immediately recalls the context of Nuclear Magnetic Resonance (NMR). In NMR, dissipation arises from the resonant energy exchange between an out-of-equilibrium 1/2-spin and the environment to which it is coupled, bringing back the system to equilibrium over a characteristic time T 1 [START_REF] Slichter | Principles of Magnetic Resonance[END_REF]. On the Bloch sphere, it corresponds to the longitudinal relaxation time that characterizes the decay of the projection of the magnetic moment on the z axis. Meanwhile, decoherence is characterized by the transverse relaxation time T 2 , characterizing the decay of the projection of the magnetic moment on the transverse plane (x, y). Thus, it is sensitive to damping processes but also to uctuations along the z axis, i.e. uctuations of the energy splitting ω 01 (ω 01 is the so-called Larmor frequency) between the two spin states that can be induced by the environment. In the frame rotating at the Larmor frequency, these uctuations will be eective only if their typical timescale is comparable with or lower than the spin dynamics timescale, i.e.

T 1 . As these uctuations cannot involve energy exchange between the spin and its environment, they only introduce what we call an adiabatic dephasing, since there is an accumulated phase ω 01 (t)dt.

Using Bloch equations describing the evolution of the magnetization, one can show that for fully dissipative processes T 2 = 2T 1 . If adiabatic dephasing is added, the transverse relaxation time will actually be shorter, leading to T 2 < 2T 1 . One can introduce the pure dephasing rate

Γ φ = T -1 2 -T -1
1 /2 that accounts only for adiabatic processes.

Bloch-Redeld theory establishes that the longitudinal relaxation time can be linked to fast uctuations of an environment variable X coupled to the spin's transverse components. Typically, treating the coupling to the environment as a perturbation, one

establishes that T -1 1 ∝ S X (ω 01 ).
Γ φ = T -1 1 -T -1
2 . Note that here there is no factor 2 as opposed to NMR. This factor appears because of the peculiar structure of the spin space, which does not come into play here.

Generally speaking, noise sources leading to frequency uctuations, e.g. 1/f noise, do not have a sharp, DC contribution, or a smoothly varying contribution spread over Γ φ , resulting in non-exponential decay. Therefore, the T 2 time has an arbitrary denition, which depends on the noise characteristics. This will be addressed below.

Meanwhile, measuring T 1 is possible through decay measurements (not to be confused with free induction decay in NMR leading to T 2 measurements): ideally, the NEMS is resonantly excited at t < 0, and relaxes to its rest position as the excitation is turned o at t = 0. Obviously, the NEMS will return to equilibrium by oscillating at its natural frequency ω 0 but with a noisy accumulated phase term ∆φ = δΩ(t)dt, where δΩ(t) = ω(t) -ω 0 . However, measuring and averaging over many realizations the decay of the squared amplitude R 2 eliminates the phase dependence, leading to a true T 1 measurement. Since the ltering reduces by many orders of magnitude the eective noise level on the gate, we put a pre-amplier after the lter so as to maintain it at a level where dephasing eects are observable, with a power gain G = 2.5 × 10

σ 2 g = BW F BW A Gσ 2 A , (4.1)
where BW F = 1.2 Hz, BW A = 10 MHz are the integrated bandwidths of the lter and the noise source, respectively. The agreement between calculated values of σ g and measured values is rather good, with at most 10 % discrepancy between the two. Moreover, since F(ω) is measured, we can quantitatively infer the voltage noise spectrum S V (ω) at the level of the gate electrode:

S V (ω) = πGσ 2 A BW A F(ω). (4.2)
Therefore, the voltage noise spectrum is only the image of the lter's spectral response. This is useful to quantitatively obtain the frequency noise spectrum (see below).

The noise on the resonance frequency is directly obtained from the gate-NEMS capacitive coupling, through Eq. (2.61):

δΩ(t) = - 1 4mω 0 ∂ 2 C g (0) ∂x 2 V 2 g = αV 2 g . (4.3)
Here the gate voltage V g is not explicit: it can either be pure noise, i.e. V g ≡ δV g or a noise added to a voltage bias V 0 , i.e. V g = V 0 + δV g . This results in either quadratic Chapter 4. A model experimental approach to classical decoherence for mechanical systems -3 dB cuto frequencies are displayed (blue and red dotted lines) with asymptots, while the dark green solid line is an interpolation function used for calculations.

0 , 0 1 0 , 1 1
or linear coupling to a Gaussian noise source, yielding qualitatively and quantitatively dierent results.

Linear coupling to a Gaussian noise source 4.2.1 Generic formalism

Let us assume that a noise source δΩ(t) featuring a Gaussian distribution and having a correlation time τ c (which can be nite) is coupled linearly to the NEMS displacement x(t), creating a stochastic restoring force: Obviously this dephasing is a stochastic quantity. The experimental approach is to measure an average one. We can use the same procedure as in Chapter 3: solving the standard equation of motion in the rotating frame leads to a mechanical susceptibility which is, in time domain, the product of the usual, certain susceptibility χ 0 (t) by an accumulated phase term. The latter is averaged over the ensemble of realizations between 0 and t: χ a (0, t) = χ 0 (0, t) e i∆φ(t) .

F g (t) = -2mω 0 δΩ(t)x(t). ( 4
(4.6)

Instead of using the path integral approach described before, we have used another equivalent technique, which is more convenient in this context. We write the exponential term in Taylor series: To calculate the term between brackets we can use the Wick formula for k Gaussian, centered variables:

e i∆φ(t) = +∞ k=0 i k k! ∆φ k (t) .
δΩ(t 1 )...δΩ(t k ) = C k i =j δΩ(t i )δΩ(t j ) , (4.9)
where the sum is taken over the ensemble C k of possible simultaneous pairing congurations for an ensemble of k stochastic processes with the same Gaussian distribution.

Here we have reduced the calculation to an integration over products of two-point correlation functions. This is quite convenient: assuming causality as well as an exponential decay for correlations, a generic two-point correlation function has the form: Then, the calculation is straightforward, and the contribution of each integrated twopoint correlator to the k th moment writes, independently of any intermediate time:

δΩ(t i )δΩ(t j ) = σ 2 exp - |t i -t j | τ c . ( 4
t 0 dt i t 0 dt j δΩ(t i )δΩ(t j ) = 2σ 2 τ c t + τ c e -t/τc -1 .
(4.12)

From Wick formula we know that only congurations with an even number of variables, i.e. even moments of ∆φ, will be non-zero. That said, since all k variables are paired, the number of pairs in a given conguration of C k must be k/2. The only unknown left is the number N k of pairing congurations in C k . To discard double occurrences, one should consider how many times one conguration can occur as pairings are identical but picked up in a dierent order. This number is equal to the number of couples permutations in an ensemble of k/2 couples, which is simply (k/2)!. Thus, the number of possible combinations contributing to an even k-th moment is:

N k = k 2 k -2 2 ... 2 2 /(k/2)! = k! 2 k/2 k 2 ! . ( 4.13)
Since all contributions have the same value calculated in Eq. (4.12), an even k-th moment of the dephasing writes:

∆φ k (t) = N k 2σ 2 τ c k/2 t + τ c e -t/τc -1 k/2 , (4.14)
while odd k-th moments are zero. Doing the replacement k → 2k in the sum, the averaged term can thus be evaluated:

e i∆φ(t) = +∞ k=0 (-1) k (2k)! (2k)! k!2 k 2σ 2 τ c k t + τ c e -t/τc -1 k . (4.15)
Finally, the averaged susceptibility in time domain is:

χ a (0, t) = χ 0 (0, t) exp -(στ c ) 2 |t| τ c + e -|t|/τc -1 , (4.16)
where we have included the case t < 0 through the use of the absolute value. The asymptotic regimes underlined in Chapter 3 can be again highlighted here: if a typical frequency jump of magnitude σ occurs such that στ c 1, the sampling time σ -1 will 4.2. Linear coupling to a Gaussian noise source 115 be small compared to τ c , and then only short times t compared to τ c will matter in the additional decay to the susceptibility. If we expand the inner exponential term in Eq. (4.16) at second order, we are left with a Gaussian decay, which yields equivalently, when translated to the frequency domain, to a Gaussian probability density for frequency uctuations:

χ IB a (0, t) = χ 0 (0, t) exp -

σ 2 t 2 2 .
(4.17)

Note that the correlation time, approximated as innite, has disappeared from the problem. This is the so-called "inhomogeneous broadening limit", where the frequency uctuations imprint directly the shape of the resonance (see Chapter 3). Inversely, if a typical frequency jump of magnitude σ occurs such that στ c 1, the inner exponential term will be very small on short timescales, while the linear part of the exponent will be large. Thus, we can approximate the average susceptibility with:

χ MN a (0, t) = χ 0 (0, t) exp -σ 2 τ c t . (4.18)
In other words, the imaginary response is Lorentzian, with a linewidth that is renormalized by a factor (1 + σ 2 τ c ). This corresponds to the "motional narrowing limit", as discussed in Chapter 3 in another context.

Results

To address the linear coupling situation, an additioner of gain 1 is inserted after the preamplier, such that the DC voltage and the voltage noise level can be tuned independently. Including the DC term in a statically shifted resonance frequency, the frequency noise in Eq. (4.3) rewrites:

δΩ = αδV 2 g + 2αV 0 δV g . (4.19)
By choosing a large enough DC bias one can favour the linear coupling to the Gaussian noise source while keeping a low enough quadratic term. Thus in this section we assume δΩ(t) ≈ 2αV 0 δV g (t), meaning that the frequency noise is essentially Gaussian, with a zero average value (hence no resonance shift due to the uctuations) and a standard deviation σ = 2αV 0 σ g . For the 300 µm NEMS used in the experiment reported in this section, we measure a coupling constant α = 2π × 13.8 Hz/V

χ(ω) = +∞ -∞ χ(ω -δΩ)p(δΩ)dδΩ. (4.20)
The p term is, in the inhomogeneous broadening (IB) regime, the probability distribution that weights the partial responses X(ω -δΩ), and is Gaussian, centered:

p IB (δΩ) = 1 σ √ 2π exp - δΩ 2 2σ 2 .
(4.21)

In the low noise limit, the resolution is not good enough to clearly make the dierence between the IB regime and the full theory. Indeed the NEMS used in the linear coupling case is slightly below average in terms of Q factor: its has a 8. [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF] Hz loaded linewidth at 100 mT, while the lter's integrated bandwidth is 1.2 Hz, leading to an estimated correlation time 0.15 s for the frequency noise. Note that the lter does not exactly shape the noise as an exponentially correlated quantity (for instance because of the low-frequency ltering), which limits in any case our experiment for a perfect match with theory. In Fig. 4.3, we see that for typical noises used in this study the dierence between the IB regime and the full expression for the average accumulated phase term quickly vanishes. The theoretical broadening is also plotted in Fig. 4.6 for an averaging in the IB approximation (black) or with the full theory (orange). The discrepancy between the two falls well within our error bars, which are set at 10 % of the initial linewidth, i.e. around 0.8 Hz.

In the inhomogeneous broadening regime, the averaged quadrature X is a Voigt prole, i.e. the convolution of a Lorentzian by a Gaussian. Therefore, the averaged response will depend on the interplay between the noise magnitude σ and the damping rate Γ , as they characterize the Lorentzian and noise distribution width, respectively. If one is large compared to the other, the narrow peak can be approximated as a δ peak with small corrections, and the response is either Lorentzian for weak noise (σ Γ ), with weakly renormalized parameters, or Gaussian (σ Γ ), i.e. dominated by noise properties.

In the case of weak noise, a typical jump in frequency due to the gate voltage noise does not exceed a fraction of the linewidth and therefore, the distribution p IB is very narrow around the drive frequency, close to a δ peak in the vicinity of the Lorentzian. As a result the average response X at a frequency ω close to the resonance remains Lorentzian at rst order and can be expanded within the integral (4.20) in powers of δΩ, which makes a calculation of noise-renormalized parameters (resonance frequency, linewidth) possible. Note that the method is fairly generic and can be used for any type of distribution that is peaked enough at zero for low noise magnitudes. If we expand Eq. (4.20) at second order:

X(ω) ≈ X(ω) -M 1 dX dω + M 2 2 d 2 X dω 2 , (4.22)
where we have introduced the n th order moments of the noise distribution p IB : M n = δΩ n = δΩ n p IB (δΩ)dδΩ. To obtain the noise-dressed resonance frequency ω res = ω 0 + ε one has to solve:

d X(ω) dω ω=ωres = 0. (4.23)
Finding ω res is easy by combining Eq. (4.22) and Eq. (4.23). Assuming the noiseinduced shift ε is small (since the noise is weak), Eq. (4.22) can then be expanded at rst order in ε around ω 0 . As X is Lorentzian, its value and that of its derivatives at ω 0 are well known. At rst order, we nally obtain:

ω res = ω 0 + M 1 .
(4.24) Chapter 4. A model experimental approach to classical decoherence for mechanical systems This result is rather simple and not surprising: in the weak noise limit, the frequency is shifted by the mean value of the distribution. If the latter is Gaussian, centered, as in this section, M 1 = 0 and the response does not experience any frequency shift.

One can also evaluate the noise-induced broadening (decoherence) in the weak noise limit where this broadening is small. The starting point is the mere denition of a FWHM:

1 2 X (ω res ) = X ω res + Γ 2 + ∆ω r = X ω res - Γ 2 -∆ω l , (4.25)
where we have introduced the left (right) hand side broadening ∆ω l (∆ω r ), in case the total broadening is asymmetric. Using Eq. (4.22), combined with expansions at rst order in ∆ω l , ∆ω r , ε around ω 0 ± Γ/2 and the fact that X is Lorentzian with known rst and second derivatives at ω 0 , ω 0 ± Γ/2, one obtains in the end that ∆ω r = ∆ω l , and the inhomogeneous linewidth ∆ω = Γ + ∆ω l + ∆ω r in the weak noise limit is then:

∆ω = Γ + 6 (M 2 -M 2 1 ) Γ . (4.26)
For the centered Gaussian noise case that we address in this section, M 2 = σ 2 and thus:

∆ω = Γ + 6σ 2 Γ .
(4.27)

An example of weakly perturbed response is shown in Fig. 4.4 where a Lorentzian and a Gaussian t are compared. The Gaussian t is much less convincing, as it does not reproduce well the experimental shape. In the weak noise limit, the relevant linewidth is therefore the Lorentzian one.

Meanwhile, the strong noise limit σ Γ cannot provide generic expressions for the resonance lines, as they will be dominated by the uctuations distribution and thus will have properties specic to p IB : as the distribution's width is controlled by σ, the Lorentzian line is thus close to a δ peak in the convolution (4.21). For the case studied here, the line will essentially be Gaussian centered around the natural frequency:

X(ω) ≈ F 0 mω 0 π 8σ 2 e - (ω -ω 0 ) 2 2σ 2 .
(4.28)

Thus, the resonance frequency still does not shift, and the linewidth is governed by the noise statistical properties. This is highlighted by the absence of the mechanical 4.2. Linear coupling to a Gaussian noise source (4.29)

A typical example is shown in Fig. 4.5, again with Gaussian and Lorentzian ts. The latter is less convincing in capturing the experimental curve, although this appears less obviously than for the weak coupling case, due to higher instrumental noise. Therefore, the Gaussian width is favored, even though the discrepancy with the Lorentzian one is not tremendous.

The picture in this situation is analogous to Doppler broadening in an atomic ensemble, where atomic spectral emission or absorption lines often have a linewidth much bigger than that given by the excited state lifetime: as the gas atoms move randomly due to thermal agitation with a Gaussian distribution on velocities v of typical spreading v th ∼ k B T /m, they see in their frame a radiation frequency corrected by a factor 1 -v/c (c is the speed of light), which increases the frequency range allowed for radiation emission/absorption. If thermal agitation is important, the "thermal width" is much bigger than the intrinsic lifetime and thus the spectral line becomes Gaussian with the so-called Doppler width [START_REF] Siegman | Lasers[END_REF].

The experimental results in the linear coupling case are summarized in Fig. The red curve is a Gaussian t, while the blue curve is a Lorentzian one.

Clearly, the Gaussian width approximation is out of range in the weak renormalization limit. In the limit where responses are dominated by noise properties, this time the Lorentzian with falls slightly below the theoretical prediction. In each range, the appropriate data are rather well reproduced by theory, within an error bar set by the natural linewidth (10 % of Γ ). Again, the full theory and the inhomogeneous broadening approximation are, within this error bar, not distinguishable. The bar set at Γ/2 underlines the transition from Lorentzian-dominated response to noisedominated properties.

Quadratic coupling 4.3.1 Framework

In this section, we do not apply a DC voltage on the gate electrode and make sure the oset voltage remains small enough compared to typical noise voltages applied.

Therefore, the frequency noise (4.3) rewrites simply: δΩ ≈ αδV 2 g . However, using the same approach as in the linear case is more dicult, as we have to consider two types of pairings in the correlation function decomposition instead of one, which makes the counting more involved. Yet, we have shown above that for typical noise values used here, the inhomogeneous broadening approximation holds reasonably well. With the bandpass lter described above (see Fig. 4.2), we make sure that the injected noise is a low-frequency one. As the voltage noise delivered by the AWG is white, Gaussian, centered, the frequency uctuation spectrum can be written from the ltered gate voltage noise δV g using Wick's formula:

δΩ(t)δΩ(t + τ ) = α 2 σ 4 g + 2α 2 δV g (t)δV g (t + τ ) 2 . (4.30)
Therefore, in the frequency domain, the frequency noise spectrum can be obtained in a similar fashion to Eq. (3.11), with a true DC term and a convolution part:

S Ω (ω) = α 2 σ 4 g δ(ω) + α 2 π +∞ -∞ S V (ω )S V (ω -ω )dω . (4.31)
The situation is slightly dierent from that of Chapter 3. The initial noise is already low frequency, which means that the convolution will entirely result in a low frequency 122 Chapter 4. A model experimental approach to classical decoherence for mechanical systems spectrum for frequency noise. We have established an expression for the voltage noise spectrum and its integrated value through Eqs. (4.1) and (4.2). Therefore, the frequency uctuations spectrum can be written in terms of experimentally measurable quantities:

S Ω (ω) = α 2 G 2 BW F BW A 2 σ 4 A δ(ω) + π αGσ 2 A BW A 2 +∞ -∞ F(ω )F(ω -ω )dω S C (ω)
. essentially slow insofar as it is weighted much more in the frequency region ≤ Γ . Note that there is also a DC contribution (in fact the rst moment) leading to a frequency shift, which is overlooked here, as we are focusing on the distinction between damping and decoherence. One can obtain the probability distribution on δΩ from the Gaussian distribution on δV g :

p(δΩ) = Θ(δΩ) 1 √ 2πσδΩ 1/2 exp - δΩ √ 2σ , ( 4
M n = +∞ -∞ δΩ n p(δΩ)dδΩ = (2n)! n! σ 2 √ 2 n . (4.34)
Therefore, the standard deviation is indeed

M 2 -M 2 1 = σ, with M 1 = σ/ √ 2 being
in fact the DC shift. Note that physically, σ is also linked to the integrated frequency uctuations spectrum:

M 2 = 1 2π +∞ -∞
S Ω (ω)dω = 3σ 2 2 .

(4.35)

Protocol for time-domain measurements

The procedure to perform time-resolved measurements is similar to that of Ref. [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Dung model[END_REF],

where it was implemented for MEMS with much larger time constants. Here the only modication is instrumental, through the use of a DAQ card with a 300 kHz sampling rate that permits to address smaller timescales. The protocol is schematized in Fig. 4.8: for t ≤ 0 the driving force is swept adiabatically (i.e. around 1 Hz/s for this experiment) up to the resonance frequency, where it stays for 1 s. In parallel, a "trigger" signal is set with an amplitude higher than a given threshold. After 1 s, at t = 0, the "trigger" signal drops below the threshold value: at this precise moment the driving force is suddenly detuned far from resonance, with a detuning frequency ∆ of typically a few linewidths. Therefore, the NEMS relaxes from its steady-state amplitude towards its rest position while oscillating at its natural frequency ω res . As the lock-in amplier reference signal is synchronized in frequency with the excitation, it records the beating between the mechanical oscillation and the local oscillator on both quadratures.

The two quadratures in time domain in the local oscillator's reference frame thus write:

X(t) = e -t/T 1 [X 0 (ω res ) cos(∆t) -Y 0 (ω res ) sin(∆t)] , Y (t) = e -t/T 1 [X 0 (ω res ) sin(∆t) + Y 0 (ω res ) cos(∆t)] . (4.36)
Therefore, the exponentially decaying envelope R(t) and thus T 1 can be obtained through R 2 = X 2 + Y 2 , if the averaging is done on R 2 rather than on X and Y separately. Otherwise, in presence of noise, such that the beating frequency is now ∆ + δΩ the time constant observed through independent averaging of the quadratures is smaller, due to averaging on noisy sinusoidal terms. We formally call this average R 2 ≡ X 2 + Y 2 Note that this can be turned into another technique to measure the T 2 time, which we will call T 2 to make the distinction with the frequency-resolved technique.

Observing the beating is the reason why the frequency detuning procedure is chosen, rather than staying locked at resonance and just switching o the excitation. Note 124 Chapter 4. A model experimental approach to classical decoherence for mechanical systems that in order to acquire a well-sampled time-resolved signal, the lock-in time constant must be much smaller than the decay time. We set a measurement time constant τ m = 100 µs. Thus, the detuning must be small enough so as to remain within the lock-in bandwidth, while being greater than the linewidth so that the NEMS truly relaxes to its rest position.

This technique can be extended without diculty to the non-linear regime. However, in the bistable regime, we took care not to park too close to the bifurcation frequency, to avoid parasitic jumps from the high to the low amplitude state, or jumps due to the shift of the hysteresis under the inuence of intrinsic frequency noise (see Chapter 6). An acquisition card NI-DAQ operating up to 300 kHz per channel was used to record data points and synthesize a trigger signal with a jitter in the trigger time smaller than 2 ms, which is reasonable as far as our NEMS is concerned. Since opening the lock-in bandwidth comes with more instrumental noise, the procedure is repeated, and the signal is averaged over typically 1000 realizations. is swept at a low speed up to the resonance frequency ω res . 2: when the resonance frequency is reached, a "trigger" signal is set at a high amplitude. It is maintained during 1 second. 3: after 1 second, at t = 0, the trigger signal goes below a threshold value, which triggers the drive frequency to be suddenly detuned at a frequency ω res + ∆. The decay is recorded with the DAQ card, and the procedure is repeated. The last row of the time sequence chart shows the frequency of the excitation (black arrow), which is synchronized with the local oscillator, at each step of the sequence.

To test this procedure while checking whether intrinsic dephasing is measurable or not, we measured decay times T 1 and T 2 both in frequency and time domain up to large displacement amplitudes: at most the displacement amplitude was four times the resonator's thickness, thus strongly in the non-linear regime. A comparison between 4.3. Quadratic coupling 125 the two measurements for the same displacement amplitude is shown in Fig. 4.9.

Each point of the frequency domain response was averaged over 3 s to be as sensitive as possible to hypothetical intrinsic dephasing. Remarkably, the beating between the NEMS and the local oscillator is a non-linear oscillation. While the period is initially equal to the detuning, it gradually becomes larger, until it reaches a value equal to [∆ + (ω res -ω 0 )] -1 , where ω 0 is the natural resonance frequency in the linear regime (mind the minus sign on ∆). This feature can be explained by the geometric non-linearity: as the motion goes to zero, the nonlinear frequency pulling gradually weakens, and the pseudo-oscillation synchronizes with the decreasing resonance frequency. Using the Lindstedt-Poincaré method of solving non-linear dierential equations, a secular perturbation theory approach detailed in Ref. [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Dung model[END_REF], one can t the non-linear oscillation with a time-varying frequency ω res (t) -ω 0 = κx 2 (t = 0) (t), where κ is left rst as a free parameter and where: While being in itself a negative result, it gives condence that our structure is ideal to model through external means the impact of frequency noise on the measured modes. Note that in Ref. [START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF], measurements on a similar SiN beam also showed that

(t) = 1 -e -2t
T 1 = T 2 .

Results

From then on, the experiment is rather simple: as we control the level of noise on the gate electrode through the voltage source, we measure response lines in the frequency domain to obtain T 2 , and decays in time domain to obtain T 1 and T 2 . Response lines (X and Y) measured at dierent noise levels are shown in Fig. 4.11.

Basically two limits exist, which are this time fully captured by the convolution picture, in contrast to Chapter 3: the correlation time of the frequency noise is assumed to be very long and thus does not appear in the calculation. For a frequency noise magnitude small compared to the NEMS damping rate, the frequency noise distribution is close to a δ function, and therefore the Lorentzian response is not too perturbed: its parameters are simply renormalized by low-order moments of the frequency noise distribution (rst for the shift, second for broadening). As the frequency noise magnitude is increased, i.e. σ ∼ Γ the response gradually deteriorates from a Lorentzian lineshape to an asymmetric resonance which is a "mix" of the bare response and the frequency noise distribution. In the limit σ Γ the bare Lorentzian response gets closer to a δ function in the vicinity of the noise distribution, and therefore the altered response is highly asymmetric, close to the noise distribution. This is reected in the fact that the asymmetry takes the form of a long tail towards higher frequencies, as a consequence of having only positive uctuations values.

From an altered line we extract the FWHM ∆ω = 1/(πT 2 ). The decrease of the decoherence time T 2 down to less than half its maximum value while increasing the level of noise applied is shown below in Fig. 4.13, left panel. The data are found to be in good agreement with the theoretical curve extracted from numerical computation of the FWHM of the altered response, Eq. (4.20). We can therefore claim that we measure the loss of phase coherence due to low frequency uctuations that are adiabatically followed by the NEMS.

We now turn to time domain measurements, with the procedure detailed above. As the level of frequency noise is increased, the ringdown time T 1 of the averaged square amplitude R 2 remains at 57 ± 5 ms, up to the highest noise applied in this work.

Thus it is truly insensitive to frequency uctuations. The ringdown curve measured for σ = 2π × 33 Hz is shown in Fig. 4.12, in magenta. Chapter 4. A model experimental approach to classical decoherence for mechanical systems As pointed out before, the ringdown technique can also be used to measure decoherence by measuring and averaging separately the two quadratures decaying and extracting a T 2 (t of the so-called R 2 . For each T 1 measured we also extracted T 2 . An example is shown in comparison with the T 1 measurement in Fig. 4.12. The measured values are reported in Fig. 4.13, where the discrepancy between T 1 and T 2 appears unambiguously even for modest levels of noise applied. This demonstrates that we indeed measure energy relaxation through the ringdown, while measuring the loss of phase coherence (which obviously includes the energy loss) with the usual frequency sweep.

0 5 0 1 0 0 1 5 0 0 1 < R > 2 < R 2 > R 2 / R 2 0 T 1 / 2 =
From then on, the pure dephasing rate

Γ φ = T -1 2 -T -1
1 accounting only for dispersive processes leading to decoherence can be deduced, whether it is obtained from T 2 or T 2 . The extracted values are shown in Fig. 4.13 right panel, with Γ φ corresponding to the pure dephasing rate calculated from T 2 . Thus, we nd a quantitative link between the analogous pure dephasing rate and the integrated power spectral density of frequency noise, as in the case of quantum bit circuits [START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF].

The agreement with theory, with discrepancies lower than ±15 %, is rather good for both methods (T 2 and T 2 ts), and validates our approach. Yet one issue remains: a characteristic time is not an absolute quantity insofar as it is dened arbitrarily, The experimental shape factors agree well with the theoretical calculation based on Eq. (4.20), and do not deviate more than 20 % from the initial value at the strongest level of applied noise. This ultimately proves the accuracy of our Lorentzian approximation. Note that as the lines in frequency domain are simply Fourier transforms of the quadrature ringdowns, the exponential approximation also holds well.

Needless to say, the shape itself under uctuations contains extremely valuable information, which is another reason why we created a non-Gaussian, asymmetric noise taking only positive values. Deconvoluting a line is a non-trivial task but remains feasible, which is of fundamental importance when intrinsic, unknown mechanisms leading to spectral broadening are addressed. A recent proposal addresses the issue in the context of nanomechanics [START_REF] Maizelis | Detecting and characterizing frequency uctuations of vibrational modes[END_REF], suggesting that computing the moments of the response could lead to the identication of these decoherence mechanisms.

Conclusion and prospects

In this Chapter, time and frequency domain responses of a mechanical mode have been measured and its relaxation T 1 and decoherence T 2 times compared so as to separate, within a framework analogous to NMR, the contribution of energy relaxation and pure dephasing among processes limiting the NEMS performance. We have established that with no external noise biases, within our resolution, our SiN nanomechanical devices are not subject to non-linear damping or dephasing over a very large range, as opposed to observations made with carbon based systems [START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF][START_REF] Ben | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF]. This issue shall be addressed further in Chapter 6. Having ruled out internal mechanisms, we have engineered slow uctuations of the resonance frequency, by capacitively coupling gate voltage uctuations controlled with an external source to the nanomechanical motion.

The coupling could be made linear, with a DC gate voltage bias, or quadratic in the voltage uctuations. With engineered slow uctuations we unravel signatures of pure dephasing. Using the analogy with studies done on quantum bit circuits, we link quantitatively the pure dephasing rate to the integrated power spectral density. The shape of noise-altered resonance lines is found to intimately depend on the nature of the coupling, i.e. on the frequency noise statistics, a feature already observed in the experiment described in Chapter 3. In a "reverse engineering" approach, this can be useful: for devices suering from actual intrinsic dephasing, the shape can provide information on the noise sources by computing its moments, which provide information on the uctuation statistics. processes that are physically taking place either in the surroundings or within the constitutive materials of the nano-electro-mechanical system. It is part of a widespread eort to acquire an exhaustive knowledge of fundamental mechanisms truly acting at mesoscopic scale and limiting the mechanical object's performance by reducing its relaxation or coherence time. Indeed these mechanisms are still far from being completely understood. Still, every friction or pure dephasing mechanism is set over a typical lengthscale, which in some cases becomes comparable with at least one of the oscillator's dimensions. Therefore, while this endeavour may suggest practical reasons, the NEMS can nevertheless be seen from the basic researcher's point of view as a tremendous tool to investigate these often complex but rich mesoscale mechanisms [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF][START_REF] Wilson-Rae | Intrinsic dissipation in nanomechanical resonators due to phonon tunneling[END_REF][START_REF] Lifshitz | Thermoelastic damping in micro-and nanomechanical systems[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF][START_REF] Lissandrello | Crossover from Hydrodynamics to the Kinetic Regime in Conned Nanoows[END_REF]. In the end, their knowledge is a prerequisite to engineer structures that are as free as possible from these mechanisms, so that other, elusive but extremely fundamental decoherence sources could be probed [START_REF] Armour | Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box[END_REF]. Note that the philosophy is rather dierent from that of the previous part: while we still attempt to stay quantitative, some strong assumptions must be occasionally made to explain our ndings, as we do not pretend to bring denite answers to questions that remain open to a large extent. In particular, some parameters strongly depend on materials and are, by essence, not completely under control. They often rely on measurements prior to this work.

Résumé en français

The rst Chapter of this part deals with an ubiquitous source of damping: even though the NEMS is put in a vacuum cell, there is always some residual amount of gas. While it can be neglected in the rst place for most of our experiments, one can wonder what happens if a portion of gas is deliberately introduced in the cell:

the gas acts as a uid interacting with the mechanical object. Such a friction process should naturally be addressed in the framework of uid dynamics, in the same fashion as vibrating wires used as viscosimeters for quantum uid studies [START_REF] Guénault | The mechanical behavior of a vibrating wire in superuid 3 He-B in the ballistic limit[END_REF]. Besides, NEMS and MEMS are now at the core of micro-and nano-uidics because of their ability to probe small lengthscales of a uid, be it for fundamental purposes tackling the renements of kinetic theory [START_REF] Kara | Generalized Knudsen Number for Unsteady Fluid Flow[END_REF] or applied ones such as biosensing [START_REF] Tamayo | Biosensors based on nanomechanical systems[END_REF].

Here, as a preamble to study more complex classical or quantum matter, we propose to investigate renements of the interaction between the NEMS probe and Helium-4 gas at cryogenic temperatures. It is a simple experiment insofar as Helium is a very well known gas, almost ideal, with tabulated properties [START_REF] Arp | Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to 2000 MPa[END_REF]. Thus, very quantitative predictions can be made to support our measurements. The relevant length to address these interactions, and especially friction processes, is the gas mean free path λ, which writes:

λ = 1 √ 2nσ He = k B T √ 2σ He P , (5.1)
where n is the gas density, and σ He the cross-section of collisions for Helium-4 atoms.

Experimental apparatus 137

The latter is computed in Ref. [START_REF] Arp | Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to 2000 MPa[END_REF], including a small temperature. Depending on the ratio between the mean free path and some geometric dimension of the device (the so-called Knudsen number), the physics at stake is indeed very dierent, and the geometry of our doubly-clamped nanobeams, suspended over a well-resolved length, make them good candidates to probe several lengthscales. While this gas damping experiment was initially dedicated to characterize the interaction between the uid and the NEMS according to well established properties, it yielded rather unexpected results at the limit of very low pressures: a reduction of damping has been observed below the standard prediction (linear in pressure). The results thus cover a wide pressure range, from the high pressure limit where the ow of gas around the NEMS is laminar and behaves well according to Navier-Stokes equations, down to very low pressures where the ow is molecular and exhibits unusual features. By "ow" we mean that even though there is no externally created ow, there is one in the NEMS reference frame since the latter is oscillating in the uid.

Experimental apparatus

Gas handling system

The experiment at stake relies on the control of pressure into a cell. Therefore, two runs have been made on two dierent helium bath cryostats in order to demonstrate the reproducibility of the results and rule out any extra geometrical features. Some characteristics are gathered in Table 5.1.

Cryostat n

• 1 Cryostat n • 2
Sample holder copper rod (massive) copper plate (thin)

Pumping line diameter

mm 5 mm

Cell diameter

cm 4 cm

Capillary tube length

1 m 1 m
Table 5.1: relevant geometrical characteristics varying from one cryostat to the other.

The experimental cell, initially under cryogenic vacuum (P < 10 -6 mbar), is connected to a capillary tube going up to an external gas handling system set at room temperature. A valve controls the connection between the capillary tube and the gas handling system. The latter is a pipe into which portions of gas are added in a controlled way by an external source made of pure (> 99.99 %) 4 He: it comes from a needle valve connected to a helium dewar evaporation. In addition, impurities nding their way towards the cell should be adsorbed on the cell's walls at cryogenic temperatures. Since pumping down to truly 10 -6 mbar at 4.2 K once helium gas has been added is time consuming, we choose to start from the low pressure range and go to high pressures by gradually increasing gas portions. The pressure is monitored from 10 -3 Torr to 100 Torr at room temperature by a Baratron gauge which truly measures pressure. It oers a resolution at the mTorr scale, and displays a zero value usually around -5 mTorr and drifting also at the mTorr scale over approximately an hour.

Since these drifts are slow and measured, they can easily be captured and corrected for the data analysis. In addition the experiments were carried out over several thermal cycles to make sure they were reproducible.

We also made sure that thermomolecular corrections all along the pumping line were negligible, so that the pressure read with the gauge is truly that of the cell. We checked that the results were the same with the cryostat full of Helium or almost empty, so that its upper part is above 4.2 K. Besides, estimates of the eect depend on tube dimensions and lead to negligible corrections [START_REF] Chernyak | Application of the variation method to the problem of thermomolecular pressure dierence in a cylindrical channel[END_REF], only becoming relevant below 10 -3 mbar. Moreover, as we will see later, measurements from one cryostat to the other yield the same results despite dierent tube diameters and cryostat sizes.

From then on the procedure is simple: when we want to increase the pressure in the cell, we rst close the valve and introduce a portion in the pipe. We then open the valve, thus circulating gas into the cell, and let the pressure reach equilibrium. Once it is reached, the valve is closed, and the pipe is pumped again to vacuum.

Samples used in this experiment

The experiments have been run on three dierent samples, each one being rather long so as to acquire a lot of signal, while reducing the amount of geometrical features: as the length is much bigger than the cross-dimensions, we can approximate our wires as innitely long strings (and innitely thin in the low pressure limit). Their characteristics are gathered in Table 5.2, and pictures are shown in Fig. 5.1. Note that the 150 µm long device has an intrinsic linewidth Γ int which is rather large, resulting in bigger error bars for very low pressure measurements than with the other devices. This is likely due to the fabrication process. Most importantly, the three devices have been designed such that their gaps g between the wire and the bottom spread over one order of magnitude, while all gaps are way larger than the devices cross-dimensions.

Usual XeF 2 does not permit an etching depth beyond 20 µm: the undercut would be too big for the structure to stay suspended (see Fig. As stated through Eq. (3.1), any gas damping contribution will be added to the intrinsic damping rate Γ int . Therefore, to adress gas damping, we simply subtract this vacuum value, using low elds so as to minimize the loading contribution and gain in resolution: the lower the intrinsic value, the better resolution is obtained since The shadowed region is a good marker of the undercut digging length.

for our lowest pressures a relative error is already existing on the vacuum linewidth.

However, all measured vacuum linewidths were found to be reproducible within 5 % of the indicated value at worst, which enable rather precise measurements. Note that all measurements were done with driving forces low enough in order to remain in the linear regime.

A typical example is shown in Fig. 5.2, with the pressure changed by one order of magnitude between the two measured resonance lines and with an excitation force adapted to obtain the same amplitude of motion. The dierence is striking, with lines much broader than in vacuum, which gives us condence concerning our resolution. 5.3 NEMS in a viscous uid: the Navier-Stokes regime

Hydrodynamic description of the NEMS damping by a uid

At a cell pressure P ∼ 10 -100 Torr, the gas mean free path λ is much smaller than the NEMS cross-dimensions w, e. Thus the NEMS can be described as an oscillating body moving in a continuous, viscous uid having a shear (or dynamic) viscosity η.

The transition regime where this approach progressively deteriorates, with so-called slippage corrections in the rst place, is addressed in Section 5.3.3. The NEMS velocity ω 0 x(ω 0 ) is at most ∼ 1 m/s, i.e. small compared to the speed of sound in helium (100 m/s), and thus second viscosity (linked to volume change) can be neglected. To derive the friction force between a uid particle of mass density ρ g and the solid body one needs to nd the uid velocity eld v g around the NEMS, which is described by the Navier-Stokes equation:

ρ g ∂v g ∂t + (v g • ∇)v g = -∇P + η∇ 2 v g . (5.2)
Such a problem for a square cross-section is addressed in Ref. [START_REF] Elie | Frequency response of cantilever beams immersed in viscous uids with applications to the atomic force microscope[END_REF]. The damping

Γ v (ω) = ρ g A c ωk (ω)Ω(ω) Ψ 2 (z)dz m .
(5.6)

Note that there is no geometric modal dependence, as m ∝ Ψ 2 (z)dz. Thus, the only mode dependence is on ωk (ω)Ω(ω).

Acoustic wave radiation

In addition to friction processes at the NEMS-uid boundary, another source of damping could come from the fact that while oscillating, the beam modulates the pressure eld around it, which is none other than sound radiation emission, the same way a guitar string produces sound in the air. Here, sound radiation is expected to be irrelevant due to the NEMS small dimensions. We can nonetheless give an estimate, assuming the NEMS is a cylinder in rst approximation. By solving the sound equation [START_REF] Morse | Theoretical Acoustics[END_REF] which yields the pressure eld P and calculating then the sound intensity I = Re [P v g ] at the NEMS/uid boundary, one obtains the radiated power, and thus the contribution to the damping [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF]:

Γ s (ω) ∼ ω 3 ρ g w 4 l mc 2 g , (5.7)
where c g ∼ 100 m/s is the speed of sound in helium. This estimate is plotted in Fig. 5.5, and appears to be at best four orders of magnitude lower than uid friction for our experiment, and thus completely irrelevant in the pressure range investigated.

Therefore it shall be disregarded in what follows.

Slippage correction

Now the damping is explicit, there should be no free parameter. This approach was indeed valid in the case of a MEMS device [START_REF] Collin | Silicon Vibrating Wires at Low Temperatures[END_REF]. However, it does not exactly match the experimentally measured damping, as shown in Fig. 5.4. In fact, the reduction of dimensions makes the nanomechanical device much more sensitive to slippage corrections. Indeed, for a bulk uid (with respect to the NEMS) the tangential velocity decreases linearly as it gets closer to the boundary between the object and the uid, and becomes zero at the boundary in good approximation because the uid is clamped on surface irregularities: this is the so-called no-slip condition [START_REF] Landau | Fluid Mechanics[END_REF]. Yet, by decreasing the cell pressure, the mean free path λ of the gas is increased and eventually becomes comparable with the device cross-dimensions e, w. In that case the approximation breaks, and tangential velocity does not necessarily go to zero at the boundary between the solid object and the uid: this is the denition of slippage. At the lowest order (i.e. before adopting a fully statistical approach, see below), a so-called slip 5.3. NEMS in a viscous uid: the Navier-Stokes regime 143 length l slip takes into account the eect [START_REF] Lilley | Velocity prole in the Knudsen layer according to the Boltzmann equation[END_REF]. This is the length over which the tangential velocity eld would go to zero if it were extrapolated beyond the NEMS-uid boundary. Following Refs. [START_REF] Carless | Vibrating wire measurements in liquid 3 He II. The superuid B phase[END_REF][START_REF] Bowley | Slip Corrections for Vibrating Wire Resonators[END_REF], we introduce a renormalization of the damping coecient by the slip length:

Λ(ω) -1 → 1 (Λ(ω) -1) -1 - 1 2u 2 l slip l slip + w/2 . (5.8)
A quantitative denition of the slip length involves the scattering processes at the NEMS-uid interface. Indeed, this length becomes relevant when λ is a fraction of e, w which means that the NEMS is able to resolve individual gas particles collisions on its surface. It is clear that the nature of these collisions will be important in the denition of the slip length. Even though microscopic mechanisms are very dicult to model, one can sort them into two classes:

• collisions where the NEMS surface essentially acts as a mirror for the incoming particle, which are called specular reections. These are called elastic, as there is no net energy transfer.

• collisions with a surface for which roughness is important due to defects, and permits many (specular) collisions before the particle bounces o the surface, with a momentum direction uncorrelated to its initial one. These collisions are said to be diusive. The relevance of this eect can be appreciated by comparing the size of typical defects, about 20 -30 nm for our samples (see Fig. 5.3), with the De Broglie wavelength for 4 He atoms λ DB = 2π / 3m g k B T [START_REF] Ziman | Electrons and Phonons: The Theory of Transport Phenomena in Solids[END_REF], which is about 0.6 nm at 4 K. No net energy is transferred, as the momentum change only aects the direction. As such, these collisions are elastic.

• collisions such that the atom "sticks" to the surface, i.e. is adsorbed. To ensure global energy conservation, this implies that another atom somewhere on the surface must be desorbed [START_REF] Sinvani | Sticking Probability of 4 He on Solid Surfaces at Low Temperature[END_REF][START_REF] Taborek | Helium on graphite: Low-temperature desorption kinetics and sticking coecient[END_REF]. By stating so, we make the assumption that the NEMS surface and the gas are at the same temperature, since the atom that is emitted is initially in equilibrium with and thus thermalized to the surface. Measurements of the sticking coecient at 4.2 K show that it can be rather high [START_REF] Sinvani | Sticking Probability of 4 He on Solid Surfaces at Low Temperature[END_REF][START_REF] Taborek | Helium on graphite: Low-temperature desorption kinetics and sticking coecient[END_REF]. Therefore it can signicantly contribute to diusive scattering.

Here we merely separate diusive from specular scattering processes with a socalled specular fraction s that is 0 for pure diusive scattering and 1 for pure specular reection. The slip length then writes [START_REF] Jensen | Boundary eects in uid ow. Application to quantum liquids[END_REF]: We see that for pure specular reection, the slip length diverges. This is expected: by denition, in the case of pure specular reection, the NEMS merely acts as a mirror and therefore does not impact the particles equilibrium properties: there is no reason why the tangential velocity eld would decrease to zero in the vicinity of the surface.

l slip ≈ 1.15 1 + s 1 -s λ.
In contrast, diusive reection, by denition, tends to slow down particles. To achieve a perfectly specular surface is challenging, and yet spectacular slippage signatures have been reported for carbon [START_REF] Holt | Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes[END_REF] and boron nitride [START_REF] Siria | Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube[END_REF] nanotubes, which likely owe to their cleanliness.

The data in Fig. 5.4 are rather well captured from 100 Torr down to about 10 Torr, where essentially λ ∼ w, with a t specular fraction s = 0.5 consistent with the sticking eect explained earlier. Unfortunately the third and fth modes damping rates could not be measured up to high enough pressures to capture the viscous regime and conrm the frequency dependence, essentially because the signal was too weak to obtain reliable data: this comes from the combination of a poor Q factor and small transmission coecients in the setup wiring. For pressures below 10 Torr the mean free path eventually becomes bigger than the NEMS cross dimensions and neither the continuous uid description nor the rst order correction (slippage) hold anymore.

Thus, for P < 10 Torr we must nd another approach, so-called ballistic, to describe the gas damping.

In fact, the transition regime between a laminar description and a ballistic one, which sets in over about an order of magnitude in our experiment, is a highly non- trivial problem, and a universal description for oscillating ows is still missing [START_REF] Bullard | Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical Resonators[END_REF].

Beyond the slippage correction, we will simply use an interpolation that makes the link between the viscous regime prediction and the ballistic one in the transition range. In the transition regime, in principle, the friction is aected by the nite response time of the uid, which leads to the denition of a generalized Knudsen number involving not only λ/w but also the penetration depth δ [START_REF] Kara | Generalized Knudsen Number for Unsteady Fluid Flow[END_REF]. Finite time eects are relevant when λ ∼ δ, which happens here around 0.5 Torr for the 100 µm device, far below the purely geometrical transition range. In addition, experimentally, such an eect would depend on the oscillation frequency, and we measured the same gas damping with the three modes, that are spaced at worst by one order of magnitude in frequency. Therefore we can safely neglect nite time eects and concentrate on nite size eects.

5.4 NEMS in a ballistic gas: the molecular regime

Statistical description of the damping

In the low pressure limit, the continuum description for the surrounding uid breaks down. It is understood here that the low pressure regime corresponds to a limit when the mean free path λ of the surrounding medium atoms is large in comparison with 146 Chapter 5. Energy relaxation in a surrounding uid the device transverse dimensions. Then, from the point of view of the device, the gas is not a homogeneous ensemble but a collection of discrete, ballistic particles carrying momentum and colliding at random times on its surface as often as between themselves. Hence one must adopt a statistical approach: the measured properties (resonance frequency, damping rate) are averaged over a gas particles distribution of velocities (or possibly, spatial coordinates), and they must give a reliable account of this interaction, which does not necessarily hold true if the gas is too rareed, in the very low pressure limit. A simple estimate can illustrate the situation with the most conservative experimentally reachable parameters: at T = 15 K, P = 10 mTorr, the number N of an ideal gas particles in a square volume V = (200 nm) 3 of lateral dimensions typical of the NEMS cross-dimensions and small compared to its length is:

N = P V k B T ∼ 50 particles, (5.10)
with a relative uctuation on the number of particles:

δN N = 1 √ N ∼ 14 %.
(5.11)

While the typical number of particles shows how far we are from the continuum description, its relative uctuations are small enough so we can use a statistical description based on gas kinetic theory. The starting point is the Maxwellian distribution on the bulk gas velocities at equilibrium:

f M (v x , v y , v z ) = m g 2πk B T 3/2 × exp - m g [(v x -v x,0 ) 2 + (v y -v y,0 ) 2 + (v z -v z,0 ) 2 ] 2k B T , (5.12)
where m g is the 4 He atomic mass: m g = 6.65 × 10 -27 kg. Here, we have considered the case of an existing ow in the gas by keeping an average velocity eld v g,0 = (v x,0 , v y,0 , v z,0 ). Indeed, one should take into account the fact that the NEMS is oscillating while immersed in the gas. Therefore, even though the gas does not experience any externally imposed macroscopic ow, there exists one in the NEMS frame, which moves at a velocity ∼ ω 0 x(ω 0 ). In practice, one can expect v z,0 to be zero and the prole (v x,0 , v y,0 ) around the beam to have the same symmetries as in the laminar case.

5.4. NEMS in a ballistic gas: the molecular regime 147

Computing the damping force

It is clear from then on that the mechanism responsible for mechanical damping through the gas is the transfer of momentum from the NEMS to the ballistic particles colliding on its surface. The average force per unit area dydz exerted by a particle colliding with the device oscillating around the n th mode resonance frequency is obtained by computing the total average momentum transfer:

∂ 2 F g (z, t) ∂y∂z = -ρ g v g Ψ n (z) ẋn (t), (5.13)
with v g the incoming gas particle velocity and the bar denoting the gas average absolute velocity. Besides, when λ ∼ w, particles start to collide with the NEMS as often as between themselves. This can be encompassed in the above description by simply stating a small correction to the usual Maxwell-Boltzmann distribution. We introduce

a modied distribution f (v x , v y , v z ) = [1 + ε(v x , v y , v z )] f M (v)
, where ε is a correction function [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF]:

ε(v x , v y , v z ) = m g 2k B T i=x,y,z a i (v i -v i,0 ) + m g 2k B T i,j a ij (v i -v i,0 )(v j -v j,0 ) + m g 2k B T 3/2 i,j,k a ijk (v i -v i,0 )(v j -v j,0 )(v k -v k,0 ) + .... (5.14)
Coecients a i... should be nonzero only in the vicinity of the NEMS, and should vanish to zero within the mean free path towards the bulk gas. Rules on the distribution allow to link some of these coecients between themselves [START_REF] Neil | Molecular ow of gases[END_REF]. First, the distribution should be normalized to 1, which leads to: i=x,y,z a ii = 0.

(5.15)

In addition, the average velocity eld v i f (v)d 3 v in each direction i = x, y and z must be equal to v x,0 , v y,0 and v z,0 , respectively, which leads to:

2a i + 3a iii + j =i a ijj = 0. (5.16)
From that, we can compute the total damping force acting on the NEMS, by integrating the total momentum transfer. Contributions in v y,0 , v z,0 can be computed but their integration over dydz will lead to a zero net contribution. Therefore, only coecients a x , a xx , a xxx are needed for the calculation. If diusive reection is indeed signicant, the equilibrium conditions of the NEMS will have an inuence on the surrounding gas due to the re-emitted particles, allegedly small since the NEMS has very small crossdimensions. Gathering the two contributions, the damping force per unit length writes: ∂F g (z, t) ∂z = -2ϑρ g v g wΨ n (z) ẋn (t),

(5.17)

with ϑ = s + (1 -s) 1 + π 4
T N /T /2. Therefore, the presence of disorder and diffusive reection at the level of the NEMS adds only a small renormalization of the damping force compared to an ideal case of pure specular reection. Note that the a i... coecients have disappeared from the expression and contribute only as corrections which involve the beam velocity. These terms are negligible, a fact that is experimentally justied: as opposed to the Navier-Stokes limit, the measured damping rate depends here neither on the mode, nor on the NEMS velocity (or current injected)

for the driving forces used, as seen in Fig. 5.5. A detailed justication can be found in Ref. [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF]. The NEMS temperature T N could be slightly dierent from the gas one due to the injected drive current, but we used drive forces small enough to stay in the linear regime, and according to our thermal model (see Chapter 2 section 2.7)

these should not heat up the device to more than 5 K when T = 4.2 K in the most conservative case, that is, when the NEMS is highly damped, and large currents are needed, in the Navier-Stokes limit. In the molecular regime the Q factor remains high and thus current-induced heating is negligible. Therefore the parameter ϑ is not very sensitive to the specularity fraction (since [1 + π/4]/2 ≈ 0.89) and will remain between 0.89 (purely diusive) and 1 (purely specular). High pressure ts, where the specular fraction plays a role due to slippage, suggest that a specularity fraction s = 0.5 is adequate [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF]. Following the same procedure as in the viscous regime, we write the dissipated power and obtain the damping rate contribution due to the rareed gas:

Γ mol = 2ϑρ g v g w Ψ 2 (z)dz m .
(5.18)

Since v g ∝ √ T is independent of P and ρ g ∝ P/T the measured broadening simply scales as P/ √ T . In addition, for a string, Ψ 2 (z)dz is 1/2 and m = M/2 for all modes. Thus, the damping in the ballistic regime does not depend on the mode for a string structure.

An example is shown in Fig. 5.5, where the gas contribution to the damping is reported for the three rst symmetric modes. The t is from Eq. (5.17) with ϑ = 0.95 in the low pressure limit (i.e. a specularity fraction s = 0.5). The rounded part between 3 and 10 Torr captures phenomenologically the transition to the Navier-Stokes regime for low velocities [START_REF] Yamamoto | Flow of a rareed gas past a circular cylinder[END_REF]. However, we see that almost as soon as we catch up with the truly molecular regime at low pressures (below 1 Torr), a reduction of damping is observed compared with the simple ballistic prediction, up to almost a factor of 10. This cannot be explained by the simple corrections to the Maxwell-Boltzmann distribution on the NEMS introduced above: these would lead to at most a 5 % deviation compared to the prediction (5.18) for the velocities used [START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF].

5.5 Unveiling the Knudsen layer at the lowest pressures

Main features and reproducibility

The reduction of gas damping below the molecular regime prediction is observed on the three devices used in this experiment, as shown in Figs. 5.5 and 5.6. Thus The solid line is the molecular regime prediction, with corrections at high pressures [START_REF] Yamamoto | Flow of a rareed gas past a circular cylinder[END_REF]. The magenta dashed line marks the pressure for which λ = g. The rather important error bars come from the signicant intrinsic linewidth, limiting the resolution. Right: same graph for the 300 µm sample (g = 40 µm). Dots are data taken with the cryostat n • 1 (see Table 5.1), while hollow squares are data taken with the cryostat n • 2. All the data are taken at

T = 4.2 K.
we demonstrate that the eect is genuine and independent of the cryostat by doing the same measurement on the same 300 µm device, on both cryostats 1 and 2 and nding the same result. Interestingly, the observations suggest that the eect, for all devices, manifests when the pressure is low enough so that the mean free path becomes comparable with the gap g between the NEMS and the bottom trench. To conrm this, we measure the eect at other temperatures, since the mean free path can also be tuned with temperature. The result is shown in Fig. 5.7 left panel for a temperature T = 1.5 K obtained by pumping on the 4 He bath (see Chapter 2, section 2.

3): at constant pressure the damping is increased along with the molecular regime prediction, by a factor 4.2/1.5 ≈ 1.67 as expected. Moreover, the departure from the standard ballistic prediction is also shifted towards lower pressures, arguing in favor of a gap-driven eect: at lower temperatures, the condition λ = g is met for lower pressures than at 4.2 K.

Up to now the temperature was homogeneous over the entire cell since it was imposed by the 4 He bath. To proceed, we realized experiments where the sample holder temperature alone was regulated above the bath temperature. The idea is that if no turbulent convection phenomenon takes place (which should be the case at such low pressures), the temperature gradient should be small enough on the scale of g to identify the gas temperature surrounding the NEMS with the sample holder temperature. A crude estimate of the thermal gradient at the level of the NEMS δT /T 0 (T 0 is sample holder temperature) can be given for the 150 µm sample which was used for temperature measurements: with a cell radius ≈ 2 cm and assuming a linear temperature gradient between the sample holder at T 0 and the cell contour at 4.2 K, the thermal decoupling is roughly δT /T 0 ≈ 7 × 10 -4 at 15 K. This is clearly negligible, and we can consider T gas = T 0 . In addition, this has been conrmed experimentally: while the surrounding helium bath was cooled down to 1.3 K, we heated the sample holder up to 4.2 K with a power ∼ 100 mW while regulating the temperature and measured the 150 µm NEMS damping in the same pressure range as before. The results for a regulated temperature are found to match perfectly those obtained by simply staying at 4.2 K, as seen in Fig. 5.7 right panel. Therefore we can claim that the properties of the gas near the NEMS are those determined by the sample holder temperature T 0 . It demonstrates that we can safely investigate the eect for temperatures higher than 4.2 K using this method.

Our ndings are strong arguments in favor of a new regime that manifests when the gas mean free path becomes commensurate with or bigger than the gap between the NEMS and the bottom trench. In other words, this corresponds to a regime where collisions with the wall become as frequent as collisions between gas particles for the gas portion in the NEMS region: the NEMS enters the so-called Knudsen layer (also sometimes referred to as "kinetic boundary layer" in the literature), of typical spread λ next to a wall, where a reduction of the (eective) viscosity is expected [START_REF] Lilley | Velocity gradient singularity and structure of the velocity prole in the Knudsen layer according to the Boltzmann equation[END_REF].

Fit procedure and results

The data measured on three dierent samples with rather dierent sizes show qualitatively similar features if we plot the gas damping contribution normalized to the molecular regime prediction. As noticed above, the discrepancy between the molecular regime prediction and the experimental data seem to become signicant around a Knudsen number Kn = λ/g equal to 1. Therefore, we try to t all the data with the same phenomenological expression, that takes the form of a polynomial fraction in the scaling variable Kn, a so-called Padé approximant P(Kn). The gas damping is t with Γ = P(Kn)Γ mol , where:

P (Kn) = 1 + c Kn 1 + (c -α)Kn + c α Kn 2
.

(5.19)

Such an expression captures both limits [START_REF] Gazizulin | Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rareed Gas[END_REF]. In the low mean free path limit g λ (i.e. around 1 Torr for the 150 µm device), the normalized damping essentially reduces to 1 + αλ/g. In the other limit λ g, it becomes α g/λ. The c factor is merely a parameter that makes the link between the two limits by rounding the shape of the t. sets are t with the Padé approximant, with dierent coecients for each device. We now turn to analyze the dependences of t coecients in temperature and gap. To minimize possible experimentalist biases we used three procedures for a t. An example is shown in Fig. 5.9 for the 150 µm (g = 20 µm) device at dierent temperatures. We rst let the computer t all the data sets with the three parameters α, α , c left free (squares). From this tentative it appears that coecients α and c are roughly temperature-independent. Then we keep c xed at the average value of the previous (all free parameters) outcome and let the computer t α and α (empty dots). Again, α stays roughly constant, with less spreading around the mean value than with the previous routine. For the last routine c and α are kept constant, with only α left as a free parameter (triangles). All three routines give results in rather good agreement and lead to α ≈ -0.3 and c = 1.2, for all temperatures and gaps. This proves that the onset of damping reduction is a purely dimensional eect, free from materials or frequency dependence: the obtained scaling appears somewhat universal at rst order in the Knudsen number. Meanwhile, the α coecient is found to scale with 1/g, as shown in Fig. 5.10.

Therefore, the normalized damping is independent of the gap g in the very low pressure (very high mean free path) limit. This is somehow natural: the mismatch between the wall and the gas equilibrium should be qualitatively set over a distance equal to a few λ. As the distance between the NEMS and the bottom becomes gradually small compared to the gas mean free path, the two lengthscales are not comparable anymore, and as far as the gas properties are concerned, the gap g is not a relevant lengthscale anymore. In fact, we can argue that in this limit the portion of the gas surrounding the NEMS has equilibrium properties that are essentially that of the wall, and therefore, the NEMS is in a sense a local probe that is directly impacted by scattering processes

1 0 1 1 0 α'
g a p ( u m ) occurring at the wall that establishes its equilibrium. Besides, the coecient α shows a huge dependence on temperature (we t a quadratic + cubic dependence, see Fig. 5.9), which strengthens this hypothesis.

Role of gas adsorption and lm growth

As explained above, our measurements point out the role of the Knudsen layer, and thus of the bottom wall, in the gas damping mechanisms. Yet, the boundary and slippage regime discussed in the literature are mostly trying to describe the mechanisms taking place when a temperature gradient is present between the wall and the uid, or when a ow is present. This is essentially the case discussed above in paragraph 5.4.2, considering the oscillatory ow around the NEMS itself, which could indeed lead to corrections. The phenomenon we discuss here is of another nature: the system is macroscopically at rest in equilibrium, but locally these conditions are broken by the presence of the boundary: if gas particles were undergoing pure specular reection at the wall, the latter would merely act as a mirror and no change in the thermodynamic and statistical properties of the gas near the wall should occur.

We must therefore invoke the diusive nature of boundary scattering at the wall, e.g. sticking events, in order to circumscribe the microscopic origin of the eect.

Indeed, the t α parameter (see Fig. 5.9 right panel) shows a dramatic temperature 5.5. Unveiling the Knudsen layer at the lowest pressures 155 dependence, especially signicant above 10 K, and since it accounts for processes very close to the wall with respect to the mean free path, it strongly suggests a surfacetriggered boundary phenomenon.

It is possible to characterize qualitatively the energy landscape at the wall as a function of temperature through the measurement of the quantity of adsorbed atomic layers. As typical binding energies of helium on a surface are known to be in the 10 -100 K range for the rst layers, we expect helium atoms to be adsorbed on the NEMS surface at our working temperatures, thus adding a mass δm to the total NEMS mass M . This will result in a change of the resonance frequency δω 0 downwards:

δω 0 = ω 0 M M + δm -ω 0 ≈ - ω 0 2 δm M .
(5.20)

One can then monitor the frequency shift as a function of pressure and temperature to compute the number of helium atoms adsorbed on the surface, and then nd the typical energy scales involved. At cryogenic temperatures, there is always one solid layer that is well anchored to the NEMS surface, even for our lowest pressures, of typical adsorption energy 70 K [START_REF] Goellner | Helium adsorption on exfoliated graphite[END_REF]. The adsorbed mass is expressed in units of adsorbed helium density (see Fig. The NEMS surface as well as the bottom trench are rather disordered. For instance, at the NEMS upper side surface Al grains have a typical size 20 µm (see Fig. 5.3).

Thus we can expect that adsorbed atoms do not complete uniformly a layer but rather ll stronger binding sites (e.g. at the junction of grains) that are somewhat equivalent to "pores" in a spongy material [START_REF] Hutson | Theoretical basis for the Dubinin-Radushkevitch (D-R) adsorption isotherm equation[END_REF]. We thus found that the adsorption isotherms (completion at a given temperature, as a function of pressure) are well described by the Dubinin-Astakhov (D-A) model, which indeed applies to disordered surfaces. In this model, the number of adsorbed atoms write:

N ads = N m exp   -   k B T ln Ps(T ) P E   n   . (5.21)
Here N m is the maximal adsorbed quantity at the saturated vapour pressure P s (T ), and E is a characteristic adsorption energy. n is a parameter that depends on the disorder at the surface, referred to in the literature as a "surface heterogeneity factor" [START_REF] Rudzinski | Adsorption of Gases on Heterogeneous Surfaces[END_REF]. The frequency shift data, converted in adsorbed atoms density for the 150 µm long device, are plotted in Fig. 5.11. Also shown are measurements on the two other devices (see Table 5.2), rescaled to the 150 µm sample dimensions. All data sets are rather well t in the low pressure region with N m = 3 layers, E = k B × 30 K, and n = 1.3. The values for P s (T ) are taken from reference works [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF]. The accuracy of 156 Chapter 5. Energy relaxation in a surrounding uid the model is surprisingly good, given that homogeneity and characteristic energies are likely to dier between the Al-coated side of the NEMS and the bare SiN one. This shall be enough, however, for a discussion on a qualitative basis. Note that above the critical point temperature of 4 He (T c = 5.19 K at P c = 2.25 bar), saturated vapour pressure is not dened anymore. Yet, we can still t consistently the data sets at higher temperatures with P s = P c . In fact, the sensitivity in P s is only logarithmic here.

1 E -3 0 , 0 1 0 , 1 1 1 0 0 , 0 2 , 0 x 1 0 Some more information on the potential landscape can be obtained: in fact, following Ref. [START_REF] Cerofolini | Electron emission from heterogeneous metal surfaces[END_REF] the D-A isotherm can be seen as an averaged property over the disordered surface, which is made of independent, homogenous domains whose adsorption isotherms are simple Langmuir ones: N ∝ (1 -P 0 /P ) -1 , where P 0 is linked to a characteristic adsorption energy ε. From then on the typical energy obtained in the D-A isotherm is then an average obtained over the distribution of domain adsoption energies [START_REF] Hutson | Theoretical basis for the Dubinin-Radushkevitch (D-R) adsorption isotherm equation[END_REF]:

g(ε) = n (ε -ε m ) n-1 E n exp - ε -ε m E n , (5.22)
5.5. Unveiling the Knudsen layer at the lowest pressures 157 where ε m is the local minimum of adsorption energy due to intermolecular forces ∼ 10 K for 4 He atoms. The distribution (5.22) is plotted in Fig. 5.12 for E = k B ×30 K

and n = 1.3: it yields a maximum at 20 K with a typical asymmetric spread ∼ 40 K around the peak value. This energy scale seems to match the growth of α with temperature reported in Fig. 5.9 left panel. This strongly suggests a link between the boundary layer eect reported here and the adsorbed surface dynamics. Note however that the isotherm plot in Fig. 5.11 probes the surface of the NEMS device, while the boundary layer depends on properties of the spongy bottom of the trench (e.g. Fig.

5

.1 right panel). The analysis done here nonetheless gives us some idea about the adsorption energies involved, and the amount of layers present on surfaces. We will come back to these issues at the very end of the Section. Of course, this description does not hold anymore at high coverages (i.e. N > N m , as surface inhomogeneities are smeared out by the now completed layers. Therefore, the potential seen by the gas atoms is not exactly that of the porous silicon substrate surface (or the NEMS one): the latter is now partly screened by the helium layers.

As atoms are adsorbed and layers are completed, the adsorption potential seen by remaining gas atoms venturing near the wall is closer to that created by helium atoms, with a typical energy scale that depends on the number of layers already completed.

In addition, layers become more uniformly completed [START_REF] Dash | Films on Solid Surfaces: The Physics and Chemistry of Physical Adsorption[END_REF]. At high pressures > 0.5 Torr for low temperatures (below typically 4 K), the number of layers is larger and exceeds N m = 3 (see Fig. 5.11). Therefore we can treat the NEMS as a uniform solid the vicinity of the wall, since in the ballistic regime the damping is a statistical quantity. With no external forcing, the evolution of the particles distribution n(r)f (v g , t) in a given volume d 3 rd 3 v g of the phase space follows the well-known Boltzmann equation:

∂(nf ) ∂t + v g . ∂(nf ) ∂r = C[nf ], (5.26)
where C is the so-called collision integral, i.e. the functional gathering the scattering processes that bring the portion of gas in the elementary volume back to thermal equilibrium. It formally writes:

C[nf ] = n 2 (r) dΩd 3 v g,1 (f f 1 -f f 1 ) K(v g , v g,1 , Ω).
(5.27)

Here K is the so-called scattering kernel, which accounts for a scattering process (v g , v g,1 ) → (v g , v g,1 ) within a solid angle Ω. The condensed notations read

f ( ) (1) ≡ f (v ( ) g(,1)
, t). In the steady-state regime, the equation to solve reduces to:

v g . ∂(nf ) ∂r = C[nf ]. (5.28)
As the dynamics at the wall is very dierent from that of the gas because of diusive scattering, there is a mismatch that reects in the collision integral (5.27). At equilibrium in the bulk gas, the collision integral is equal to zero, from which we deduce the Maxwell-Boltzmann distribution f M (5.12) as the equilibrium distribution. At the wall in the steady-state regime the collision integral is not zeroed anymore, which implies that the distribution f = f M on the boundary [START_REF] Cercignani | Mathematical Methods in Kinetic Theory[END_REF].

Therefore, the dynamics at the wall introduces a deviation to the Maxwellian distribution that propagates o the surface within a characteristic length comparable with the mean free path λ. The latter argument is justied a posteriori by our experiment, but also qualitatively: if there is a gradient of the distribution, there should be one for thermodynamic quantities that are derived from the distribution through its moments.

From this non-equilibrium steady state, the return to equilibrium in the gas is dened by scattering processes that will naturally take place over a typical distance λ.

Note that the deviation from the Maxwell-Boltzmann distribution is not necessarily small. There is currently no exhaustive knowledge of the microscopic mechanisms occurring at the wall, and the statistical description far within the Knudsen layer remains an extremely challenging problem, largely unsolved theoretically. The key feature here is that at the onset of the boundary layer (λ/g 0.1), the NEMS, which is closer to the bulk spatially, is essentially sensitive to small reminiscences of this gradient, which appear to be universal, since the α parameter is independent of the gap and the cell temperature.

160

Chapter 5. Energy relaxation in a surrounding uid While we do not attempt to solve the Boltzmann equation (5.28), which would be a considerable theoretical challenge far outside the scope of this work, we can use again the approach developed in the previous section, in paragraph 5.4.2, to describe the onset of the damping reduction, from a modication of the equilibrium distribution f M . This is justied insofar as the behaviour seems universal and the departure from the molecular prediction is not too big in the region λ/g 0.1. We use the polynomial correction (5.14) to the Maxwell-Boltzmann distribution, and keep average velocities v x,0 , v y,0 , v z,0 in the expansion. The coecients a i... are not explicited so far, but they should be space-dependent, as the average velocities should also. As seen earlier, it is possible to connect these coecients through sum rules [START_REF] Neil | Molecular ow of gases[END_REF]. Moreover, the average kinetic energy denes the temperature through m g |v g -v g,0 | 2 /2 = 3k B T /2. This relation imposes additional sum rules on coecients because of the averaging. In addition, symmetries of the problem are such that there should not be any dependence of thermodynamic quantities on y, z. Very conveniently, the rst three moments of the collision kernel are zeroed because of mass, momentum and energy conservation, which leads to the well-known transport equations. The mass conservation equation is the moment of order zero of Eq. (5.28) projected along x and writes:

∂ ∂x f (v x , v y , v z )v x n(x)d 3 v g = 0.
(

In fact, from the application of sum rules, this moment of order zero is already zero.

The rst moment gives the conservation of momentum along the x direction:

∂ ∂x f (v x , v y , v z )(m g v x )v x n(x)d 3 v g = 0.
(5.30)

The second moment yields the conservation of energy:

∂ ∂x f (v x , v y , v z ) 1 2 m g v 2 x v x n(x)d 3 v g = 0.
(

This set of transport equations, through the application of sum rules, leads to 2 equations on only four independent variables among the a i... coecients, which are functions of x, identied to g the position of the NEMS in the vicinity of the bottom (at which x = 0). These four coecients are the only ones relevant to our problem, when the expansion 5.14 is truncated at third order. The next step is to write thermodynamic quantities as slightly perturbed at the onset of the Knudsen layer n = n 0 + δn(x) and T = T 0 + δT (x). From the hierarchy of conservation equations (5.29),(5.30),(5.31), we can then link the quantity δn/n 0 to δT /T 0 and the a i... .

Coming back to the Boltzmann equation, we can follow the Chapman-Enskog method [START_REF] Chapman | The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diusion in Gases[END_REF] for the left-hand side term of Eq. (5.28): the a i... coecients are de- do have a physical meaning: they express the near-eld induced deviation to bulk equilibrium due to the scattering processes at the wall. With such a denition, a i... coecients will be nonzero at the onset of the Knudsen layer, and zero in the bulk when

x → +∞. The temperature gradient is also developed as δT /T 0 = d 1 Kn + d 2 Kn 2 + ....

Obviously, the knowledge of the collision integral is required from now on. The problem can be dened in a self-consistent way by also writing the collision integral in powers of the velocity eld, with coecients that are also powers of Kn:

C[nf ] ≈ n 0 m g 2πk B T exp - m g (v x -v x,0 ) 2 2k B T × κ 1 (Kn)(v x -v x,0 ) λ + κ 2 (Kn) m g 2k B T 1/2 (v x -v x,0 ) 2 λ + κ 3 (Kn) m g 2k B T 3/2 (v x -v x,0 ) 3 λ + ... (5.32)
with κ i (Kn) = ν

i Kn + ν

i Kn 2 + .... Such a development seems licit, since we are close to the bulk with Kn 0.1, where the collision integral should be zero as the distribution tends towards f M . In addition, constraints can be obtained on these parameters, using the fact that the three rst collision kernel moments should be zero (conservation laws). Using this phenomenological approach permits to overlook the scattering kernel, which is elusive since we do not know in detail the wall's dynamics:

the only quantities one can extract are characteristic energies, as shown in the previous paragraph, and are meant to bring only qualitative arguments here.

From then on, the problem is self-consistent if we truncate expansions at order n+1 in velocities and order n in Kn on the left-hand side of Eq. (5.28), n + 2 in velocities and n + 1 in Kn in the collision integral C[nf ]. We chose to stop at n = 3, after which the problem is not easy to handle regarding the calculation. The full procedure is tedious and carried out in Mathematica, and we nally obtain that the average velocity eld is zero: v g,0 = 0, while all introduced coecients can be expressed as functions of parameters b (j) i... . The next step is to compute macroscopic parameters, evaluated at the position of the NEMS x = g, i.e. Kn = λ/g. We already have δT /T 0 and δn/n 0 . Meanwhile the kinetic pressure tensor component P ij is by denition the ow of momentum m g v i in the direction j:

P ij = nm g v i v j f (v x , v y , v z )d 3 v
, from which we obtain the thermodynamic pressure P = (P xx + P yy + P zz )/3. Note that in the end we still have P = n 0 k B T , but each of the tensor components does not necessarily follow the same law. Again, for symmetry reasons only P xx = P xx,0 + δP (Kn) should be aected, with P xx,0 the bulk value. The calculation shows that:

• the kinetic pressure eld remains actually constant, i.e. δP (Kn) = 0. This is reassuring, as a deviation would mean that in general no manometer could be accurate as soon as the surface is not perfectly specular.

• there is a temperature gradient, only quadratic in Kn and b (j) i... coecients, which should thus be weak enough so it can be disregarded in the rst place.

Note that this temperature drop is known to occur in the boundary layer, and takes the form of a temperature jump at the wall, since the wall itself should be thermalized at T 0 [START_REF] Neil | Molecular ow of gases[END_REF][START_REF] Cercignani | Mathematical Methods in Kinetic Theory[END_REF].

• the density gradient is a rst order correction in the Knudsen number, and in the b

(n) i... : δn(Kn) n 0 = - 1 2
2b (1) xx + 6b (1) xxxx + b (1) xxyy + b (1) xxzz Kn.

(

This last point shows that the observed reduction of damping is the result of a rarefaction phenomenon (see schematic in Fig. 5.13), induced by the dynamics at the wall which translates into a deviation to the Maxwell-Boltzmann distribution f M in the Knudsen layer. As the eective density in the gas is reduced, so is the number of collisions on the surface of the NEMS: this ultimately explains the reduction of damping observed in our experiments. Indeed, since the damping rate calculation involves an average over the particles distribution, we can write an eective near-eld coecient ϑ nf instead of ϑ in the damping rate expression (5.18), which incorporates the density reduction term δn(Kn)/n 0 to match the observed damping reduction:

ϑ nf (Kn) ≈ ϑ + 1 2 s + (1 -s) π 4
T N T 0 + 1 2 δn(Kn) n 0 - 1 8 (1 + s)b (1) xxxx Kn ≡ ϑαKn . (5.34)
It goes without saying that the expression into brackets in Eq. (5.33) can only constitute a single free parameter, because of the lack of information on scattering processes at the wall: thus we recover the P function in the low Knudsen number limit, through ϑ nf ≈ ϑ(1+αKn), with alpha the t parameter already introduced in section 5.5.2 and Eq. (5.34). Therefore, assuming s = 0.5 holds in the whole temperature range and taking again T N ≈ T 0 , only the b

(1) i... are free parameters to describe the damping reduc- tion, which accounts for surface scattering. The link between the measured eect and surface properties suggests that it could depend on the sample. Yet, the ts presented in paragraph 5.5.2 suggest that the deviation at rst order, i.e. the α parameter, is independent of the device. Nevertheless, we can argue that all three devices are made of the same materials (SiN thin lm on silicon substrate), and suspended the same way with the XeF 2 etching, which results in a similar spongy bottom for all the devices (see Fig. 5.1). This may explain the absence of discrepancy in α from one sample to the other. The other explanation would be that the rst order deviation is not aected by the details of the diusive scattering mechanisms at the wall, making this term truly universal. This hypothesis is favored by the fact that we obtain an α which is temperature independent. This suggests once more that the coecients b

(1) i... originate only from a mismatch in the distribution which has nothing to do with temperature, i.e. with the excitations at the wall, since these excitations depend on temperature.

Thus, higher order coecients b

(2),(3).. i... must depend on temperature and bear the information on the wall excitations, since α strongly depends on temperature.

xxxx Kn/8 for the former damping reduction. Since this term comes from an expansion at order 4 in velocities, one can argue that it should be small compared with the lower order coecient b

(1) xx appearing in the density reduc- tion. This leads us to the introduction of an eective density dened in the Knudsen regime (when λ > g), following Eq. (5.18):

n ≈ mΓ mol P(Kn) 2ϑm g w Ψ 2 (z)dz = n 0 P(Kn).

(

This recalculation operated on the measured data is represented for three dierent temperatures in Fig. 5.14 and t with n = n 0 P(Kn). Therefore, deep in the boundary layer, that is, close to the wall, the density should evolve as n w ≈ n 0 α g/λ, i.e.:

n w ∝ α (T 0 )P 2 /T 2 0 .

(

This anomalous pressure dependence is the dashed asymptote represented in Fig. 5.14 in the low pressure regime for each temperature [START_REF] Gazizulin | Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rareed Gas[END_REF].

0 , 0 1 0 , 1 1 0 2 0 1 0 2 1 1 0 2 2 1 0 2 3 1 0 2 4 T = 1 . 7 K T = 4 . 2 K T = 1 5 . 0 K D e n s i t y ( m -3 )
P r e s s u r e ( T o r r )

Figure 5.14: eective density recalculated from the damping data at 3 dierent temperatures as a function of the cell pressure, for the 150 µm device. Thin solid lines are calculated with the ts from Eq. (5.19), while asymptotes are shown in the low (∝ P 2 ) and high (∝ P , bulk density n 0 ) pressure regimes.

Eq. (5.36) is in fact a powerful result, because of the huge dependence of α in temperature (Fig. 5.9 left panel). If α were temperature independent, the transition range between the anomalous P 2 and the standard ∝ P regime would increase linearly with the temperature, since the physics would be set only by the mean free path λ ∝ T 0 /P . The observed behaviour is more complex: at 15 K, the density reduction is still close to the standard bulk prediction even at rather low pressures (below 0.1 Torr), and the anomalous regime is not truly reached: the cusp is drifting up, but is less salient, and the data would reach the asymptote at even lower pressures. We believe that it is the result of the competition between the mean free path increase and the dramatic change of energy landscape at temperatures comparable with the typical adsorption energies, which is monitored through the strong dependence of α in temperature (see Fig. 5.9 left panel). We thus stress out the fact that this eect is truly visible only at low temperatures, meaning temperatures smaller/comparable to the adsoption energy. It originates from mechanisms that take place at low energy scales, and which should be suppressed at room temperature. Our observations regarding the transition range from n ∝ P to n ∝ P 2 might explain why higher temperature experiments failed to reproduce this Knudsen layer signatures [START_REF] Kara | Nanouidics of Single-Crystal Diamond Nanomechanical Resonators[END_REF]: even though this eect should be present, it would manifest itself at pressures that are not attainable experimentally.

Another aspect of this scaling near the wall is that at very low pressures the damping does not depend on the gap g anymore. In Fig. 5.15, we indeed see that the damping data (rescaled in units of eective density) for all three devices in the asymptotic limit collapse on the n ∝ P 2 curve. In that sense, we believe that in this limit we are truly sensing the wall's properties, and that the anomalous pressure dependence is a direct signature of its dynamics. This last point highlights the advantage of using a nanomechanical string within the Knudsen layer. Indeed its cross-dimensions are much smaller than the relevant lengthscales. Therefore, while the Knudsen layer is an out-of-equilibrium property (since it involves gradients of thermodynamic quantities), a local equilibrium can be dened at the level of the NEMS, which then acts as a non-invasive probe for a non-equilibrium medium.

Conclusion and prospects

Summary of the results

The aim of the measurements reported in this Chapter was to characterize one ubiquitous damping source for nanomechanical devices, namely the friction due to the uid in which it is immersed. In particular, the NEMS small cross-dimensions make it particularly sensitive to features that are otherwise negligible with larger oscillators, such as slippage. In contrast with measurements performed in air at room temperature, we realized a model experiment were an almost ideal gas was used ( 4 He around a Kelvin), which properties are tabulated. We thus have measured the gas damping contribution by varying the pressure in the experimental cell over ve orders of magnitude and the temperature between 1.5 and 15 K, thus tuning considerably the gas mean free path.

At low pressures when the gas becomes rareed, i.e. has its mean free path comparable with the NEMS cross-dimensions, slippage features become visible for the NEMS, and for even larger mean free paths (even lower pressures), a description relying on kinetic theory for a ballistic gas accounts for the damping properties. Unexpectedly, we observed a reduction of the damping up to one order of magnitude below the simple 166 Chapter 5. Energy relaxation in a surrounding uid ballistic prediction. Possible experimental artefacts have been ruled out by measuring this deviation for three dierent samples, in two cryostats having dierent congurations. We interpret this reduction as a nite-size eect due to the proximity of the bottom trench when the mean free path is comparable with or bigger than the distance between the NEMS and the bottom trench. We show that this geometric region, referred to in the literature as the Knudsen layer, exists because of the diusive scattering processes occurring at the wall for incoming gas particles, and qualitatively put an energy scale on them by monitoring the gas adsorption on surfaces for dierent temperatures and pressures. We show with a qualitative transport model that these processes create a non-equilibrium region near the wall where the gas statistical properties, and thus its thermodynamic parameters, are renormalized. The most striking feature is a gas density reduction near the wall, which causes directly the damping reduction. We also emphasize the crucial role played by low temperatures, which favor adsorption mechanisms. These measurements demonstrate the importance of surface coatings, while showing that the NEMS can be used as an ecient near-eld probe for surface science.

0 , 0 1 0 , 1 1 0 2 1 1 0 2 2 1 0 2 3 g = 4 µ m g = 2 0 µ m g = 4

Prospects

Obviously, many aspects have been overlooked, which deserve consideration: rstly, the transition range between the continuous uid description and the statistical one, and its consequences on friction, still lacks of a universal description beyond the slippage correction, despite recent investigations [START_REF] Bullard | Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical Resonators[END_REF][START_REF] Lissandrello | Crossover from Hydrodynamics to the Kinetic Regime in Conned Nanoows[END_REF][START_REF] Kara | Generalized Knudsen Number for Unsteady Fluid Flow[END_REF]. Meanwhile, more quantitative information might be obtained on surfaces by combining our measurements with AFM studies which could detail the granular structure at the NEMS surface or the spongy one at the bottom.

In addition, the sensor aspect was not exploited further from computing the number of adsorbed atoms. With a better resolution one could push it to study the structural properties of the formed lm, as proposed in Ref. [START_REF] Tavernarakis | Atomic Monolayer Deposition on the Surface of Nanotube Bibliography Mechanical Resonators[END_REF], or the coupling between the resonator and the lm at the single excitation level, as proposed in Ref. [START_REF] Harris | Laser cooling and control of excitations in superuid helium[END_REF]. Besides, our study was restricted to a Helium gas. It instantly comes to mind that the same experiment could be done with a dierent gas, as what was done in Ref. [START_REF] Kara | Nanouidics of Single-Crystal Diamond Nanomechanical Resonators[END_REF] at higher temperatures, or with a liquid [START_REF] Gil-Santos | High-frequency nano-optomechanical disk resonators in liquids[END_REF]. Since we showed the crucial role of low temperatures, however, this appears to be dicult: most elements will be condensed at helium temperatures, with the exception of hydrogen at 20 K. Such an experiment thus requires an update of the apparatus in order to be able to measure at this temperature while operating in safe conditions.

Finally, a most exciting endeavour is the use of a nano-probe to investigate yet unknown mechanisms in quantum uids such as superuid 4 He or 3 He. Some pre- liminary results have been obtained in 4 He, which are not shown in this manuscript.

Other recent results have highlighted the potential of nanobeams for such purposes, through the mapping of the superuid transition in 4 He [START_REF] Bradley | Operating Nanobeams in a Quantum Fluid[END_REF]. A collaboration with Lancaster University has allowed to immerse our NEMS in superuid 4 He down to mK temperatures, revealing the damping due to the superuid excitations [START_REF] Guénault | Probing Superuid 4 He using Nanobeams at mK Temperatures[END_REF]. The case of superuid 3 He is even more appealing, because of its much richer phase diagram [START_REF] Vollhardt | The Superuid Phases of Helium 3[END_REF]. Indeed, superuid 3 He is a BCS superuid (atoms form Cooper pairs) with a coherence length ("size" of a Cooper pair) typically lying in the 10 -100 nm range.

Using a probe with cross dimensions comparable with the coherence length could open a new range of investigation on the intimate structure of superuid 3 He, with an un- precedented accuracy [START_REF] Defoort | Probing Bogoliubov Quasiparticles in Superuid 3 He with a `Vibrating-Wire Like' MEMS Device[END_REF]. The results shown in this Chapter also highlight their potential to study the superuid in conned geometries, where elusive topological features such as Majorana states are theoretically predicted to exist [START_REF] Wu | Majorana excitations, spin and mass currents on the surface of topological superuid $^3$He-B[END_REF]. This of course requires a huge experimental development far beyond the results of this Chapter, with e.g. the use of a nuclear demagnetization stage [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF][START_REF] Guénault | The mechanical behavior of a vibrating wire in superuid 3 He-B in the ballistic limit[END_REF] to cool 3 He down to 100 µK, where most of its thermal excitations are removed. Internal energy relaxation and frequency noise in nano-mechanical systems Contents The last Chapter of this manuscript focuses on intrinsic sources of mechanical damping and frequency noise. It is motivated by a puzzling fact: the Q factor of nanomechanical oscillators, that is, the ratio between the stored energy and the dissipated energy over an oscillation cycle, is found to roughly decrease with the decreasing volume [START_REF] Ekinci | Nanoelectromechanical systems[END_REF].

While proving a major limitation for applications (more energy is required to enable motion if the Q factor is low) as well as for studies of quantum mechanical eects (a good Q factor is needed to resolve tiny eects), it also poses very fundamental questions: such features must be caused by eects that take place at mesoscale, as the competition between reduced dimensions and the typical lengthscales involved in these processes becomes signicant.

To address this process, we remove all known external contributions (e.g. loading [START_REF] Cleland | External control of dissipation in a nanometerscale radiofrequency mechanical resonator[END_REF], or gas [START_REF] Gazizulin | Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rareed Gas[END_REF]) and take advantage of low temperatures, where already known multiphonon processes are suppressed, leaving less and less uncontrolled thermally activated microscopic degrees of freedom [START_REF] Cleland | Foundations of Nanomechanics[END_REF]. In the same spirit, internal spectral diusion processes activated by temperature, such as the one described in Chapter 3 are negligible. Yet, we will see that mechanisms that limit the NEMS lifetime and coherence still exist at low temperatures. These are commonly assumed to be some sorts of defects in the materials that can transit between equivalent position states, thus causing retarded stress response to a strain perturbation, hence dissipation. In nanomechanical structures, there are only a handful of studies attempting to clarify their role, which is made even more unclear due to the size reduction. We will see in this chapter that these defects might also explain internal frequency noise features by measuring not only the damping, but also frequency uctuations at slow timescales thanks to a newly developed technique. Thus, we begin with a detailed account on how this technique is used and applied to "high" (4 K) temperature frequency uctuations.

Then, in order to investigate more their microscopic origin, we take advantage of the dilution temperatures to cool the NEMS down to 10 mK, where the behaviour of these hypothetical defects is thought to be dominant and their signatures on the damping striking, even though complexity (or physical richness) is added by the metallic layer of the NEMS. We thus compare our damping data to frequency noise measurements made in this range and discuss a possible mechanism.

6.2 Measuring intrinsic frequency noise

Dynamical bifurcation as a tool to measure frequency noise

To address intrinsic frequency noise we use a technique adapted from what was proposed in Ref. [START_REF] Aldridge | Noise-Enabled Precision Measurements of a Dung Nanomechanical Resonator[END_REF]: since the jump associated to a bifurcation event is sharp by contruction, the bifurcation frequency is extremely well dened, with a resolution arbitrarily set by external, user-dened parameters. These bifurcation frequencies can be linked readily to resonance frequencies. Indeed the bifurcation frequency on the upper branch writes:

ω bif,↑ = ω 0 + 2β|x bif,↑ | 2 - β|x bif,↑ | 2 2 - Γ 2 4 . (6.1)
The case of the lower branch is slightly more involved. As one increases the driving force, the lower branch amplitude at its bifurcation point x bif,↓ initially decreases below x s , and the bifurcation frequency follows the same law as the upper branch bifurcation frequency, see Eq. (6.1). Once x bif,↓ = 3 1/4 x s / √ 2, a turning point is reached and increasing the driving force results in an increase of x bif,↓ . In that case, the bifurcation frequency writes:

ω bif,↓ = ω 0 + 2β|x bif,↓ | 2 + β|x bif,↓ | 2 2 - Γ 2 4 . (6.2)
For large displacements, the upper bifurcation frequency merges asymptotically with the resonance frequency that shifts as in Eq. (2.70) (reminder: ω bif,↑ ≈ ω 0 + βx 2 bif,↑ ), while the lower bifurcation frequency follows asymptotically ω bif,↓ ≈ ω 0 + 3βx 2 bif,↓ . A representation of the bifurcation frequencies diagram is shown in Fig. 6.1.

The key message is that there is a simple correspondence between both bifurcation frequencies and the resonance frequencies through Eqs. (6.1) and (6.2), and therefore, measuring the bifurcation frequency uctuations is a way to access frequency noise on ω 0 if we assume that thermal uctuations on position are small, which is the case at 4 K, and that force or damping uctuations are negligible, a point which shall be discussed explicitly below. A practical consequence is that the bifurcation frequency cannot be dened with a better resolution than that obtained for the resonance frequency.

We performed experiments measuring the statistics of the two bifurcation points mostly on a 250 µm high stress SiN device. The procedure to measure the bifurcation frequency noise is schematized in Fig. 6.2: for the upper branch, the frequency sweep starts from low frequencies with a large sweep step and a high speed to enable reasonable measurement time, up to about half the hysteresis: from then on, the step is set at a low enough value so as to minimize sampling noise on the resulting bifurcation frequencies, and the sweep is done at a speed ∼ 0.1 -1 Hz/s enough to be in an adiabatic regime. The last point is important, as it is known that too large speeds exacerbate escape events from the metastable state, which are not bifurcation events [START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF]. We have veried that for speeds up to one order of magnitude dierent from the reference one used, the results were identical within our error bars. When the bifurcation event occurs, the amplitude switches sharply from the metastable state (high amplitude) now made unstable to the only remaining stable one, which is a low amplitude one:

as the amplitude goes below a user-xed threshold value, the frequency at which the amplitude crosses the threshold is recorded, and the frequency sweep is stopped. The sequence is then repeated, and a track and a histogram of bifurcation frequencies is eventually built. Therefore, we have an access to the resonance frequency noise, as discussed above. We checked that the procedure was threshold-independent as long as the threshold remains within the amplitude region between the two states.

Besides, another source of frequency uctuations could come from the jitter on the internal clock of the generator, due e.g. to room temperature uctuations. We characterize it using one generator's signal on the input port of the lock-in amplier while using another identical generator as a reference signal. Once the certain shift due to the discrepancy between the two reference clocks is corrected, we observe a phase drift ∼ 1 mHz over a few minutes at 1 MHz, and ∼ 10 mHz over a few minutes at 10 MHz. Assuming the two generators undergo a similar phase noise, we can assume that the latter is evenly distributed, which leads to a frequency noise in the ppb range, that is, two orders of magnitude below the lowest frequency noise values reported in our experiments.

The procedure allows to record a bifurcation frequency with an arbitrary precision, with measurement times ranging from 10 s to 1 min, and is repeated over a few hours to acquire enough statistics.

Note that we described the measurement operated with the upper state branch, i.e. with the upper bifurcation frequency and higher amplitude. Obviously, it can be operated without supplementary eort on the lower state branch by reversing the protocol. A typical frequency noise track taken while operating on the lower branch is shown in Fig. 6.3, with a reference frequency (the average one) subtracted. From such a track one can extract a few key informations, which are described below.

Analysis procedure and spectral features

A few steps are necessary to extract information from a frequency noise track. A typical frequency noise such as in Fig. 6.3 indeed features fast jumps, on the order of the measurement times, but also slow drifts that make a brute force estimate of the noise magnitude unreliable since they forbid reproducible measurements. First we can obtain a quantitative spectral information by performing a Fast Fourier Transform (FFT) on the measured signal autocorrelation. This yields the frequency uctuations spectrum S f (ω). The power spectral density associated to the frequency noise displayed in Fig. 6.3 is shown in Fig. 6.4.

The spectrum clearly is that of a so-called non-stationary noise, t with a 1/f 1+µ -type law, where µ is an exponent smaller than 0.5. Possible origins of such a noise will be addressed later, but for the time being, it makes the problem a little harder: one cannot estimate precisely a frequency noise magnitude by integrating the spectrum (which yields the variance), as this integral strongly depends on the time over which a track is taken: this measurement time imposes a low cut-o frequency ω low . Besides, the sampling time (time between two neighbour frequencies) also imposes a high cut-o frequency ω high , even though it does not crucially aect the integral here since the weight is bigger at low frequencies. For a noise power spectral density S f (ω) = A 0 /ω 1+µ , the variance is:

σ 2 f = 1 π ω high ω low S f (ω)dω = A 0 πµ ω µ low 1 - ω low ω high µ , (6.3)
where we have assumed that the spectrum was symmetric in zero, hence the integral over positive frequencies and the factor 1/π instead of 1/(2π). Note that the case µ = 0 is divergent in the last expression in Eq. ( 6.3), but the limit µ → 0 is easily found, yielding also a strong dependence in the measurement bandwidth:

1 0 -4 1 0 -3 1 0 -2 1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 ω l o w
F r e q u e n c y n o i s e P S D ( a . u . / H z ) F r e q u e n c y ( H z )

∝ 1 / f 1 , 4
ω h i g h σ 2 f = A 0 2π ln(ω high /ω low ). Since our measurement technique does not allow to always have xed measurement and sampling times from one frequency track to another, we cannot extract reproducible quantities from the bare spectrum. Therefore, to get rid as much as possible from edge eects, one can conveniently compute the statistics on jumps δf (t i ) from a frequency f i ≡ f bif (t i ) at a time t i to the next one measured f i+1 . In fact, this is almost equivalent to apply a high-pass lter to the frequency noise track, since we can write in the limit where the sampling time ∆t min → 0:

δf (t) ≈ df bif (t) dt ∆t min . (6.4)
From then on, the original spectral information is kept. Indeed, the spectral density on jumps writes:

S δf (ω) ≈ (∆t min ω) 2 S f (ω).

(6.5)

The ω 2 illustrates the fact that we have applied a high-pass lter to the raw frequency noise by computing jumps, since it puts more weight on high frequencies. Long term drifts are thus canceled and do not aect our estimates of frequency noise. The inte-Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems grated value of the jumps spectrum is calculated as follows, and can be considered as the characteristic magnitude of frequency noise:

σ 2 δf = 1 π ω high ω low S δf (ω)dω = A 0 (∆t min ω high ) 2 πω µ high (2 -µ) 1 - ω low ω high 2-µ . ( 6.6)
This expression can be simplied: rst, since µ is close to 0, the second term into brackets is very small, as the two experimental cut-o frequencies are always separated by at least two orders of magnitude. Second, the Fourier transform properties impose that ∆t min ω high = π. Thus, the jumps variance is:

σ 2 δf ≈ πA 0 ω µ high (2 -µ) . (6.7)
We see that this variance is rather weakly dependent on the sampling frequency, since µ is small. We can estimate the discrepancy between two successive measurements with only dierent sampling frequencies, in the most extreme case: the maximum µ observed was 0.4 and sampling times varied between 10 and 60 seconds at most. This leads to discrepancies of at most (σ

δ f,2 -σ δ f,1 )/σ δ f,1 = (ω high,1 /ω high,2 ) µ/2 -1 ≈ 38 %,
and usual sampling times are less spread. Therefore, our measurements combined with the jump analysis provide safe grounds for a quantitative study of intrinsic frequency uctuations. Note that there is no dependence on the higher cut-o if µ = 0. Besides, a 1/ω-type spectrum is rather convenient in that we do not need fast measurement times, since the weight is on low frequencies.

Usually, we compute the discretized version of Eq. (6.6), which is easily accessible through the jumps histogram. If we assume that jumps are independent events, they should reasonably follow a Gaussian distribution of width σ δf . This width is the socalled Allan deviation [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF], and its experimental version for a nite number N of samples is expressed as follows:

σ δf = δf 2 -δf 2 = 1 N -1 N -1 i=1 (f i+1 -f i ) 2 . (6.8)
Note that we have assumed that on average δf = 0 to write the last equality. The frequency histogram associated to the frequency noise track of Fig. 6.3 is shown in Fig. 6.5. left panel. The histogram is not clearly well t by a Gaussian, which is likely due to the slow drift component. In fact, most histograms taken from raw data do not feature a clear Gaussian shape. The jump histogram is plotted in Fig. 6.5 right panel for comparison. The Gaussian t is good, and yields a typical width that is 1) reproducible and 2) smaller than the one obtained from raw data, which is expected since we have removed the slow drift. Besides, as a proof of robustness, insets show that the jumps are kept around zero with time, and that the spectrum of jumps is well tted with Eq. (6.5), where the frequency noise spectrum is the one t in Fig. 6.4. This demonstrates that nite time corrections which were neglected to write Eq. (6.4) can be overlooked in our analysis. F r e q u e n c y ( H z ) Remarkably, if one can obtain a reliable estimate of σ f through the histogram, it provides a good way to predict and t the exponent µ in the spectrum. Indeed, just as the spectra on jumps and on frequency noise can be linked, the two variances can be related combining Eq. (6.8) and Eq. (6.3):

σ 2 f = 2 -µ π 2 µ ω high ω low µ -1 σ 2 δf . (6.9)
This relation strongly depends on µ, which is the only free parameter if the two variances are extracted from histogram computations. Applying this relation to the two magnitudes measured in the case of Fig. 6.5, we nd that µ = 0.4 with a 10 % margin on the value if we assume a 15 % error on the measured estimates. This value is consistent with the 1/ω 1.4 t in Fig. 6.4. Albeit appealing, this method suers from a major drawback: as we said above, most of the time, the distribution on frequency noise is not Gaussian enough to properly extract σ f because of slow drifts.

Note that using the nonlinear properties of the resonator should not involve signicantly the Brownian motion transduction mechanism described in Chapter 3: rst, the resulting statistics on the frequency noise would be strongly asymmetric [START_REF] Maillet | Nonlinear frequency transduction of nanomechanical Brownian motion[END_REF]. On the other hand, a quick estimate, given typical nonlinear coecients for our beams and the true thermal uctuations at 4 K yield a spectral broadening in the µHz range.

To make sure that the interplay between nonlinearities and uctuations is negligible, Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems we measure frequency noise with an added noise force: the statistics starts to be altered for equivalent temperatures in the 10 6 K range. Thus this mechanism is here irrelevant.

Results and discussion

Since the technique is demonstrated to be reliable, we use it to characterize thoroughly frequency noise in our device [START_REF] Maillet | Measuring frequency uctuations in non-linear nano-mechanical resonators[END_REF]. First we measure by operating on the upper branch, at high motion amplitude. The results yield similar frequency power spectra with a 1/ω 1+µ characteristic, where µ = 0.4 ± 0.05 for all measurements on the same 250 µm long device, and a Gaussian distribution on frequency jumps.

-4 0 -2 0 0 2 0 4 0 0 An example is shown in Fig. 6.6. Unexpectedly, the Allan deviation is much bigger than the one measured while operating on the lower branch. We thus have measured the frequency noise over a large range of motion amplitudes (i.e. of driving forces), operating both on the upper and lower branches. The Allan deviation appears to scale quadratically with the motion amplitude on the upper branch and seems to remain rather constant on the lower branch, as seen in Fig. 6.7.

Note that there is a clear oset, which is well captured by the lower branch measurements. In addition, we tuned the Q factor of the NEMS with the magnetic eld, only to nd that the Allan deviation depends linearly on the Q factor at xed motion amplitude. The measurements were also carried on a 15 µm long SiN device with the same characteristics, apart from the length. The same dependence is observed, although with bigger error bars. Furthermore, we measured, at a high constant displacement on the upper branch and high Q factor, the dependence on temperature from 4 to 20 K. Within error bars, the Allan deviation is not clearly aected by temperature. Of course this does not mean that there is no temperature dependence, but this should be small and uncorrelated to the observed nonlinear features. To further investigate a possible temperature dependence, we measure frequency noise by operating on the lower branch, where the motion amplitude is the smallest (x ≈ x s ) and the Allan deviation rather insensitive to it.

The result is plotted in Fig. 6.8. We observe a linear dependence of the Allan deviation in temperature from 1 to 30 K. An oset of 0.16 Hz remains if one extrapolates the linear dependence to T = 0 K. The observed linear dependence is consistent with the work of Ref. [START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF], where a similar long, stressed SiN device was studied, but the frequency noise value we measure at 4 K is one order of magnitude higher than that reported in this work. The main dierence is the presence, in our case, of the Aluminium layer. In the aforementioned work, the origin of frequency noise is thought to be relaxation of two-level defects in the materials, that translates into uctuations of stress or Young's modulus [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF]. Here, it is likely that another mechanism, perhaps of the same nature, takes place in the metallic layer. This could be linked with the work done in Refs. [START_REF] Olkhovets | Actuation and internal friction of torsional nanomechanical silicon resonators[END_REF][START_REF] Collin | Metallic coatings of microelectromechanical structures at low temperatures: Stress, elasticity, and nonlinear dissipation[END_REF][START_REF] Seitner | Damping of metallized bilayer nanomechanical resonators at room temperature[END_REF] that report the dependence of damping on the thickness of the metallic layer, pointing towards metal-dominated dissipation mechanisms. We will focus on this below. This is further supported by the fact that our intrinsic Q factor is about 600 000, hence almost four times lower than that of the device used in Ref. [START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF] which is 400 µm long (we expect a linear dependence of Q in length, see Ref. [START_REF] Quirin | Damping of Nanomechanical Resonators[END_REF]). For the time being, we see that the slope in temperature is rather weak, which explains why we do not see it by measuring at high amplitudes if we consider the Q.x 2 dependence itself as being temperature independent. Note that the extrapolation to zero temperature is such that σ δf (T = 4 K, x ∼ x s )/σ δf (T = 0, x ∼ x s ) ≈ 2, which again is close to what authors in Ref. [START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF] have obtained.

While the temperature dependence had already been reported in Ref. [START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF], the nonlinear dependence in the motion amplitude we observe has not been observed so far to our knowledge. The question now is: where does it come from ? The dependence in the Q factor could originate from the bifurcation process: indeed, the spinode point in the (x, ω) plane writes x s = 1 3 1/4 Γ/β. Therefore, the prefactor in the quadratic dependence could be linked to the mechanical energy at the spinode point ∝ x 2 s . If so, the quadratic dependence should only appear for displacements above the spinode point, which makes the t shown in Fig. 6.7 meaningless for x → 0. Fig. 6.9 shows an attempt to plot the data rescaled to the spinode point energy on the X axis, with the Allan deviation extrapolated for the spinode subtracted (what is called here x s ∼ 0). The data for all Q factors are rather well captured by the t, and we see that all data taken while sweeping on the lower branch are gathered at the origin. ). The green solid line is a simple linear curve of slope 0.13 Hz. Inset: same graph for the 15 µm device, with a slope 0.7 Hz.

It is nonetheless rather hard to imagine a mechanism linked to the bifurcation process that could amplify frequency uctuations. It is thus quite natural to proceed by assuming that bifurcation is ideal (i.e. occurring truly instantaneously within our resolution), and that the extra noise source has to be sought in the bifurcation position expression ω bif -ω 0 ∝ βx 2 bif . This means that either β or x bif has to be the noisy variable. We can rule out β, which is ∝ E/ω 0 , since a Gaussian noise on the Young's modulus E having the observed signatures (mode-dependent, proportional to Γ -1 ) seems unphysical.

Writing ω bif -ω 0 ∝ βx 2 bif × (1 + δx bif /x bif ), we realize that the right scaling is obtained if δx bif /x bif = -δΓ/Γ , i.e. if the eect is produced by amplitude uctuations generated by a uctuating damping rate. To be quantitative, we recall that [START_REF] Dykman | Theory of uctuational transitions between stable states of a nonlinear oscillator[END_REF]:

                     x bif,↑ ≈ x max , x bif,↓ ≈ x max Γ 4β 1/3 , x max = F 0 mω 0 Γ , (6.10)
Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems which leads to:

ω bif,↑ ≈ ω 0 + δω + βx 2 max 1 - δΓ Γ , (6.11
)

ω bif,↓ ≈ ω 0 + δω + 3βx 2 max 1 + o δΓ Γ , (6.12
)

ω s ≈ ω 0 + δω + √ 3Γ 2 1 + δΓ Γ . (6.13)
We thus justify why the lower bifurcation point is amplitude independent, and why the upper branch leads to a βx 2 max /Γ scaling of the additional noise: this shall be understood as an amplication of the eect of the damping uctuations on the postition of ω bif,↑ , which at the spinodal point add up with actual frequency uctuations through Eq. (6.13). For our measurements, we are always far from the spinodal point, thus Eqs. (6.11) for the upper branch and (6.12) for the lower one do apply instead of Eq. 6.13. Their extrapolations at x max → 0 do not match the spinodal expression, which is why we quoted x s ∼ 0 in the graphs of Figs. 6.8 and 6.9. As a nal result, the damping noise is simply √ 3/2 times the slope extracted from the linear t of the data in Fig. 6.9.

The nonlinear dependence was also measured on the third and fth modes of the 250 µm beam. The quadratic behaviour is again observed on both, again with osets.

The tted frequency noise parameters are reported for the three modes in Table 6.1.

We nd out that damping uctuations are roughly 2.5 times smaller than frequency uctuations, regardless of the intrinsic value of Γ . This means that for our highest Q devices Q factor uctuations can be as large as 5 %, while for most devices they are of order 0.1 % or smaller. Similarly, for our best devices the frequency noise σ δf represents up to 20 % of the resonance width Γ/(2π). This means that in frequency domain measurements, frequency noise can contribute to decoherence with a slight broadening of the resonance peak (see Chapter 4). While it is a small eect, the inuence of damping noise to decoherence remains always within our error bars: dening a T 2 from x max using x max = F 0 /(mω 0 Γ ) (see Chapter 4), we see that δΓ impacts it marginally (5 % at most). Similarly, in a time domain measurement we can wite the averaged susceptibility in complex form:

χ a (t) = exp [iδωt] exp iβx 2 max (t) 2δΓ Γ t χ N L (t), (6.14)
with χ N L (t) the non-linear, deterministic susceptibility in time domain, and dened in Eq. (4.37) as the time-domain non-linear oscillation factor. Assuming the statistics is Gaussian on a decay timescale on both noises, the two averages can be calculated explicitly:

χ a (t) = exp - σ 2 f t 2 2 exp -βx 2 max (t) σ δΓ Γ 2 t 2 χ N L (t), (6.15)
the uctuations being taken in the inhomogeneous broadening limit (see Chapters 3 and 4). This last point seems licit to us, as the uctuations observed have a The values measured at the lowest displacement are presumably the true intrinsic frequency noises. Indeed, the typical relative frequency noise σ δf /f 1 for the fundamental mode lies, at 4.2 K, in the 0.1 -0.01 ppm range, which is fairly consistent with values reported earlier in the literature. Some of these values are listed in Table 6.2 for comparison.

However, no damping uctuations have been reported to our knowledge, and the value quoted for the goalpost device is recalculated from Ref. [START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF]. Interestingly, the ratio of frequency noise to damping noise seems to be also around a factor of 3 for this device, hinting that this ratio exists regardless of the dierences in the resonator's nature (cantilever-type versus doubly-clamped) and materials (silicon versus silicon nitride). In addition, the spectrum measured is always of the form 1/ω 1+µ with no change in µ for the same device, regardless of the displacement amplitude and hence, of the uctuation type. Therefore, it is tempting to claim that uctuations in frequency [START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF][START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF][START_REF] Sansa | Frequency uctuations in silicon nanoresonators[END_REF]). Here d.c. stands for "doubly-clamped". For the Si/Al goalpost, the values of σ δΓ have been recalculated from Fig. IV.19 in Ref. [START_REF] Defoort | Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF].

and damping are linked through the same microscopic mechanism, even though more experimental insights are needed.

We are left with the question of the actual microscopic origin of the intrinsic fre-6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 187 quency noise (and damping) measured. The temperature dependence is not unexpected: a linear dependence in the range considered could be produced through a model featuring defects acting as thermally switching two-level systems [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF][START_REF] Cleland | Foundations of Nanomechanics[END_REF]. As these interact with a strain eld, they relax towards equilibrium, changing the local stress or the local Young's modulus as proposed in Ref. [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF]. Since we are dealing with highly-stressed beams, we do believe that for our structures (and the one of Ref.

[146]) it is more likely a stress uctuation that is generating frequency noise. Therefore, we could imagine that uctuations in the stored stress on long timescales (unlike dissipation, see below) result in the observed ∼ k B T /ω spectrum [START_REF] Dutta | Low-frequency uctuations in solids: $1/f$ noise[END_REF]. Note that the exponent is rather dierent from 1, ranging between 1.2 and 1.4, which poses questions on whether the mechanism at stake can be simply explained by TLS. The next sections are focused on the interaction of these two-level systems with strain elds, and measurements on dissipation and frequency noise at lower temperatures are meant to clarify their role.

6.3 Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM)

Structural two-level defects (TLS) in amorphous materials

We turn to analyze theoretically and experimentally microscopic sources of damping, that might also be responsible for frequency noise. For a start, it seems licit to rule out clamping losses, as they strongly depend on the beam's aspect ratio: at worst for our structures, a damping dominated by clamping losses [START_REF] Wilson-Rae | Intrinsic dissipation in nanomechanical resonators due to phonon tunneling[END_REF] would lead to Q factors in the 10 7 -10 8 range, while we do not measure, for the structures of interest here, Q factors higher than 10 6 . Once external contributions like gas damping [START_REF] Gazizulin | Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rareed Gas[END_REF] and high-order phonon processes leading to thermoelastic [START_REF] Lifshitz | Thermoelastic damping in micro-and nanomechanical systems[END_REF] or Akhiezer [START_REF] Cleland | Foundations of Nanomechanics[END_REF] damping are removed by lowering temperature and pressure, mechanical damping (and decoherence) at very low temperatures is thought to originate mostly from the interaction between the mechanical mode and Two-Level Systems. These low-energy excitations are described in the framework of the Standard Tunneling Model (STM). The STM was introduced in the early 70s by P. W. Anderson [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF] and W. A. Phillips [START_REF] Phillips | Tunneling states in amorphous solids[END_REF] to explain apparent anomalous features in thermal properties of amorphous solids at low temperatures [START_REF] Zeller | Thermal Conductivity and Specic Heat of Noncrystalline Solids[END_REF], and as it was found later, in their acoustic properties. The main assumption is that the disordered lattice typical of amorphous systems allows for its atoms (or some small group of atoms) to possess two nearly equivalent metastable position states separated by an energy barrier of height V 0 over a distance d and hav-Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems ing an asymmetry energy ∆. These atoms can go from one potential well to the other by quantum mechanical tunneling [START_REF] Phillips | Two-level states in glasses[END_REF] or thermal activation [START_REF] Faust | Signatures of two-level defects in the temperaturedependent damping of nanomechanical silicon nitride resonators[END_REF] as shown in Fig. 6.11. From then on, the acoustic (or mechanical) and thermal properties of amorphous solids are determined by the interaction between these TLS and phonons.

Potential energy

Configurational coordinate QT TA Figure 6.11: schematic representation of a two-level system with asymmetry ∆ and barrier height V 0 . The motion of the atom from one metastable well to the other is induced either by quantum tunneling (QT) or thermal activation (TA).

While the most salient signatures are now rather well known and characterized in bulk structures, their exact impact on nanomechanical damping is still poorly understood despite a broad range of studies, for instance Refs. [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Hoehne | Damping in high-frequency metallic nanomechanical resonators[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF][START_REF] Lulla | Dissipation and nonlinear eects in nanomechanical resonators at low temperatures[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF], to mention a few. This lack of consistency from one study to the other seems to originate partly from the additional complexity brought by dimensionality aspects. For instance, while TLS are usually treated as an energy continuum distribution, it is not clear whether this description still holds in the context of nanomechanics, where the structure is so tiny that only a small number of TLS are present [START_REF] Remus | Damping and decoherence of a nanomechanical resonator due to a few two-level systems[END_REF]. Furthermore, many experiments have been realized on crystalline samples (mono or poly), but the results are still treated within the STM; to be consistent, one has to invoke defects 6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 189 that act as TLS [START_REF] Hoehne | Damping in high-frequency metallic nanomechanical resonators[END_REF], as opposed to the constitutive TLS of glasses, or a modied version of the STM that is consistent with long-range order present in crystals [START_REF] Phillips | Comment on "Two-Level Systems Observed in the Mechanical Properties of Single-Crystal Silicon at Low Temperatures[END_REF].

In addition, the small size of NEMS devices also makes them hard to address with non invasive probes: for instance, our magnetomotive scheme involves a metallic layer with electrons that can also act as an additional bath for TLS [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF], increasing further the complexity of the problem. In this section we address the dissipation at very low temperatures of a 15 µm long nanomechanical resonator, and compare it with results obtained in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF] and on a 50 µm long sample with otherwise identical characteristics embedded in a microwave cavity for a new (microwave) optomechanics experiment. We treat here the quantum mechanical aspect of TLS, as thermal activation is exponentially suppressed with temperature and thus nearly irrelevant at T = 1 K or below.

Single TLS Hamiltonian

A generic TLS is pictured in Fig. 6.11 in the previous section. The tunneling amplitude is nonzero due to the overlap between the two potential wells wavefunctions and the nite barrier height. It is physically relevant to consider a nite overlap, as far as the spacing between two positions for an atom does not exceed a few Å. This overlap Ψ L |Ψ R is modelled to vanish exponentially in the Gamov factor λ. The latter is obtained within the WKB approximation as a function of the barrier height V 0 and the distance d, in the congurational coordinate space, between the two potential wells: λ = d √ 2m a V 0 / , with m a the mass of the tunneling entity. Thus, the tunneling amplitude between the two wells, i.e. their coupling energy, writes:

∆ 0 ≈ Ωe -λ . (6.16)
In the position space, the Hamiltonian of a single, isolated two-level system is then:

H 0 = 1 2 ∆ ∆ 0 ∆ 0 -∆ = ∆ 2 σ z + ∆ 0 2 σ x , (6.17)
where we have introduced Pauli matrices σ x,z , which highlight the formal analogy with a 1/2-spin. Diagonalizing the Hamiltonian (6.17) is straightforward, and one obtains the two TLS energy eigenstates:

ε ± = ± 1 2 ∆ 2 + ∆ 2 0 . (6.18)
In the following, we will often refer to the energy spacing ε between the two eigenstates to characterize the TLS: Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems treatment that is usually disregarded in the literature which essentially applies the bulk, 3D case.

Relaxational TLS-phonon interaction

TLS energy modulation by an applied strain eld By denition, atomic positions will be perturbed by any externally applied strain modulation. Thus, the potential landscape seen by an individual TLS will be modied, which puts the TLS out of equilibrium as it modies its energy splitting. Since the wavelengths associated to a mechanical drive are much bigger than the distance between the two potential wells (which is typically in the Å range), at rst order the driving eld only modies the asymmetry, hence the energy splitting of the TLS. The TLS will then relax towards equilibrium by exchanging energy with its environment, e.g. thermal phonons in any materials [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF], or conduction electrons in metallic glasses [START_REF] Hunklinger | Chapter 3: Thermal and Elastic Anomalies in Glasses at Low Temperatures[END_REF].

We shall focus rst on phonon-driven relaxation of TLS, which is ubiquitous and thought to be responsible for a large part of our experimental ndings. Indeed, since the TLS returning to equilibrium involves tunneling, hence atomic motion in the disordered lattice, it modies the local stress. Apart from the driving eld, thermal phonon modes are assumed to be at equilibrium. Besides, phonon modes that are resonant with the TLS energy splittings involved are assumed to have wavelengths much bigger than the interwell spacing. In fact, this can be justied easily: at low temperatures, well below the Debye temperature, modes of wavevectors ∼ 1/d, which are at energies much bigger than k B T , are not thermally activated and thus do not interact with TLS. As a result, any strain eld E, at rst order, is a tiny linear perturbation of the TLS potential landscape: the potential barrier is almost unchanged, and strain only couples to the TLS asymmetry. Therefore, the interaction Hamiltonian is diagonal in the position basis (|Ψ L , |Ψ R ):

H ph,p int = - → γ • - → E σ z , (6.25)
where γ j = 1 2 ∂∆/∂E j , j = 1...6 (to match the denition of strain as a 6-vector component [START_REF] Cleland | Foundations of Nanomechanics[END_REF]) is the TLS asymmetry change per unit strain, i.e. the TLS-phonon interaction strength. Here we have kept the directional aspect of the coupling, due to non-zero transverse strain components in a real material with a non-zero Poisson coecient experiencing longitudinal strain.

The interaction Hamiltonian (6.25), treated as a weak perturbation, can be rewritten in the energy eigenbasis (| -ε/2 , |ε/2 ) ≡ (|g , |e ): 12: schematic representation of the relaxational interaction, detailed for one TLS: as the beam is undergoing a deformation when resonantly excited, the resulting local strain eld changes the interatomic potential and modulates the TLS asymmetry. The dots represent defects randomly distributed in the beam, the dashed line on the beam is the neutral axis, where the applied strain eld is zero. The TLS equilibrium conguration is the dashed potential, while the full line represents the strain-modulated potential.

H ph,ε int = - - → γ • - → E ε (∆ σ z + ∆ 0 σ x ) .
This Hamiltonian is the building block of the physical picture: it describes both the strain modulation of TLS and the relaxation of TLS to phonons via thermal strain elds.

Damping rate and frequency shift due to the relaxational interaction

The relaxational interaction can be divided into two parts, that are both described by the interaction Hamiltonian (6.26). It can be treated in a similar fashion to magnetic or dipolar moments interacting with an externally applied eld. We proceed through a semi-classical treatment that is close to that proposed by Philips [START_REF] Phillips | Two-level states in glasses[END_REF]. However here we 1) keep the vectorial nature of the strain eld and the couplings while taking into account macroscopic constraints such as small cross-dimensions or modal shapes [START_REF] Remus | Damping and decoherence of a nanomechanical resonator due to a few two-level systems[END_REF], and 2) lead the derivation to yield, at the end of the calculation, the damping rate and the frequency shift that are our measurable quantities. The quantum aspect of the treatment will reside in the calculation of the TLS relaxation time that appears in the TLS susceptibility, i.e. the change in polarization due to an applied strain eld.

We derive rst the response of a TLS to a classical strain eld that arises from our Laplace excitation, with a Zener-like model [START_REF] Zener | Elasticity and Anelasticity of Metals[END_REF]. This yields the power dissipated by the modulating strain eld, and this power can be linked to the NEMS damping. At rest, the TLS polarization p 0 = p e -p g , where p e,g are the states occupation probabilities, is easily obtained, as the TLS is in equilibrium with its thermal bath:

σ z 0 = tanh[ε/(2k B T )],
where σ z 0 = Tr e -H 0 /(k B T ) σ z /Z denotes the ensemble average at equilibrium, Z being the TLS partition function. First, the TLS energy is Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems modulated by a classical oscillating macroscopic strain eld -→ E = -→ E 0 cos(ωt). Therefore, the polarization p = p 0 + δp is shifted from its equilibrium by the applied strain eld as the energy levels are modulated: the associated change in energy in a time interval δt, written in the classical limit from Eq. (6.26), is:

δU = - ∆ ε - → γ • - → E ṗ δt. (6.27)
The evolution of the TLS polarization can be written in the relaxation time approximation, i.e. for strain perturbations that do not bring the system too far from thermal equilibrium:

ṗ = - p -p st τ rel , (6.28)
where p st is the steady-state polarization for a given strain -→ E , and τ rel is the characteristic time over which the out-of-equilibrium polarization goes back to the steady state value. The latter can be linked to the thermal equilibrium polarization, at rst order in the strain eld:

p st ≈ p 0 + ∂p 0 ∂∆ ∂∆ ∂ - → E • - → E = p 0 + - → γ • - → E k B T ∆ ε sech 2 ε 2k B T . (6.29)
Meanwhile, we have δp = ṗ since p 0 is time-independent. As the polarization does not follow instantly the strain oscillation, we decompose the change in polarization in an in-phase and quadrature component: δp = δp c cos(ωt) -δp s sin(ωt). Rewriting Eq. (6.28), we have two equations on (δp c , δp s ), and solving them yields the TLS susceptibility -

→ χ T LS dened through δp c + iδp s = - → χ T LS . - → E 0 :              δp c = 1 1 + ω 2 τ 2 rel • - → γ • - → E 0 k B T ∆ ε sech 2 ε 2k B T , δp s = - ωτ rel 1 + ω 2 τ 2 rel • - → γ • - → E 0 k B T ∆ ε sech 2 ε 2k B T . ( 6
= - → γ • - → E 0 2 k B T ∆ ε 2 sech 2 ε 2k B T ω [sin(ωt) cos(ωt) -ωτ rel cos 2 (ωt)] δt 1 + ω 2 τ 2 rel . (6.31)
From then on, we can write the power dissipated through the modulation of a single

P c = - → γ • - → E 0 2 2k B T ∆ ε 2 sech 2 ε 2k B T ω 2 τ rel 1 + ω 2 τ 2 rel . (6.32)
The last step is to sum this dissipated power over an ensemble of two-level systems that are distributed in our resonator. The distributions introduced in paragraph 6.3.3 assume that those are numerous enough so that they indeed feature a continuous distribution per energy bandwidth per unit volume. However, in a nanomechanical resonator, the dimensions reduction might be dramatic enough so that nite size effects dominate. There is no reason not to assume that TLS are randomly distributed, and the strain dependence makes the problem strongly dependent on local properties.

Thus we cannot make a quantitative guess with a model featuring a discrete ensemble of TLS. Therefore, we will keep a continuous distribution for the time being, keeping in mind that the size reduction might limit our accuracy. Introducing dimensionless variables u = ε/(k B T ), v = ∆/ε and using the distribution (6.24), the dissipated power per unit volume thus writes:

P mic,V = P 0 ω 2 - → γ • - → E 0 2 ∞ u min du 1 0 dv v 2 1 -v 2 sech 2 u 2 ωτ rel (u, v, T ) 1 + ω 2 τ 2 rel (u, v, T ) I(T )
. (6.33) This form is convenient: the prefactor does not depend on temperature, so that the dependence on temperature only comes from the TLS relaxation time τ rel . Keeping a quantity per unit volume is crucial here, as there is a dependence on the local strain prole which must be taken into account when integrating over dimensions. In our experiments, only the rst out-of-plane exural mode is excited. The resulting strain eld amplitude is then (written in the 6-vector form [START_REF] Cleland | Foundations of Nanomechanics[END_REF]):

- → E 0 = x 0 x t ∂ 2 Ψ(z) ∂z 2 (ν, ν, -1, 0, 0, ν), (6.34)
where ν is the Poisson's ratio (ν = 0.3 for silicon nitride, 0.34 for Aluminum, so we consider it uniform in our beam) and we have noted x t the transverse coordinate to avoid any confusion with the displacement amplitude. In addition, we have considered the possibility of cross-coupling TLS-strain terms by including shear. In our denition

x t = 0 along the neutral line of the beam. The product -→ γ • -→ E 0 2 can thus be written:

- → γ • - → E 0 = x 0 x t ∂ 2 Ψ(z) ∂z 2 [(γ x + γ y + γ s )ν -γ z ] , (6.35)
Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems where γ s is the coupling associated to the shear strain component of the exural mode, i.e. ∂∆/∂E 6 . Injecting the result in the power per unit volume (6.33) and integrating over the NEMS volume (for x t ranging from -e/2 to e/2), one obtains the power dissipated by the strain modulation of TLS energies:

P mic = [γ z -ν(γ x + γ y + γ s )] 2 we 3 12
P 0 ω x 2 0 2 I(T ) l/2 -l/2 ∂ 2 Ψ(z) ∂z 2 2 dz. (6.36)
This expression is obtained from microscopic arguments. It can be equated to the power dissipated per unit length of the NEMS over one oscillation cycle, written in a macroscopic fashion with the damping rate from the NEMS dynamics equation:

P mac = 1 2 mω 2 x 2 0 Γ rel , (6.37)
where we have introduced the damping rate Γ rel , that is, the contribution to the linewidth due to strain-modulated TLS relaxing to equilibrium. Equating the two expressions and identifying the moment of area I = we 3 /12 leads to a damping rate:

Γ rel = P 0 γ 2 I mω I(T ) l/2 -l/2 ∂ 2 Ψ(z) ∂z 2 2 dz, (6.38)
where γ = γ z -ν(γ x + γ y + γ z ) is an eective coupling parameter. This is formally equivalent to the formula proposed in Ref. [START_REF] Quirin | Damping of Nanomechanical Resonators[END_REF] where the authors introduce ad hoc a retarded (imaginary in frequency domain) Young's modulus E 2 , whose microscopic origin is addressed in our work. For our pre-stressed beam taken in the analytic string limit, it can be further simplied since for the fundamental mode Ψ(z) = cos(πz/l) and m = ρewl/2 = M/2:

Γ rel (T ) = ω 3 P 0 γ 2 I ρewc 4 f I(T ) = C ω I(T ), (6.39)
with c f = σ 0 /ρ the sound speed for our high-stress beam exural modes, σ 0 being the in-built stress. We have used the fact that our exural modes have a linear dispersion relation in the string limit, i.e. ω = πc f /l for ω close to the fundamental exure resonance. The constant C is dened so in order to make the parallel with the usual C derived for bulk amorphous solids. We can now predict the two limit behaviours expected for the damping. A TLS relaxation time should depend on its energy conguration and temperature, since the latter is characteristic of a bath into which a TLS relaxes energy. In the most canonical case, e.g. a bath of phonons, environmental degrees of freedom are switched o as temperature is decreased, resulting in an increase of the relaxation time with decreasing temperature. Since we have a 6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 197 nite number of TLS, there must be at least one with a minimal relaxation time. In the limit where ωτ rel,min 1, the ensemble of TLS never reaches thermal equilibrium.

As a result, TLS do not have time to exchange energy with their surroundings, which limits their ability to rearrange the local atomic conguration. Therefore, the resulting damping must decrease as we approach this limit, by decreasing temperature. In the limit where ωτ rel,min 1, the integral I(T ) can be evaluated analytically for simple expressions of the TLS relaxation time. The latter will be derived after. In the "slow" limit ωτ rel 1, the TLS have time to relax to equilibrium in between successive strain increments. Therefore the relaxation time should not be relevant anymore, and indeed, I(T ) can be evaluated analytically, so that the resulting damping rate is simply a constant in temperature, the so-called "plateau":

Γ rel = ωτ rel 1 πωC 2 . (6.40)
Practically, the temperature above which the damping is constant is always found around 1 K in the literature on sound attenuation in glasses, with a C constant in the 10 -4 -10 -3 range for almost all glasses [START_REF] Hunklinger | Chapter 3: Thermal and Elastic Anomalies in Glasses at Low Temperatures[END_REF]. This similarity found in many works has led the community to speculate about a possible unversality of the glassy behaviour at low temperatures. Note however, that the usual C constant to which it has been hitherto referred is dened as C bulk = P 0 γ 2 /(ρ c 2 l ), with c l the longitudinal sound speed (or transverse one, which is of the same order), because it is derived in the case of bulk matter. Our calculation highlights the role of dimensionality, such that the C constant for our high-stress beam is renormalized:

C beam = πe l 2 • E 12σ 0 C bulk , (6.41)
while in the beam limit it reduces to C beam = C bulk because of the dispersion relation 2.10 that essentially compensates the aspect ratio. Provided that the theory applies to our system and reasonably matches the experimental data, it shows that the TLS contribution to the damping rate in the plateau region, that is, for most experiments between 1 and 10 K, scales quadratically with the aspect ratio e/l, and is proportional to the ratio between Young's modulus and in-built stress. This is an important result, as it shows explicitly with a minimal set of assumptions the role of dimensionality jointly with in-built stress in the damping of nanomechanical resonators. In particular, we see that Γ ∝ C beam ω ∝ 1/ √ σ 0 : in-built stress reduces the damping in the highstress limit.

In the opposite regime of strain oscillation ωτ rel 1 where the TLS relax slowly compared to the strain modulation speed, the damping rate (6.39) writes: Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems

Γ rel = ωτ rel 1 ∞ u min du 1 0 dv v 2 1 -v 2 sech 2 u 2 Cτ -1 rel (u, v, T ). (6.42)
Meanwhile, the reactive contribution, i. e. the stored energy associated to the strain modulation of a TLS energy writes:

U (t) = - ∆ ε - → γ • - → E δp. (6.43)
Injecting the expression for δp in the case of an oscillating strain and integrating over an oscillation cycle, we have

U = - → γ • - → E 0 2 2k B T ∆ ε 2 sech 2 ε 2k B T 1 1 + ω 2 τ 2 rel , (6.44)
which is to be equated to δkx 2 0 /2, the change in potential energy due to the reactive contribution, where δk ≈ δω 0 /(2ω 0 )k 0 is a spring constant modulation due to the real part of the TLS susceptibility (i.e. the instantaneous stress change due to an oscillating modulation strain). In the end, after integration over all TLS and over the volume as we did for the dissipated power, the relative change in the NEMS resonance frequency writes in the string limit:

δω 0 ω 0 rel (T ) = ω 2 0 P 0 γ 2 I ρewc 4 f R(T ) = CR(T ), (6.45)
where R(T ) is dened analogously with I(T ), as the averaging over all TLS susceptibilities' real parts:

R(T ) = ∞ u min du 1 0 dv v 2 1 -v 2 sech 2 u 2 1 1 + ω 2 τ 2 rel (u, v) . ( 6.46)
Phonon-driven TLS relaxation rate in a nanostring at low temperatures

We now turn to a derivation of the TLS relaxation rate τ -1 rel . To do so we must make reasonable assumptions on the bath to which an out-of-equilibrium TLS relaxes energy. First we investigate phonon-driven relaxation: its role is rather well known [START_REF] Phillips | Tunneling states in amorphous solids[END_REF][START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF][START_REF] Phillips | Two-level states in glasses[END_REF], and in our case the metallic layer can be made superconducting at very low temperature, suppressing the electron bath. However, in our case, dimensionality plays a crucial role: indeed most studies consider bulk phonon modes, that are irrelevant in the case of a truly 1D beam [START_REF] Cleland | Foundations of Nanomechanics[END_REF] Populations p e , p g for the two states |g , |e follow standard rate equations in the limit of weak perturbation from equilibrium:

   ṗg = -τ -1 g→e p g + τ -1 e→g p e , ṗe = -τ -1 e→g p e + τ -1 g→e p g , (6.47)
with p e + p g = 1. The total rate τ -1 rel at which the TLS polarization p = p e -p g reaches equilibrium is the sum of both processes rates: τ

-1 rel = τ -1 e→g + τ -1
g→e . Since the equilibrium polarization writes p 0 = tanh[ε/(2k B T )], we have by solving the system (6.47) at equilibrium the detailed balance between the two rates (due to the possibility of spontaneous emission of a phonon), leading to: τ -1 rel = τ -1 e→g (1 + e -ε/(k B T ) ). Thus, we only have to derive the emission rate. Initially, phonon populations are in thermal equilibrium at temperature T . Therefore, each mode ω k,s (s corresponds to a given family: torsion, longitudinal or one exural) has a population determined by the Bose factor n k,s = exp

ω k,s k B T -1 -1
, and Fermi's Golden rule associated to an emission process writes:

τ -1 e→g = k,s 2π g, n k,s + 1 H ph,ε int e, n k,s 2 δ (ε -ω k,s ) . (6.48)
From reference books [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Cleland | Foundations of Nanomechanics[END_REF], one can write the longitudinal strain eld due to propagating phonons at thermal equilibrium in a simple second quantized form:

-

→ E k,l = 2ρVω k,l a † -k,l + a k,l ik l (-ν, -ν, 1, 0, 0, 0), (6.49)
where V is the NEMS volume. The a ( †) k,l operator is the annihilation (creation) operator for a phonon in the mode k l , such that n k,l = nk,l ≡ â † k,l âk,l . From the denitions of the strain operator (6.49) and Pauli matrices, the squared matrix element in (6.48) 200 Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems can be evaluated:

g, n k,l + 1 H ph,ε int e, n k,l 2 = k 2 l 2ρVω k,l γ 2 ∆ 2 0 ε 2 (n k,l + 1). (6.50)
Here we have assumed that γ was the parameter introduced above, which was then taking into account the shear strain component. However, this shear component has a negligible contribution if we consider TLS: it is unlikely to be large, as strain is directed mainly towards z. Thus we neglect it, as well as torsional modes which emerge from shear. Going to the continuum limit for phonon modes, one needs to introduce the phononic density of states (DOS), g( ω k,l ). Since phonon modes considered are those who match the TLS energy splitting by energy conservation, i.e. ω k,l = ε/ , the TLS relaxation rate is easily calculated:

τ -1 rel (ε, ∆ 0 ) = π∆ 2 0 γ 2 g(ε) 2 ρVεc 2 l coth ε 2k B T , (6.51)
where we have used ω k,l = c l k l . Thus, in a similar fashion to phononic specic heat, the phonon-driven relaxation rate of two-level systems is proportional to the phonon DOS, which becomes crucial when the system studied has small characteristic dimensions. In addition, the quadratic dependence on the tunneling amplitude highlights the genuine role played by quantum tunneling of the atom in the process.

As we said earlier, the dominant phonon wavelength is roughly comparable with the structure's cross-dimensions in the 0.1 -1 K range. In this situation, connement imposes a lower energy cuto on the available phonon states, and only the longitudinal polarization survives. For 1D phonons in the pure linear dispersion relation case, the Debye DOS is constant [START_REF] Ashcroft | Solid State Physics[END_REF]: g 1D (ε) = l/(πc), and therefore the TLS relaxation rate τ -1 rel ∝ ∆ 2 0 /ε instead of ∆ 2 0 ε in the case of 3D phonons. In reduced units:

τ -1 rel (u, v, T ) = γ 2 k B T 2 ρewc 3 l (1 -v 2 )u coth u 2 . (6.52)
In contrast to bulk structures where τ -1 rel ∝ T 3 , we obtain a relaxation rate that is linear in temperature, as a direct consequence of dimensionality reduction. Note also that the dependence in (u, v) is quite dierent. Thus, in one dimension, we can expect that the TLS contributing the most to the response are not the same as in 3D. A 2D plot of TLS contributions is shown in Fig. 6.13: we plot the real and imaginary part magnitudes of the TLS susceptibility normalized to the prefactor

-→ γ • -→ E 0 2k B T
, as a function of the TLS parameters in reduced units, at 10 K (i.e. ωτ rel 1) and 0.1 K (i.e. ωτ rel 1).

As expected, the contribution of large tunneling amplitude TLS is almost inexistant at high temperatures: the TLS who dominate are those with ∆ ∼ ε, simply because the TLS relaxation to the bath, which destroys the TLS coherence, is too fast compared to the drive to favor TLS with large tunneling amplitudes. At low temperatures, roughly T < 1 K, TLS with signicant tunneling ∆ 0 ∼ 0.7 ε dominate the response. Again, this is expected, since this time strain modulation prevents TLS to relax to the bath as it is faster than relaxation processes, thus protecting the TLS quantum coherence.

However, we see also that low energy TLS ε < k B T are dominant in the relaxational response, in stark contrast to structures in the 3D phonon limit where dominant TLS are those whose energy is comparable to k B T [START_REF] Feerman | The Low Temperature Acoustic Properties Of Amorphous Silica And Polycrystalline Aluminum[END_REF].

Now that we have the explicit TLS relaxation rate 6.52, we can calculate analytically the damping rate of the NEMS (6.39) in the limit ωτ rel 1:

202 Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems

Γ rel (T ) = ωτ rel 1 π 2 γ 2 Ck B T 3 2 ρewc 3 l . (6.53)
We have not considered the contribution of exural modes in this model. With the same reasoning, however, and adapting the strain eld expression to exural modes [START_REF] Remus | Damping and decoherence of a nanomechanical resonator due to a few two-level systems[END_REF], a suppelementary ε 2 dependence appears in the relaxation rate (6.51), and thus one is left with a T 3 dependence which is masked by the linear dependence (6.53) below 1 K, so we discard it: TLS relaxation to these modes is too slow compared to relaxation to longitudinal modes. Furthermore, the macroscopic mode which is addressed (exure) couples mostly through the strain eld along the z direction. It is rather reasonable to imagine that the family of TLS sensitive to these deformations is not reacting much to shear. We can thus assume γ s ≈ 0 and discard the torsional family of phononic modes in our discussion (which produce only shear [START_REF] Cleland | Foundations of Nanomechanics[END_REF]).

Meanwhile, the frequency shift is suppressed for T 1 K, and exhibits a logarithmic dependence for T 1 K:

δω 0 ω 0 rel (T ) = 0 for ωτ rel 1, = -C ln T T 0 for ωτ rel 1, (6.54)
with T 0 a reference temperature in the limit ωτ rel 1, that is, typically above 1 K, and ω 0 ≡ ω 0 (T 0 ) the corresponding reference frequency.

Resonant TLS-phonon interaction

Assuming that TLS are numerous enough to feature a continuous distribution, it follows that a collection of TLS have a splitting that matches the energy of a phonon from the excited mode at frequency ω close to ω 0 (ε = ω), and thus resonant scattering of phonons by TLS has to be considered. The description of the resonant process is more straightforward than the relaxational one: the applied phonons are directly scattered by TLS at equilibrium through resonant absorption and emission, and we did not consider any eect of dimensional connement on the distribution of TLS as soon as it is taken continuous. The number of phonons n ph scattered by the whole ensemble of TLS per unit volume evolves as:

ṅph,V (ω) = - ∞ ε min ε ∆ 0,min p e τ -1 em -p g τ -1 abs δ(ε -ω)P (ε, ∆ 0 )dεd∆ 0 . (6.55)
The absorption and emission rates τ -1 abs , τ -1 em are again calculated with Fermi's Golden 6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 203 rule, see Eq. (6.50), except this time we sum over TLS states instead of phonon ones, and the phonon number is not the thermal equilibrium one but the number of applied phonons. In addition, we have to consider here phonons from the excited exural mode instead of the longitudinal ones. The strain eld amplitude for the excited mode, in a second quantized fashion, writes [START_REF] Remus | Damping and decoherence of a nanomechanical resonator due to a few two-level systems[END_REF]:

- → E 0 = 2mω 0 a † + a x t ∂ 2 Ψ(z) ∂z 2 (ν, ν, -1, 0, 0, ν). (6.56)
Knowing the expression of the strain eld, we can apply Fermi's Golden Rule to obtain τ em and τ abs . In the end after integration over the volume we have the following evolution for the phonon number:

ṅph = - ∞ ε min ε ∆ 0,min πω 3 γ 2 I ρewc 4 f (n ph p 0 -p e ) ∆ 0 ε 2 δ(ε -ω)P (ε, ∆ 0 )dεd∆ 0 . (6.57)
The solution to this equation yields a decay rate that writes:

τ -1 res = ∞ ε min ε ∆ 0,min πω 3 γ 2 I ρewc 4 f p 0 ∆ 0 ε 2 δ(ε -ω)P (ε, ∆ 0 )dεd∆ 0 . (6.58)
As pointed out earlier, the equilibrium TLS polarization is p 0 = tanh ε 2k B T . Integrating over the tunneling energies is straightforward and can be taken easily in the ∆ 0,min → 0 limit. That also permits to extend the other integral to ε min → 0, which vanishes with the Dirac function that imposes ε = ω. In the end, the damping rate is directly the phonon relaxation time:

τ -1 res = πωC tanh ω 2k B T .

(6.59)

In the limit where ω k B T it is negligible. Yet the change in the sound speed (the instantaneous modulus) is obtained through the Kramers-Kroenig relation [START_REF] Phillips | Two-level states in glasses[END_REF]:

δc f = c f π +∞ 0 τ -1 res (ω ) ω 2 -ω 2 dω , (6.60)
Since there is an integration over all frequencies, the change in the sound speed is not necessarily negligible. Finally, the frequency shift (i.e. the change in the sound speed) relative to a reference frequency at an arbitrary temperature T 0 , for temperatures such that k B T ω, writes:

δω 0 ω 0 res (T ) = C ln T T 0 , (6.61)
204 Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems with again T 0 the reference temperature. Interestingly, the resonant and relaxational contribution to the frequency shift cancel each other above 1 K, leading to a zero net contribution. At very low temperatures however, the resonant contribution should be the only one left, and its observation a clear signature of a TLS-like behaviour.

Note that the C constant appears as a single t parameter for the two dierent processes: as a slope for the frequency shift and the plateau value of the damping.

Therefore the values found should be reasonably similar to conrm the theory. At the same time, the damping rate on the NEMS at low temperatures, Eq. (6.53) involves C, known macroscopic mechanical parameters and the coupling strength γ. By tting damping and shift in the dierent regimes, it is thus possible in ne to extract the density of TLS P 0 and the eective coupling strength γ. This is what we shall discuss now on the basis of our experiments.

Results and discussion

Measurement protocol

The experimental procedure is simple. For a given temperature T , we measure the resonance frequency and mechanical damping using the magnetomotive technique, and subtract rst the capacitive and resistive loading contributions (see Chapter 2

Section 2.6.3). A few precautions need to be taken: rstly, we measured in both normal and superconducting states of the metallic layer by tuning the magnetic eld (see Fig. 6.14), searching for signatures of normal-state electrons contribution to the damping, following Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. The magnetic eld is parallel to the thin metallic lm, leading to high enough critical applied magnetic elds so we can safely investigate the superconducting regime for temperatures below 1 K. The critical temperature of the Al thin lm is found to be T c = 1.4 ± 0.05 K, typical of thin lms whose critical temperature is usually higher than for bulk Al [START_REF] Townsend | Superconducting Behavior of Thin Films and Small Particles of Aluminum[END_REF] where it is 1.2 K [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF]. Incidentally, the temperature was monitored by two thermometers anchored to the mixing chamber plate of the cryostat with a regulation loop so as to ensure that our measurements were made at a stable temperature.

In the superconducting state, a damping rate non-linear in the displacement amplitude is observed, as reported in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. We do not focus on this feature here, which might arise from a departure from the small strain perturbation limit used in our modelling and calls for further investigations. Thus, we measure a resonance with a decreasing driving force, and in case the non-linear dependence is signicant, we extrapolate the behaviour at zero force. In particular, additional non-linearities distorting the resonance line come into play below 100 mK, that might originate from vortices unpinned in the metallic layer (thin lm Aluminum is in practice a type-II superconductor) [START_REF] Brandt | Drastic increases of frequency and damping of a superconducting vibrating reed in a longitudinal magnetic eld[END_REF]. They are rather well captured by a phenomenological Dung- like non-linearity, explained in greater detail in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. These non-linearities are not addressed here. We simply measure the response down to the lowest force possible to obtain a reliable extrapolation, using our calibrations to convert the resonance amplitude to a displacement and thus to a damping rate through x max = F 0 mω 0 Γ . We use the 15 µm beam because its cross-over temperature between the plateau regime at high temperatures and the power law one at low temperatures is higher than for, say, the 250 µm device used for frequency noise measurements. Thus, the investigation range for TLS behaviour is larger in temperature.

Results for the damping

With these precautions we can reach a resolution of 10 % on the extrapolated NEMS damping rate and frequency shifts in the superconducting state. The result for the damping rate is shown in Fig. 6.15, for both normal and superconducting state, down to 30 mK. Below this temperature, the signal is not clear enough to obtain reliable information.

At high temperatures (T 2 K) the damping rate exhibits a small linear dependence, that was measured already for all our devices and is attributed to friction at grain boundaries. The slope is very small (∼ around 3 K at 850 ± 25 Hz which we think is the plateau value (6.53) due to the contribution from all TLS relaxing faster to the environment than the strain variation: as temperature is increased, their interaction with the environment is stronger so they relax faster. Below 2 K, the damping rate decreases rather quickly as the NEMS is cooled down. Below T c = 1.4 K and down to about T ≈ 800 mK, it follows a rather weak power law Γ ∝ T 0.5±0.05 for both normal and superconducting states, i.e. a sublinear dependence that is qualitatively close to results previously obtained in the literature for NEMS at least partially metallic [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF]. Note, however, that these results were obtained with unstressed devices, which might limit our comparison as far as relaxation to phonons is concerned: indeed, it is argued in Ref. [START_REF] Seoánez | Surface dissipation in nanoelectromechanical systems: Unied description with the standard tunneling model and eects of metallic electrodes[END_REF] that exural modes with quadratic dispersion relations instead of linear (our case, because of high in-built stress) dominate over the other modes in the TLS relaxation, which ultimately leads to a T 1/2 damping rate.

Results and discussion 207

Nevertheless, below 800 mK, the damping rate in the superconducting state departs from its normal state counterpart, following instead a linear (Γ ∝ T 1.06±0.1 ) law down to 30 mK. Meanwhile the normal state damping follows the same law as above, down to 200 mK where it saturates around 250 Hz. The damping in the superconducting state is rather close to our expectations: its temperature dependence compares well qualitatively with the expression (6.39) derived above, as well as with the results obtained on a similar sample (dimension-wise), made of superconducting Aluminum in Ref. [START_REF] Sulkko | Strong Gate Coupling of High-Q Nanomechanical Resonators[END_REF]. Note that we do not reproduce the T 3/2 law reported in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF] for a goalpost cantilever structure: if we allow ourselves to speculate a little, such a feature might owe to the goalpost peculiar geometry and modal decomposition [START_REF] Collin | Modal Decomposition in Goalpost Micro/Nano Electro-Mechanical Devices[END_REF], combined with the fact that according to Ref. [START_REF] Seoánez | Surface dissipation in nanoelectromechanical systems: Unied description with the standard tunneling model and eects of metallic electrodes[END_REF] exural modes with quadratic dispersion relations (as in the goalpost cantilever or unstressed beam cases) should dominate the phonon-driven relaxation of TLS, leading to non-integer power laws of temperatures. One can remove the tiny slope at high temperature so as to leave only what is thought to be the damping due to TLS relaxing. The result is shown in Fig. 6.16.

If we take as a given the plateau value predicted by the STM at 625 Hz, obtained 208 Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems from frequency shift measurements (see Fig. 6.18 in the next section, C = 2.3 × 10 -5 should be the same for the plateau and the frequency shift below 1 K), we observe a discrepancy between the STM model prediction (yellow curve in Fig. 6.16) and the measured plateau value at 850 Hz, of about 25 %. One can argue that the plateau is measured for temperatures at which the aluminum layer cannot be superconducting, and that a contribution from electrons might renormalize the expected plateau value.

This possibility is discussed later. We t the data to the expression (6.39) with the 1D phonon-driven relaxation rate (6.52) with a coupling strength γ = 10 eV and a TLS density P 0 = 1.9 × 10 44 J -1 .m -3 , thus assuming that the plateau value that truly counts with respect to our modelling is the one in agreement with the slope of the frequency shift, while the measured value depends on electron-mediated mechanisms (see our discussion in the next section).

A comparison can be made with a 50 µm NEMS mounted for microwave optomechanical experiments and measured with our collaborators Xin Zhou, Rasul Gazizulin and Dylan Cattiaux. Except for its length and its read-out system (a superconducting microwave LC circuit capacitively coupled to the NEMS thermomechanical motion [START_REF] Teufel | Dynamical Backaction of Microwave Fields on a Nanomechanical Oscillator[END_REF]), its mechanical characteristics are identical to the 15 µm device. Due to the experiment conguration, it was not possible to measure the device at higher temperatures than 500 mK, as quasiparticles start to alter the microwave cavity functioning above this temperature. For the same reason, we cannot measure the damping rate in the normal state because a magnetic eld would also destroy the cavity superconductivity which is necessary. Thus, we only have the data in the superconducting regime in the slow TLS limit, where it initially follows the linear slope but rather quickly curves:

if we take as a plateau value the constant extracted from frequency measurements (see Fig. 6.18 inset in the following section), however, we see that this is perfectly consistent with the tunneling model if we take roughly the same strain-TLS coupling constant as for the 15 µm, γ = 8.5 eV but increase the density of TLS by almost one order of magnitude P 0 = 1.2 × 10 45 J -1 .m -3 . Considering the discrepancy usually reported on this parameter, which spreads over a couple of orders of magnitude, this seems acceptable: the two beams have not been realized at the same time, and did not undergo the same process. The 50 µm higher density of TLS might as well be explained by all the additional steps that were required to fabricate an embedded sample.

Besides, it might be that with submicronic structures we truly reach the regime where the discrete nature of the TLS distribution come into play: in that case, it makes a description in terms of P 0 rather inaccurate, as there is not enough volume to really consider a signicant number of TLS so as to form a continuous distribution. Usual densities reported for P 0 ∼ 10 44 J -1 .m -3 are in the 10 19 -10 20 m -3 range, [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Hamoumi | Microscopic nanomechanical dissipation in gallium arsenide resonators[END_REF], which leads to an estimate of the number of TLS as small as ∼ 10 -100 in our beams (there is only roughly a factor of 3 in length between our two beams). Such an estimate poses a limitation to really quantitative modelling, as this number is likely to vary a lot in relative units, depending on the materials and the fabrication process.

Results and discussion

209

In 1D, the cross-over from fast to slow TLS (relative to the drive frequency) happens to be rather smooth, which explains why the data remain still a factor of 2 below the predicted plateau. Despite this, they are well captured by the t, and deep enough in the superconducting regime so we do not see any quasiparticle eect that might lead to a disagreement with the model as opposed to the case of the 15 µm beam.

The discrepancy between normal and superconducting state damping, on the other hand, is qualitatively consistent with previous measurements reported in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. In addition, we report a similar power law as in this work and that of Refs. [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF],

where measurements on metallic beams were made in the normal state. One notable exception is the work of Ref. [START_REF] Hoehne | Damping in high-frequency metallic nanomechanical resonators[END_REF], where no sublinear dependence was observed in the damping, falling linearly even in presence of normal-state electrons. However, the authors raised the possibility that kinks on dislocations were more likely candidates than TLS in their polycrystalline materials, leading to dierent relaxation mechanisms.

This exception left aside, the similarity between several measurements in rather dierent conditions (dierent metals, partially or fully metallic beam, bending and string limit) but all having in common normal-state electrons, added to the comparison with the superconducting state in our case, points towards an electron-assisted mechanism rather than what Ref. [START_REF] Seoánez | Surface dissipation in nanoelectromechanical systems: Unied description with the standard tunneling model and eects of metallic electrodes[END_REF] proposes, i.e. a pure phononic relaxation leading to a sublinear temperature dependence. This is further supported by the fact that no difference between the two states is observed above 800 mK: it seems natural insofar as in the superconducting regime, the fraction of electronic quasi-particles becomes dominant for temperatures above T c /2 [START_REF] Tinkham | Introduction to Superconductivity[END_REF]. Electron-driven mechanisms are discussed later in this section.

The saturation observed in the normal state at 200 mK and below is not specic to our measurements: it has been measured also in Refs. [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. However, in the case of Ref. [START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF], this saturation occurs at lower temperatures, around 30 mK. One should notice that in this work care was taken to lter both injection and detection lines, which is not the case here or in Refs. [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. Thus, a possible explanation is simple thermal decoupling from the cryostat temperature due to electromagnetic radiation carried out by unltered injection and detection lines, which heats up electrons in the metallic layer. Since at such temperatures phonons and electrons are still well coupled, it means that the environment temperature, whether it is phonons or electrons for the TLS, remains at 200 mK and progressively gets decoupled from base temperature as the latter is lowered.

Results for the frequency shift

The resonance frequency is also systematically measured as a function of temperature, both in normal and superconducting state for the 15 µm device. Again, all known extra-contributions (Dung-type non-linearities, magnetomotive loading) were removed. The result is shown in Fig. 6.17. Above 1 K, the resonance shifts downwards Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems quickly, as already measured for all our samples (see Fig. 2.18). We t the data to our usual aT + bT 3 law (see Chapter 2 section 2.7): in this limit, the frequency shift seems only to originate from the interface mismatch between Al and SiN generating dierential stress. The fact that no other contribution is observed above 1 K is consistent with our modelling, which predicts that the resonant (6.61) and the relaxational contribution (6.45) of the STM should cancel each other in this limit.

.17: resonance frequency as a function of temperature for the 15 µm device, with the same color conventions as in Fig. 6.15. The dashed line is a t of the type aT + bT 3 .

Below 1 K, however, we observe the usual logarithmic dependence. Note that this behaviour is rather weak and is not directly observable in the superconducting state because of non-linear features [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF], which explains the rather large error bars.

From 1 K down to 200 mK, in contrast with damping data, there is no discrepancy between the normal and superconducting states, as observed in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF]. This seems reasonable, as the dominant process, i.e. resonant, is dierent in nature from the relaxational one which dominates the damping: in the resonant interaction picture, the applied phonons are directly scattered by TLS, without any intervention of the electrons, and thus the frequency shift should not depend on the electronic state since the relaxational interaction contribution (6.54) is essentially zero in this temperature range. Below 200 mK the normal state shift departs from the superconducting one.

However, the deviation is much "softer" than the kink observed when saturation occurs for the damping data. It indicates that the mechanism behind the observed saturation might be a true physical eect, and not a simple thermal decoupling. This calls for further work with proper shielding from electromagnetic radiation, to decide between The t to the data using Eq. (6.61) for the logarithmic slope yields a parameter C = 2.3 ± 0.3 × 10 -5 . This is a little low compared to the usually expected glassy range plateau, which situates it more in the 10 -3 -10 -4 range. However, with the aspect ratio renormalisation of C developed in our modelling, it makes more sense, and compares well with the values obtained for nanomechanical resonators in Refs. [START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF].

It leads through Eq. (6.41) to a comparative bulk C constant in the 10 -4 range, which is very reasonable with respect to the hypothesis of the universality of glasses.

For the 50 µm long NEMS, it is also reasonable, as it leads through renormalization to a corresponding C bulk ∼ 10 -3 , still well in the glassy range. For a TLS density P 0 ∼ 10 44 J -1 .m -3 in accordance with literature [START_REF] Feerman | The Low Temperature Acoustic Properties Of Amorphous Silica And Polycrystalline Aluminum[END_REF], we assume that the TLS-strain coupling constant γ is in the 10 eV range, that is, one order of magnitude higher than usually reported constants. Note that such a high value has been recently reported for GaAs optomechanical structures [START_REF] Hamoumi | Microscopic nanomechanical dissipation in gallium arsenide resonators[END_REF]. Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems 6.4.4 Discussion on the contribution of normal state electrons

Even though the role of normal state electrons is qualitatively evidenced, the microscopic mechanism remains elusive. One rst striking fact is that this eect is only visible in such manner (discrepancy between normal and superconducting state + sublinear in normal state) for sub-micronic devices: experiments performed on micron-size structures [START_REF] Feerman | Elastic properties of polycrystalline Al and Ag lms down to 6 mK[END_REF] and macroscopic vibrating wires [START_REF] König | Tunneling systems in polycrystalline metals: Absence of electron-assisted relaxation[END_REF] did not reveal such a dierence.

In Ref. [START_REF] Haust | Low Temperature Acoustic Properties of Poly-Crystalline Aluminium[END_REF], a discrepancy between normal state and superconducting frequency shifts was reported for a vibrating Al reed, though nothing could be concluded on the inuence over the damping due to the presence of large eddy current damping.

The question of the interplay between electrons and two level systems in metallic glasses has attracted a consequent number of studies ever since the beginning of the Standard Tunneling Model as a realistic theory to explain low temperature properties of amorphous solids. Notably, sound attenuation experiments [START_REF] Hunklinger | Chapter 3: Thermal and Elastic Anomalies in Glasses at Low Temperatures[END_REF] have established that for bulk metallic glasses at low temperature, the temperature dependence of the TLS relaxation rate was much weaker than that observed for the same experiments in dielectric materials where they were explained by the usual TLS-phonon coupling.

However, this discrepancy is not expected in our experiments, as the NEMS cross dimensions, below 1 K, are smaller than the dominant phonon wavelength, thus leading to much weaker dependences in T of the TLS relaxation rate (see section above).

Besides, the Al lm structure is more polycrystalline than amorphous, though it is claimed in Ref. [START_REF] Esquinazi | Acoustic properties of amorphous SiO2 and PdSiCu, and of crystalline Ag, NbTi and Ta at very low temperatures[END_REF] that it is irrelevant. We can assume that our measurements in the superconducting state of the metallic layer below T c /2 give us access to a pure phononic contribution to the TLS relaxation, as the Cooper pair condensate is not scattered by TLS [START_REF] Black | Inuence of the Superconducting State upon the Low-Temperature Properties of Metallic Glasses[END_REF] and the quasiparticles density is small. Note that in principle, electron-driven relaxation is irrelevant at high temperatures since the damping does not depend anymore on the TLS relaxation times. However, it is impossible, with our data, to nd a quantitative agreement between the slope of the frequency shift and the plateau value to more than 25 % (see Fig. 6.16). One can argue that in the plateau region, we cannot have access to a pure phononic contribution as the layer is not superconducting, which means that the role of electrons might extend to more than a simple change in the TLS relaxation rate.

A simple microscopic picture can describe the basic electron-driven relaxation of TLS: while tunneling, a TLS changes the local atomic conguration, which causes a local variation of the Coulomb atomic potential that scatters conduction electrons:

this can be seen as scattering of the electrons by TLS mediated by virtual phonons, resulting in the following eective Hamiltonian in the position basis, which takes into account the electron spin s:

H el,p int = k,q,s V z k,q σ z + V x k,q σ x c † k,s c q,s , k,s . Note that this interaction conserves the electron spin, ruling out at rst order mechanisms of magnetic origin to explain our observations. Considering only conduction electrons in a narrow band on the surface of the Fermi sea, since the TLS have energies ε F , we can approximate V x,z k,q ≈ V x,z . Since the Fermi wavelength ∼ 1 -10 nm is smaller than the aluminium layer cross-dimensions, the electronic density of states n is assumed to be 3D. As our working temperature is much lower than the Fermi temperature T F ∼ 10 5 K for aluminium, we assume that the electronic DOS n e is slowly varying in the vicinity of the Fermi-Dirac distribution. In other words, n e (ε) ≈ n e (ε F ) ∝ √ ε F .

Therefore, the wavelengths at stake are this time a few Å at most, so at rst glance there is no reason to neglect electron-assisted tunneling, i.e. the σ x term in the coupling Hamiltonian (6.62). Note that in principle, the tunneling amplitude is due to the two-well wavefunctions overlap, which is by denition small: thus, we do not expect electron-assisted tunneling to play a signicant role in the TLS relaxation to electrons but keep it in the calculation of the TLS-electron bath relaxation rate.

a) b)

Figure 6.19: diagrams of the rst order processes leading to electron-TLS energy exchange as calculated in Eq. (6.63).

Using again lowest order perturbation theory, Fermi's Golden Rule provides a relaxation rate for weak electron-TLS coupling, accounting for processes depicted in Fig.

6.19:

τ -1 el (ε) = 2π k,q,s e, N q,s + 1, N k,s -1 H el,ε int g, N q,s , N k,s 2 δ (ε -ε k + ε q ) + g, N q,s + 1, N k,s -1 H el,ε int e, N q,s , N k,s 2 δ (ε -ε q + ε k ) , (6.63) where H el,ε int is the interaction Hamiltonian with the fermionic bath re-written in the

f (ε k ) = 1/ 1 + exp ε k -ε F k B T
. Summing over all possible initial (nal) states |k, s (|q, s ) allowed by energy conservation and taking into account spin degeneracy, we obtain: (6.64) where h(ε) is an overlap integral of Fermi-Dirac distributions obtained by summing over all nal electron states of energies ε q = ε k -ε (or ε k + ε if energy is yielded by the TLS rather than absorbed):

τ -1 el (ε) = 4πn 2 e (ε F )V 2 (∆ 0 /ε) 2 |V z | 2 + 1 -(∆ 0 /ε) 2 |V x | 2 × [h(ε) + h(-ε)] ,
h(±ε) = +∞ 0 f (ε k ) [1 -f (ε k ± ε)] dε k . (6.65)
This integral is analytic, and in the limit ε F k B T , the electron relaxation rate yields:

τ -1 el (ε) = 4πn 2 e (ε F )V 2 (∆ 2 0 /ε)|V z | 2 + ε -(∆ 2 0 /ε) |V x | 2 coth ε 2k B T . (6.66)
This result is rather disappointing: writing τ -1 el in reduced units makes a linear de- pendence in temperature appear, which is no dierent from phonon-driven relaxation.

Therefore, it is not enough to explain our T 1/2 dependence.

However, this is only true if we sum over the usual distribution of TLS energies. It might as well be that the collection of TLS that eectively contribute to the electron-TLS interaction is not the usual distribution. Theoretical attempts have been made to explain anomalous sound attenuation in metallic glasses [START_REF] Coppersmith | Low-temperature acoustic properties of metallic glasses[END_REF], which invoke a renormalisation of the TLS distribution due to second-order perturbation contributions in the electron-TLS interaction, which appear to be strong in most experiments. The argument, developed by Kondo [START_REF] Kondo | Two-Level Systems in Metals[END_REF], is that the second order term of the electron-TLS interaction depends logarithmically on the ratio between the conduction bandwidth and thermal energy k B T . Therefore, at low temperature (analogous to temperatures below a Kondo temperature for resistivity), there is no reason to neglect this secondorder term, leading to qualitative features in stark contrast with what we obtained through Eq. (6.66). In Ref. [START_REF] Coppersmith | Low-temperature acoustic properties of metallic glasses[END_REF], a calculation based on dissipative tunneling theory [START_REF] Leggett | Dynamics of the dissipative two-state system[END_REF] leads the authors to propose the following explanation: the strong coupling of a TLS to the fermionic bath leads to a renormalisation of the tunneling amplitudes ∆ 0 → ∆ r = ∆ 0 (∆ 0 /D) K/(1-K) , where D is a high-energy cut-o and K ∝ n e (ε F )|V z | 2 is a normalized electron-TLS coupling strength. This renormalization aects substantially low-energy TLS, whose energy lies within the electron conduction band. In the end the TLS relaxation time, and thus the damping scales as:

Γ rel ∝ T 1-2K for ωτ rel 1, = πCω 2(1 -K)

for ωτ rel 1, (6.67) This is valid in the incoherent tunneling regime, that is, as long as Kk B T /∆ r 1,

which is always true in our case. Thus, the strong interaction between electrons and TLS would not only renormalize the low-temperature damping rate, but it would also aect the plateau value (see the data and the black horizontal solid line in Fig. 6.16).

The K constant is left as a free parameter, and we nd out that both the plateau and the damping rate in the normal state are well captured for K = 0.25, as shown in Fig. 6.16. This is consistent with the fact that the normal state and superconducting state give equal damping rates above 800 mK, because the distribution of TLS is renormalized the same way independently of the relaxation channel. Indeed, above T c /2 the quasiparticles density is signicant enough to ensure that the distribution is renormalized by the strong electron-TLS coupling. Then it leads also to a renormalized damping rate for TLS-phonon coupling since it involves a sum over the TLS distribution. In that case it would explain why such a discrepancy is observed only for low dimensional structures: we have seen in section 6.3.5 that if the phonon bath is reduced to one dimension, the TLS that are most aected by energy relaxation to the bath are the low energy ones ε k B T , that is, the ones more likely to be aected by a strong coupling to electrons. In short, the mechanism we propose is not directly electron-driven relaxation; it is rather the relaxation to phonons of TLS with a distribution renormalized by the TLS interaction with electrons, which somehow screen the TLS seen by the phonon bath, in analogy with conduction electrons spins screening the spin of a magnetic impurity in the Kondo problem. Note that discrepancies in the K constant, which is material-dependent, might explain the variety of sublinear laws observed in the literature, between 0.3 and 0.7 [START_REF] Zolfagharkhani | Quantum friction in nanomechanical oscillators at millikelvin temperatures[END_REF][START_REF] Venkatesan | Dissipation due to tunneling two-level systems in gold nanomechanical resonators[END_REF][START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF].

Discussion on the TLS location

While the modelling seems robust, and the results reveal many signatures of a TLSlike behaviour, the coupling strength extracted is not very satisfying. Besides, our calculation assumed for simplicity that TLS were uniformly distributed spatially in the beam. This last assumption, for our actual beams, might well be wrong, because of the beam's bilayer nature: while a TLS model seems licit if we address the dissipation arising from the SiN amorphous layer, it is more dicult to justify in the metallic layer because of the latter's polycrystalline structure. Yet, the discrepancy we observe between normal state and superconducting damping clearly favours TLS present in the metallic layer, as the nitride is electrically insulating. In Ref. [START_REF] Phillips | Comment on "Two-Level Systems Observed in the Mechanical Properties of Single-Crystal Silicon at Low Temperatures[END_REF] it is argued that Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems TLS-like behaviour might still emerge in a crystal, though the excitations would be very narrowly distributed in energy around zero due to the long-range order present in a crystal. This so-called Phillips model leads to a linear temperature dependence of the TLS relaxation rate with a logarithmic relative change in frequency, which would be in agreement with our measurements and the results of Ref. [START_REF] Sulkko | Strong Gate Coupling of High-Q Nanomechanical Resonators[END_REF] where the beam was metallized. However, this is inconsistent with the observation of Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF] where both the metallic layer and the insulating one were crystalline, with the same deposition method as the present work, and where a damping ∝ T 3/2 was observed.

In the end, a reasonable assumption would be that TLS be located in the oxide Al 2 O 3 layer that inevitably grows after a few minutes on the aluminum layer [173]. As this occurs essentially on the surface, the TLS that form in this oxide matrix couple much more to the strain eld, which is larger on the surface (that is, the furthest from the neutral axis) according to Eq. (6.34). Besides, the surface TLS can show rather dierent natures than their bulk counterpart: they can be e.g. nanovoids due to the deposition process. Such a dierence might explain the dierence in coupling strength.

As our modelling shows, the low-dimensional character of the beams makes them much more sensitive to the surface contribution, while the volume contribution is smeared out: this is somewhat the spirit of the renormalisation on C presented in Eq. 6.41: if the structure was truly bulky, the C constant would not likely lie in the 10 -5 range. This is because the volume contribution would dominate over the surface one, since the strain eld would gradually be reduced to the bulk modes, leading to enhanced couplings to bulk TLS rather than surface ones and reducing to the usual plateau constant of glasses. Note that the oxide layer explanation also reconciles qualitatively our measurements in the superconducting state with those obtained in Ref. [START_REF] Lulla | Evidence for the Role of Normal-State Electrons in Nanoelectromechanical Damping Mechanisms at Very Low Temperatures[END_REF], as it is certain that the same oxide was present. It is also consistent with the electronic relaxation scenario, as the oxide layer is only a few atomic layers thick above the conducting aluminum layer, and maybe not uniform.

Link to frequency noise

The last part of this chapter shows our experimental attempt to demonstrate the link between TLS features in the nanomechanical damping and the frequency noise of nanomechanical structures at low temperatures. Using the technique described earlier in this Chapter to measure frequency noise, by systematically counting bifurcation events on the lower branch in order to be sensitive only to actual frequency noise, we measure the frequency uctuations of our device in an unprecedented way, separating normal state data from superconducting ones. The spectrum for this sample scales as 1/ω 1.2 regardless of the electronic layer state. The result for the frequency noise magnitude, plotted in terms of Allan deviation to bypass non-stationary contributions (see the rst section of this Chapter), is shown in Fig. 6.20, along with damping uctuations measurements that also reveal a discrepancy (see Fig. 6.20 inset). Once again, a clear discrepancy is visible between the frequency noise in the normal state and in the superconducting one, appearing again around T ≈ 700 -800 mK.

Besides, the frequency noise saturates in the superconducting state while spectacularly increasing when we go down to 100 mK as T -1/2 when the metallic layer is normal. Below 100 mK data are not shown: in the superconducting state, the additional nonlinearity pollutes the signal, while in the normal state, the current required to be strongly bistable heats too much the device. One fact is particularly striking: the solid lines used as ts are mere tentatives, and if we take the measured damping rates and their ts for both normal and superconducting state as a given, the law used to t the frequency noise data simply scales as Γ/T for both regimes, without any adjustment. We thus strongly believe that the observed excess frequency noise in the normal state has the same microscopic origin as the normal state damping.

Qualitatively, one can explain frequency uctuations due to tunneling TLS: the latter are in equilibrium with a phonon bath, and these phonons are atomic displacements, thus causing strain eld uctuations, translated as TLS polarization uctua-Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems tions by the TLS susceptibility (6.30), and this noise causes in turn uctuations in the stored stress of the mechanical mode, that is, in its resonance frequency. Through the uctuation-dissipation relation (3.3), the spectrum of polarization uctuations of a TLS is directly linked to the imaginary part of the TLS susceptibility, that is responsible for the NEMS damping. We measure very low frequency uctuations in the mHz -Hz range, that is, frequencies that are well within the regime ωτ rel 1. In this limit the imaginary part of the TLS susceptibility is a constant, as seen through Eq. (6.40). It results through the uctuation-dissipation theorem in a spectrum of polarization uctuations, and thus, frequency uctuations, scaling as 1/ω in the low frequency range.

Conclusion and prospects

Summary of the results

We have presented a technique adapted from Ref. [START_REF] Aldridge | Noise-Enabled Precision Measurements of a Dung Nanomechanical Resonator[END_REF] based on the bifurcation phenomenon for a NEMS in the bistable regime in order to measure frequency noise with an arbitrary precision. We have used it to extract Allan deviations that are reliable estimates of the frequency noise magnitude, and found out it was quadratically dependent on the motion amplitude, for two very dierent samples. This can be explained by the properties of the bifurcation points: on the high amplitude branch bifurcation, we believe that uctuations of the damping rate are amplied, masking the true frequency noise. Operating on the low amplitude branch, we are insensitive to these damping uctuations, thus truly measuring the frequency noise. We have then used it to measure the temperature dependence of the frequency noise from 1 K to 25 K, which is linear with an oset. The results are consistent with the ongoing literature on nanomechanical frequency noise at low temperatures. It agrees qualitatively with a model of thermally switching two-level-systems, which predicts a linear temperature dependence [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF][START_REF] Dutta | Low-frequency uctuations in solids: $1/f$ noise[END_REF]. To investigate further this possibility, we have measured the NEMS damping from 30 mK to 20 K, where two-level systems that tunnel quantum mechanically instead of thermally switching are thought to be a dominant source of nanomechanical damping. We have derived a semi-quantitative model that takes into account the dimensionality of the beam, the role of internal stress as well as its modal decomposition, showing that the usual 3D model (with respect to phonons) of tunneling TLS is not accurate when describing the interaction of a truly highly pre-stressed nanomechanical beam at low temperatures with its constitutive TLS. Our measurements of damping rate and frequency shift are rather well explained by this model when the metallic layer is superconducting. In particular, the shift at low temperatures is undoubtedly logarithmic, providing a strong signature of the role of TLS.

The results are compared with another device of similar characteristics except for its length, yielding again a rather good agreement with our model. However, the model 6.5. Conclusion and prospects 219 fails to explain the damping observed in the normal state that is similar to that in Ref.

[28], even with the standard electron-driven relaxation model of TLS. A qualitative argument which invokes second-order processes in the electron-TLS interaction, leading to renormalized TLS distributions and relaxation rates, might explain our ndings, even though more theoretical work is required to get deeper into quantitative considerations. We nally use the technique developed to measure frequency noise in the superconducting and normal state regimes of the metallic layer: again, a discrepancy is observed between the two, starting from 800 mK where the fraction of quasiparticles in the superconducting regime begins to signicantly decrease towards low temperatures.

Tentative ts show that the magnitude of the frequency noise scales as Γ/T for both superconducting and normal state regimes. Thus we believe it is a good evidence of the role of tunneling TLS in the internal frequency noise of NEMS at low temperature.

It makes the link with what is observed at higher temperatures up to 30 K and more in Ref. [START_REF] King | Frequency and phase noise of ultrahigh q silicon nitride nanomechanical resonators[END_REF], where TLS that thermally switch between states instead of tunneling are thought to be the dominant contribution.

Prospects

The large dependence of the Standard Tunneling Model on materials-dependent properties makes it uneasy to study in a fully quantitative way. However, to elude the major problem of renormalization, our setup might be adapted so the NEMS is controlled capacitively [START_REF] Quirin | Universal transduction scheme for nanomechanical systems based on dielectric forces[END_REF], with no need for a metallic layer otherwise mandatory for magnetomotive measurements. This would be convenient for another reason: indeed, an issue that remains unsolved beyond educated guesses is the location of TLS. While we know that surface TLS are more involved due to larger stress at the surface of the beam, it is not clear whether these TLS are more in the metal (oxidized) surface or the nitride one. Pure silicon nitride is expected to give high Q factors [START_REF] Quirin | Damping of Nanomechanical Resonators[END_REF], and we do not measure so high values, which makes us think, complementarily to [START_REF] Collin | Metallic coatings of microelectromechanical structures at low temperatures: Stress, elasticity, and nonlinear dissipation[END_REF][START_REF] Seitner | Damping of metallized bilayer nanomechanical resonators at room temperature[END_REF], that the TLS that contribute the most are hosted in the metallic layer. Measurements on bare SiN samples might help to clarify the discussion. Note that the problem is not specic to silicon nitride, and that experimentally, the best way to address it is to painfully acquire statistics on the low-temperature damping properties of uniform beams made of dierent materials. If the results were strikingly dierent, it would then pose the very interesting question of the reality of TLS found in metallized beams, which are not supposed to exist in a polycrystal in their usual form [START_REF] Hoehne | Damping in high-frequency metallic nanomechanical resonators[END_REF] and with their usually assumed distribution [START_REF] Phillips | Comment on "Two-Level Systems Observed in the Mechanical Properties of Single-Crystal Silicon at Low Temperatures[END_REF].

Besides, the question of renormalization by normal state electrons is still bound to qualitative considerations only, and clearly requires some theoretical input, in link with Kondo physics. In particular, it is interesting to clarify the role of dimensionality in the renormalization procedure, as it seems that this discrepancy was observed only for resonators with submicronic cross-dimensions. Note that this model does not explain nano-mechanical systems the saturation observed at 200 mK in the normal state, which cannot yet be explained for certain with thermal decoupling. Other models of electron-mediated interactions could be considered. It might be for instance that the electrons in the normal state introduce correlations between TLS that are conned in the nano-string, in analogy with the RKKY interaction. More experiments are thus clearly required. Besides, most experiments at low temperatures are easily performed at 4 K, where not all BCS superconductors are below their critical temperature. Thus, understanding the interplay between electrons, phonons and TLS is also of practical interest.

Main achievements

This thesis was concerned with behaviours in nano-mechanical systems that arise from uctuations or disorder and their interplay with non-linearities. On the purely macroscopic aspects, we have demonstrated an example of spectral diusion that is nontrivial, arising from the presence of both Brownian motion and non-linearities: its observable eects are more or less important depending on whether the noise correlations are kept slow or not. We have showed the limitations it can put on sensing applications through the derivation of a fundamental limit to frequency resolution. We have demonstrated that additional complexity arises from the application of a strong driving, with Brownian motion of the driven motion that becomes squeezed and slowed down. In complement, we have performed time and frequency resolved measurements at the mechanical resonance with a similar but externally created frequency noise to model the dierence between damping and decoherence in a framework inspired by NMR.

Concerning microscopic aspects, we have demonstrated, when the NEMS is surrounded by a rareed gas, the existence of a boundary (Knudsen) layer near the sample wall, spread over a distance comparable with the atoms mean free path. We have shown that it is the result of a deviation to the well-known Maxwell-Boltzmann equilibrium distribution in the bulk, due to diusive scattering of gas particles at the sample wall. This deviation is shown to lead to a rarefaction of the gas near the wall, which results in a renormalized, reduced damping of the NEMS (up to almost a factor of 10), validating its use as a non-invasive probe for a medium in conned geometries and paving the way towards studies of the intimate behaviour of quantum uids. Finally, we have developed a new technique of measuring intrinsic frequency noise which relies on the bifurcation phenomenon (made possible through the Dung Chapter 7. General Conclusion non-linearity). We applied it to the characterization of our nitride beams, and discussed the results in the framework of the Two-Level Systems model also invoked to explain low-temperature mechanical damping. The latter was measured so as to assess the validity of the Standard Tunneling Model at low temperatures. We found additional evidence of the role of stress, structure dimensionality, and of the electron bath in TLS mediated dissipation, and extended these measurements to frequency noise. A qualitative explanation involving renormalisation of the TLS distribution seen by the phonon bath was proposed.

Outlook

These results are performed in a purely classical fashion: neither is the system quantum, nor is its measurement apparatus, unlike e.g. in hybrid opto-and electromechanical systems [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF][START_REF] Arcizet | A single nitrogen-vacancy defect coupled to a nanomechanical oscillator[END_REF][START_REF] Yeo | Strain-mediated coupling in a quantum dotmechanical oscillator hybrid system[END_REF]. We currently work on a more demanding experiment that aims at cooling down to the quantum regime a high frequency nanomechanical beam coupled to a quantum-limited superconducting microwave cavity. While this has been achieved already for almost a decade [START_REF] O'connell | Quantum ground state and single-phonon control of a mechanical resonator[END_REF][START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF], it has never been performed for a MHz range NEMS having a good Q factor, without active cooling. Here we intend to benet from ultra-low temperature techniques to cool down the sample, by using nuclear adiabatic demagnetization [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF] of a copper stage to which the sample is thermally anchored. At the lowest temperatures, such that ω m ∼ k B T , quantum eects start to play a role in the dynamics of the resonator: the thermal occupation becomes vanishingly small as temperature is lowered, and only the mode's quantum uctuations remain. In this limit the classical treatment exposed in the above sections fails to describe the behaviour of the resonator. The macroscopic collective variable that is used to describe its dynamics is that of a true quantum harmonic oscillator. It represents an exciting endeavour insofar as almost nothing is known, beyond speculation, on macroscopic quantum coherence for a mechanical degree of freedom.

It will require further developments allowing the coupling of the resonator to e.g. a quantum bit, in order to generate entanglement and witness decoherence of quantum states of motion [START_REF] Armour | Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box[END_REF].

Even though the results are not reported in this manuscript, a microwave optomechanical setup embedding a 50 µm long NEMS resonating at 4 MHz has been designed in our team. It has been cooled down passively and operated successfully down to 10 mK. Results already obtained in the eld, such as the use of dynamical back-action of the micro-wave eld on the nanomechanical beam to cool down or heat up the mechanical mode [START_REF] Teufel | Dynamical Backaction of Microwave Fields on a Nanomechanical Oscillator[END_REF], or the opto-mechanically induced transparency observed for optomechanical systems [START_REF] Weis | Optomechanically Induced Transparency[END_REF], have been reproduced. Besides, the data in dissipation and frequency shift, that are interpreted in the framework exposed in Chapter 6, are very convincing proofs that the NEMS thermalizes to the dilution base 7.2. Outlook 223 temperature, which make this device a good example of phononic thermometer when its coating is superconducting. It bypasses saturation problems encountered with electronics-based thermometers [START_REF] Meschke | Electron Thermalization in Metallic Islands Probed by Coulomb Blockade Thermometry[END_REF], where electron-phonon decoupling is a crucial bottleneck. Nuclear demagnetization has been performed with a base temperature of 700 µK reached. While the results are encouraging, the analysis is more involved, which is why no colder attempt has been made on this setup. In particular, the NEMS thermal decoupling is a highly non-trivial problem in the 1-10 mK temperature range, and nothing is really known about the thermalization of a NEMS below 1 mK. Our fabrication process make 50 MHz NEMS attainable, so it is vital that the NEMS thermalize around 1 mK at least in order to reach the quantum regime of the mechanical resonator.

  -electro-mechanical systems: sensors and versatile platforms for basic research . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Context and structure of this thesis . . . . . . . . . . . . . . . 4 1.1 Nano-electro-mechanical systems: sensors and versatile platforms for basic research

Figure 2 . 1 :

 21 Figure 2.1: left: global picture of a typical nanofabricated sample measured in this thesis.Each pad is annoted according to the bonding scheme. Right: zoom on the central part, with a Scanning Electron Microscope picture of the NEMS. The beam is here 10 µm long, with a gate electrode designed in its vicinity (see Section 2.8.3). Note the spongy silicon bottom, which is a consequence of XeF 2 etching.

Chapter 2 .Figure 2 . 2 :

 222 Figure 2.2: the initial wafer, made of bulk silicon with a thin lm of stressed, amorphous silicon nitride, is coated with a PMMA resist, made uniform by spinning and baking.

Figure 2 . 3 :

 23 Figure 2.3: schematics of the electron beam lithography step, with darker red representing the insulated parts of the resist, removed after developing in MIBK/IPA.

Figure 2 . 4 :

 24 Figure 2.4: schematics of the evaporation and lift-o steps.

Figure 2 . 5 :

 25 Figure 2.5: left: reactive Ion Etching of the structure, selectively removing the exposed nitride layer. Right: Xenon Fluoride etching of the exposed silicon layer.

Figure 2 . 6 :

 26 Figure 2.6: tilted SEM picture of a nanofabricated NEMS of dimensions 15 µm long × 130 nm thick × 300 nm wide. The SiN thickness is 100 nm, with a 30 nm Aluminium on top.

2. 3 . 13 2. 3

 3133 Cryogenics and thermometry Cryogenics and thermometry 2.3.1 Measurements at liquid helium temperatures A signicant part of the results reported in this manuscript have been obtained at moderately low temperature on a standard liquid 4 He bath cryostat. Multiple reasons explain this choice, compared to room temperature measurements:

Figure 2 . 7 :

 27 Figure 2.7: picture of the sample holder, a copper rod connected to the upper ange of the cell. A typical chip, containing several samples, is glued on the nger.

Figure 2 . 8 :

 28 Figure 2.8: left: schematic basic operating principle of a dilution unit. Right: overview of the Bluefors dilution unit used for our experiments.

Figure 2 . 9 :

 29 Figure 2.9: coil setup, with the still screen on and a zoom on the cold nger inserted into the coil.

Figure 2 . 10 :

 210 Figure 2.10: schematic clamped-clamped (or doubly-clamped) beam, with oriented axes.

(2. 7 )

 7 This ansatz can be, by construction, a solution of Euler-Bernoulli equation for particular values of wave vectors λ set by boundary conditions. Applying boundary conditions (2.3) to the ansatz displacement eld, we obtain the condition on λ, which is a transcendental equation: cos(λ n l) cosh(λ n l) = 1.

Figure 2 . 11 :

 211 Figure 2.11: functions cos(λ n l) (blue) and 1/ cosh(λ n l) (yellow). Eigenmodes correspond to the intersection points.

Figure 2 .

 2 Figure 2.13: generic schematic picture of the wiring. Depending on the experiments, arrangements are made, which will be detailed in the appropriate chapters.

Figure 2 .

 2 Figure 2.14: schematic representation of the magnetomotive actuation and detection principle, for the rst exural mode.

Figure 2 .

 2 Figure 2.15: electro-mechanical in-phase (blue dots) and in-quadrature (red dots) responses to a sinusoidal magnetomotive excitation swept in frequency around the resonance, as detected at the lock-in amplier level. Solid lines are ts from Eq. (2.22) in Volt units.

. 36 )

 36 As for the electrical environment, we can use the modelling of Section 2.6.1, represented with Norton convention in Fig.2.16.

Figure 2 . 16 :

 216 Figure 2.16: equivalent electrical circuit in Norton representation, with C l the capacitive part of the loading impedance.

  o n a n c e F r e q u e n c y ( H z ) M a g n e t i c F i e l d ( T )

Figure 2 .

 2 Figure 2.18: left: measured resonance frequency of a 300 µm NEMS as a function of the cell temperature. A t (dashed line) of the form a(T -T 0 ) + b(T -T 0 ) 3 is used as a guide for the eyes (see text). Right: Measured damping rate of the same device as a function of temperature, with a linear t as a guide for the eyes (note the weak slope).

Figure 2 . 19 :

 219 Figure 2.19: shift of a 300 µm long NEMS resonance frequency as a function of the 50 kHz heating current amplitude. The solid line is the application of Eq.(2.50), with a and b parameters determined above (see Fig. 2.18), left panel.

I n j e c t e d h e a t i n g c u r r e nR e s c a l e d h e a t i n g c u r r e nFigure 2 .

 2 Figure2.21: left: shift of the resonance frequency as a function of the injected heating current applied at dierent frequencies (see legend), without applying a rescaling factor. Right: shift of the resonance frequency as a function of the same heating currents, with a rescaling factor. For each heating current frequency, a dierent rescaling factor is applied, so that all the data collapse on the DC heating curve. The line is the same as in Fig.2.19.

(2. 53 )

 53 Knowing the injection factor |G I (ω)|, we can now express the actual force exerted on the beam in real units. And since we know the global factor as well, we also have a direct access to the detection coectient |G D (ω)|: |G D (ω)| = |G(ω)| |G I (ω)| .

Figure 2 . 22 :

 222 Figure 2.22: Injection losses as a function of frequency with a 300 µm long NEMS resonating at 591 kHz, with a resistance in the metallic layer R N = 1.1 kΩ. The data are obtained from the scaling factors used to produce Fig. 2.21, with corresponding colors and shapes for data points. The dashed line is a t from a RC circuit modelling, with a capacitance t parameter C = 550 pF.

Figure 2 . 23 :

 223 Figure 2.23: Typical linear response curve of a 300 µm NEMS. The red solid line is the application of Eq. (2.23), with a measured Q = 12800 and spring constant k = 0.41 N/m.Points at high amplitudes (≥ 50 nm) are taken in the nonlinear regime (see Section 2.9).

Chapter 2 .-

 2 Experimental and theoretical tools o n a n c e F r e q u e n c y ( H z ) D C G a t e V o l t a g e V 0 ( V ) O f f s e t V o l t a g e i d u a l F i t E r r o r ( H z ) D C G a t e V o l t a g e ( V )

Figure 2 . 25 :

 225 Figure 2.25: Behaviour of a local portion of the beam under deformation. The dotted line is the neutral axis, whose points only displace in the x direction [43].

Fig. 2 .

 2 Fig. 2.25 illustrates the elongation eect when considering a small portion of the beam: if one neglects shear eects, the neutral axis should be displaced only in the

Figure 2 . 27 :

 227 Figure 2.27: Typical shift of the resonance frequency due to the Dung interaction (same device as the previous gure). The red solid line is a t from Eq. (2.70), with an extrapolated zero-displacement frequency 707516 ± 5 Hz and a Dung coecient β = 1.33 mHz/nm 2 .

  l i t u d e ( n m ) D r i v e f r e q u e n c y ( H z )

  m o d e f r e q u e n c y ( H z )

  v e f r e q u e n c y ( H z )

Figure 2 . 31 :

 231 Figure 2.31: typical resonance lines measured on a 300 µm NEMS in the parametric amplication (φ = -π/4, left panel) and squeezing (φ = +π/4, right panel) regimes, for the same pumping factor h, with resonant displacement amplied by a factor 4.75 and squeezed by a factor 1.75, respectively. Solid lines are ts from Eq. (2.79) with h = 0.77.

3. 7 . 1 62 Chapter 3 .

 71623 Beyond linearization . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.7.2 Slowing down the decay of position noise correlations near bifurcation points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.7.3 Asymmetric amplication of uctuations in the phase space . . . 100 3.8 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . 102 Interplay between non-linear dynamics and position uctuations 3.9 Résumé en français . . . . . . . . . . . . . . . . . . . . . . . . . 104

(3. 4 )

 4 On the other hand the imaginary part Im[χ(ω)] of the mechanical susceptibility in frequency domain corresponds to the retarded part of the susceptibility in time domain: thus it represents dissipation. Note that integration of the relation (3.3) leads to the equipartition theorem relation(3.2). It is also worth noticing that in the linear response regime one should have for the power spectrum:

  relation time τ b of any bath's degrees of freedom is assumed to be very small compared to the NEMS decay rate. Using the fact that Im[χ(ω)] = mωΓ |χ(ω)| 2 , or integrating Eq. (3.5), one obtains a quantitative expression for the spectrum of the equivalent Langevin force F th exerted by the bath constituents on the resonator, responsible for 3.3. Dispersive coupling to Brownian motion through geometric nonlinearities 65 thermal uctuations. This relation is often referred to as a second version of the FDT:

Figure 3 . 1 :

 31 Figure 3.1: Schematic spectrum of the n th mode frequency uctuations induced by nonlinear coupling to the m th mode Brownian motion.

3. 3 .Figure 3 . 2 :

 332 Figure 3.2: schematics of the frequency noise in the inhomogeneous broadening situation.

Chapter 3 . 1 FFigure 3 . 3 :

 3133 Figure 3.3: schematics of the frequency noise in the motional narrowing situation.

Figure 3 . 4 :

 34 Figure 3.4: left: phase accumulated through the nonlinear mechanism along dierent Brownian trajectories (inset) of the rst mode of the sample "250 µmn • 1". The thick black line is an average over these trajectories. Right: standard deviation around the average accumulated phase path. The red dashed line is a t ∝ √ t following diusion laws. The accumulated phase is shown in units of Brownian motion as measured at the lock-in level with the technique described in section 3.5.

FDT relation ( 3 . 6 )Figure 3 . 5 :

 3635 Figure 3.5: schematics of the experimental setup. Note that the sine wave injection can be a signal at the rst mode frequency, or the third one, or removed if we are only interested in the position uctuation spectrum.

Fig. 3 . 6 ,

 36 Fig. 3.6, left panel. Besides, as shown below, high eective temperatures (10 6 -10 10 K) are needed to witness e.g. inhomogeneous broadening. Reaching such eective temperatures requires large noise powers. Thus, to avoid overheating of the sample and selectively put the rst mode at high eective temperatures, a bandpass lter is inserted after the generator. The lter gain curve is shown in Fig. 3.6, right panel. Its integrated bandwidth is 0.7 MHz, with a maximum transmission factor 0.95.

Figure 3 . 6 :

 36 Figure 3.6: left: AWG output voltage noise histogram for an applied noise 1 V pp , with a Gaussian t. Right: amplitude gain of the bandpass lter. The dashed line corresponds to the resonance frequency of a 300 µm beam.

7

 7 75 15 Hz/m 2 n = 1 250 µm-n • 1 0.45 N/m 0.9 MHz 190 Hz β 1,1 = 8.5 10 15 Hz/m 2 n = 3 250 µm-n • 1 4.0 N/m 2.MHz 25 Hz β 3,1 = 1.6 10 16 Hz/m 2

2 . 4 .

 24 Results in the driven case 79 3.4.2 Measured resonance lines under nonlinearity-induced frequency uctuations The derived accumulated dephasing term (3.29) acts as a slow modulation of the time domain signal, both in amplitude and frequency. In frequency domain, the measured averaged response is then the standard frequency domain susceptibility, with a noisy eigenfrequency, convoluted with a complex-valued pseudo-distribution which is the Fourier transform G(ω) of the average accumulated phase term (3.29):

2 FFigure 3 . 7 :

 237 Figure 3.7: Fundamental mode resonance lines in phase (blue) and quadrature (red) measured with the device 300 µm n o 2 (see Table3.1), in the motional narrowing (left panel) and inhomogeneous broadening (right panel) regimes in the case of intra-mode, Dung mediated frequency uctuations, at a constant driving excitation. Solid lines are direct applications of Eq. (3.34). The grey line is a measurement of the standard response at the same drive tone for comparison, with complex Lorentzian ts (dashed lines).

3. 4 .

 4 Results in the driven case 81

Figure 3 . 8 :

 38 Figure 3.8: frequency shift (left) and linewidth broadening (right) of the fundamental mode resonance as a function of its own Brownian motion amplitude, with the two discussed regimes denoted. The solid lines are theoretical curves extracted numerically from Eq. (3.34), while dashed lines show the laws obtained in the motional narrowing regime (Eqs. (3.31) and (3.32) for shift and broadening respectively). Insets in both graphs are zooms on the data obtained in the motional narrowing regime. The pink line roughly marks the transition between the two regimes at α 1,1 = 1.

  p l . ( n m )

Figure 3 .

 3 Figure 3.10: frequency shift (left) and linewidth broadening (right) of the third mode resonance as a function of the fundamental mode Brownian motion extension, with the two discussed regimes denoted. The red solid lines are the parameters extracted numerically from Eq.(3.34), while dashed lines show the laws obtained in the motional narrowing regime.

3. 5 . 83 2

 583 Observation of non-linear position noise spectra

  3.13). Note that points taken at very high eective temperatures (∼ 10 8 -10 10 K) are already in the Dung regime concerning the spectra lineshape (see paragraph below).

3. 5 . 87 6

 587 Observation of non-linear position noise spectra

Figure 3 . 88 Chapter 3 .

 3883 Figure 3.13: integrated value of the Brownian motion spectrum in real units, with its corresponding eective mode temperature, as a function of the force noise spectrum in real units. The solid line follows the equipartition relation (3.2) even though the spectrum is distorted (see below), with mode parameters measured independently.

( 3 . 2 /Figure 3 .

 323 Figure 3.14: spectral peak height in raw units as a function of the magnetic eld. "Main" refers to the Brownian motion spectrum, "Background" is the sum of ohmic, capacitive and uncorrelated contributions, and "Cross term" is the contribution arising from the crosscoupling between the ohmic/capacitive noise and the Brownian motion. With a transmission line capacitance estimated at C eq = 500 pF, which is consistent with the disadapted cable length (a few meters), all tted contributions are captured by the model within 15 %. The blue solid, red solid and black dashed lines reproduce Eqs. (3.42),(3.43), and (3.44) respectively.

Figure 3 .

 3 Figure 3.15: height (left panel, normalized to the maximum value) and width (right panel, normalized to the NEMS linewidth) of the spectrum peak, as a function of the lock-in amplier bandwidth normalized to the NEMS linewidth, at 0.84 T (black), 0.6 T (blue), 0.42 T (green), and 0.3 T (red), hence for damping rates 140, 70, 36 and 20 Hz respectively. Solid lines are guides, while the dashed line shows the optimal bandwidth with respect to the electromechanical linewidth.

2 / 2 /

 22 Figure 3.16: position noise spectra of the fundamental mode obtained in the motional narrowing (left) and inhomogeneous broadening (right) regimes. Solid lines are the application of Eq. (3.37). The grey line is a spectrum obtained in the linear regime, for low enough eective temperatures so that the eect of Dung non-linearities can be neglected, while the dashed line is a Lorentzian t with the linewidth corresponding to that of the driven response.

  Figure 3.18: thermal frequency dressing (left) and broadening (right) for the fundamental exure, with units shown in gures, as a function of the dimensionless stress parameter u = σAL 2

3. 7 .Figure 3 . 19 :

 7319 Figure 3.19: amplitude at resonance of the rst mode sine-wave response experiencing a constant position noise ∆x 2 1 = 2.2 × 10 -15 m 2 (T eff = 7 × 10 7 K), as a function of the driving force amplitude, for the sample 250 µm n • 1. The black solid line is the linear drive prediction, while the dark yellow dashed line is a guide for the eyes. Inset: in-phase and quadrature responses for an applied force F 1 = 6.1 pN.

. 52 )

 52 To simplify, we introduce the variable Z(Ω) = Z(Ω)e -iφ 0 (Ω) , such that the steady state solution is real-valued. If now we introduce a small perturbation (created e.g. by the Langevin force g th ), we can linearize Eq. (3.51) in the uctuations around the steady-state solution. We introduce ZR = Re[ Z] and ZI = Im[ Z] the quadratures of the motion along the reference phase φ 0 , and allow them to uctuate (in practice, they do so because of the Langevin force): ZR = Z0 + δZ R and ZI = δZ I (since the steady state solution is real). Using the real and imaginary parts of Eq. (3.52) for the steady state solution multiplied by e -iφ 0 , and injecting in Eq. (3.51) one obtains the dynamics equation for the uctuations (δZ R , δZ I ):

  2 ], and we see clearly that Re[v 1 ] = Re[v 2 ] even for moderately distorted lines. This appears clearly in Fig.3.20: one relaxation rate increases while the other decreases. Let us mention that analogies can be made with Landau's theory of phase transitions[START_REF] Landau | Statistical Physics, Third Edition[END_REF]: for a given force close to the critical force (when considering a sweep in frequency) or a given frequency close to the spinodal point (when considering a force ramped through the critical force), the quartic Hamiltonian (3.7) in displacement is on the verge to allow a new stable motional state to develop. Therefore, according to Landau's theory of phase transitions we can expect uctuations near this critical point to diverge, while their dynamical timescales are slowed down, even though the analysis in terms of scaling laws and is unfortunately beyond the scope of this work and calls for further analysis.

(3. 55 ) 97 -

 5597 Therefore, in the eigenbasis, one quadrature can be discarded while the other has very slow timescales, thus reducing the physics to a simple one dimensional problem. Since the resonance frequency asymptotically merges with the upper bifurcation frequency at very large forces, the range where the timescales depart reduces as the force is increases, as seen in Fig.3.20. This timescale separation allows for instance 3.7. Anomalous uctuations for non-linear driving elds

Figure 3 . 98 Chapter 3 .

 3983 Figure3.20: mechanical responses in dimensionless units (see text, the dashed parts represent the unstable solution) plotted with relaxation rates -Re[v 1,2 ] in dimensionless units computed from Eq. (3.54) as a function of the reduced driving frequency, for reduced driving forces g 0 = 0.3 (weakly non-linear), g 0 = 1, g 0 = g c = 1.241 (critical force where the hysteresis opens), g 0 = 1.6 (strongly non-linear, bistable). On the last panel, arrows represent the frequency sweep direction.

Figure 3 .

 3 Figure 3.21: setup with a parallel opposition circuit used to cancel the resistive background (attenuators and lters not shown). The components used have values L op = 0.19 mH, R b,op = 1 kΩ, and the tunable capacitance can be adjusted between C op = 0 and 200 pF.The cable length is adjusted to optimize the noise cancellation. V 0 and δV in generate the driving force and noise respectively, while V is the detected voltage fed to the lock-in.

Figure 3 . 100 Chapter 3 .

 31003 Figure 3.22: left: noise correlator C YY (t) on the Y quadrature in raw units for a force F 0 = 17 pN close to the critical point and a phase +45 • compared to the reference phase yielding in-phase and quadrature responses. The blue line is an exponential t ∝ e -t/τ , with a comparison to the pure linear case (dashed red line). The inset shows the intercorrelation function C XY Right: relaxation times obtained on both quadratures correlators as a function of the driving frequency, for the same applied force. The dashed vertical line marks the spinodal point in amplitude, while the horizontal one marks the relaxation time in the linear regime.

3. 7 .Figure 3 .

 73 Figure 3.23: uctuating quadratures correlators for a force F 0 = 42 pN, with a driving frequency a few Hz detuned from ω bif,↑ . The phase is set at +45 • from the reference phase in which we observe the in-phase and quadrature responses. The dashed line is the exponential decay expected in the linear regime.

Figure 3 . 24 :

 324 Figure 3.24: uctuations in the quadratures space for a force F 0 = 42 pN > F c , with a driving frequency a few Hz detuned from ω bif,↑ . The phase is set at +45 • from the reference phase in which we observe the in-phase and quadrature responses. Squeezing is represented by the oval shape. Inset: cubic pseudo-potential representation in the 1D approximation.

102

 102

(Στ c 1 )

 1 , where uctuations correlations decay too fast to have a full impact on the probe mode's dynamics. The two limits are analytically encompassed in a path integral model of the susceptibility's expectation value. This model reproduces well the experimental data within 20 % at worst without free parameters.In addition, we have developed a measurement technique that allows us to obtain phase-resolved position noise spectra. It allowed us to observe the same signatures 3.8. Conclusion and prospects 103

  Dans ce chapitre nous montrons comment le bruit de position d'un mode mécanique, c'est-à-dire son mouvement Brownien, est transduit en bruit en fréquence sur la résonance du mode lui-même ainsi que des modes auxquels il est couplé, par le biais de la non-linéarité géométrique introduite dans le Chapitre précédent. Ce bruit en fréquence est basse fréquence, hautement structuré, non-Gaussien, et sa largeur spectrale est xée par le temps de relaxation du mode uctuant en position. Nous montrons que ce bruit induit une dynamique non-triviale du mode sondé subissant le bruit en fréquence : il y a une compétition entre le temps de corrélation (mémoire) ni du bruit en fréquence et la diusion de la phase du mode sonde (diusion spectrale). Nous mettons en évidence deux limites importantes. La première concerne des uctuations grandes, telles qu'un saut en fréquence provoqué par le bruit dans le référentiel tournant est échantillonné sur un temps susamment court pour que l'auto-corrélation du signal de saut en fréquence soit encore importante: cela provoque un décalage en fréquence de la raie (car le bruit en fréquence est de moyenne non nulle), et un élargissement inhomogène de la raie de résonance dû à la diusion spectrale. Dans le cas inverse, lorsque les corrélations du bruit s'eacent sur des temps très courts devant le temps nécessaire au mode sonde pour suivre le saut en fréquence, seule une partie des uctuations est eectivement intégrée par le mode sonde et la diusion spectrale est donc limitée : c'est le pénomène de rétrécissement par le mouvement, qu'on retrouve également sous une autre forme en Résonance Magnétique Nucléaire. Les deux limites sont capturées par une approche théorique reposant sur une intégrale de chemin introduite dans[START_REF] Zhang | Spectral eects of dispersive mode coupling in driven mesoscopic systems[END_REF]: la phase accumulée par le mode sonde est moyennée sur les diérentes réalisations de 3.9. Résumé en français 105 trajectoires Browniennes, en tenant compte de la relaxation du mode subissant un mouvement Brownien. L'expérience est réalisée sur plusieurs NEMS de tailles diérentes, et nous observons l'eet dans trois congurations essentielles: pour un couplage entre un mode Brownien et un mode sonde forcé diérents, dans le cas où le mode sous forçage est couplé à ses propres uctuations (via l'interaction de Dung) et dans le cas où les uctuations de position sont susamment grandes pour induire une distorsion du spectre de bruit de position lui-même. An de pouvoir observer de façon certaine l'eet du bruit de position tout en palliant la faible sensibilité de la mesure magnétomotive, nous avons soumis le mode sous mouvement Brownien à un bruit en force externe équivalent à une température de mode eective. Par ailleurs, nous avons développé une technique utilisant la détection synchrone pour mesurer la densité spectrale de bruit de position. Les résultats expérimentaux, reportés dans[START_REF] Maillet | Nonlinear frequency transduction of nanomechanical Brownian motion[END_REF], sont en bon accord avec l'approche théorique en intégrale de chemin dans tous les cas de gure, lorsque l'excitation sinusoïdale est susamment faible pour rester en régime linéaire.Si cette condition n'est pas remplie, nous observons un écart signicatif à la théorie, que nous pensons dû à l'action en retour de l'oscillation forcée sur les uctuations de position. Ce phénomène est étudié expérimentalement de façon qualitative dans le cadre de la théorie de la bifurcation dynamique, en lien avec les phénomènes critiques : nous observons en particulier un ralentissement de la dynamique du système ainsi qu'une compression asymétrique des uctuations de position près du point critique d'ouverture d'une hystérésis et près des points de bifurcation.Nous abordons également un aspect plus appliqué, qui concerne la limite xée sur la résolution en fréquence due au phénomène étudié: en eet, la non-linéarité géométrique ainsi que les uctuations thermiques de positions sont des caractéristiques intrinsèques au système. Ainsi, tout système nanomécanique subira les uctuations de fréquence telles que nous les avons décrites. Nous montrons par des estimations quantitatives que la contrainte interne dans le matériau permet de réduire l'eet, et que des structures à faible contrainte interne comme les nanotubes de carbone sont particulièrement sensible à ces uctuations, ce qui explique certaines observations d'élargissement spectral, qui invoquaient déjà le mécanisme de façon qualitative[START_REF] Eichler | Symmetry breaking in a mechanical resonator made from a carbon nanotube[END_REF].

  2), which has an intrinsic linewidth of 3.6 Hz, clean signals are obtained with a eld B = 100 mT, which yields a loaded linewidth Γ = 2π × 5.6 Hz (i.e. a decoherence time T 2 = 57 ms). Therefore, the mode remains mostly mechanical with respect to damping/dephasing mechanisms. This is crucial if one wants to unveil dephasing mechanisms of mechanical origin, for instance in the non-linear regime.

Figure 4 . 1 :

 41 Figure 4.1: schematics of the experimental setup. The DAQ card enables fast communication between the computer software and the generator, which permits to perform well resolved time domain measurements.

Figure 4 . 2 :

 42 Figure 4.2: power gain characteristic of the low-frequency lter used in this experiment.-3 dB cuto frequencies are displayed (blue and red dotted lines) with asymptots, while the dark green solid line is an interpolation function used for calculations.

(4. 7 )dt 1

 71 Each k term of the sum, i.e. k th moment of the dephasing, can be explicited: δΩ(t 1 )...δΩ(t k ) .

4. 2 .Figure 4 . 3 :

 243 Figure 4.3: averaged dephasing term ("blurring") in (4.20) for dierent noise magnitudes with a noise correlation time τ c = 0.15 s. Dashed lines are the corresponding Gaussian approximations, i.e. the average dephasing term in Eq. (4.28).

Figure 4 . 4 :

 44 Figure 4.4: resonance line (X quadrature) measured for an applied DC gate voltage V 0 = 1.37 V and a voltage noise σ g = 17.3 mV rms , hence a frequency noise magnitude σ = 4.12 rad/s. The red curve is a Gaussian t, while the blue curve is a Lorentzian one.

4 . 6 .

 46 Chapter 4. A model experimental approach to classical decoherence for mechanical systems p l i t u d e ( n m ) D r i v e F r e q u e n c y ( H z )

Figure 4 . 5 :

 45 Figure 4.5: resonance line (X quadrature) measured for an applied DC gate voltage V 0 = 5.29 V and a voltage noise σ g = 78 mV rms , hence a frequency noise magnitude σ = 72.4 rad/s.The red curve is a Gaussian t, while the blue curve is a Lorentzian one.

  a d e n i n g ( H z ) F r e q u e n c y n o i s e m a g n i t u d e ( r a d / s ) Γ/2

Figure 4 . 6 :

 46 Figure 4.6: broadening of the resonance line as a function of gate-controlled frequency noise.Blue dots are obtained from the FWHM of a Lorentzian t of the resonance line, red dots come from the FWHM of a Gaussian t. The error bars represent 10 % of the bare resonance linewidth. Dashed lines are the two asymptotic behaviours of the linewidth broadening, i.e. applications of Eq. (4.27) (blue) and Eq. (4.29) (red). The solid black line is the broadening extracted from the IB approximation, while the orange line comes from the full theory. The dashed vertical line is a guide marking the transition from weak renormalization to noisedominated response.

  (4.32) A numerical computation of S C (ω) is shown in Fig.4.7. The uctuations remain

Figure 4 . 7 :

 47 Figure 4.7: convolution part of the frequency uctuations power spectral density S c (ω), recalculated from the low-frequency lter gain characteristic, with colored area.

Figure 4 . 8 :

 48 Figure 4.8: protocol for time-domain measurements, with a frequency response in the left panel as a guide, and the time sequence represented in the right panel. 1: the excitation tone is swept at a low speed up to the resonance frequency ω res . 2: when the resonance frequency is reached, a "trigger" signal is set at a high amplitude. It is maintained during 1 second. 3: after 1 second, at t = 0, the trigger signal goes below a threshold value, which triggers the drive frequency to be suddenly detuned at a frequency ω res + ∆. The decay is recorded with the DAQ card, and the procedure is repeated. The last row of the time sequence chart shows the frequency of the excitation (black arrow), which is synchronized with the local oscillator, at each step of the sequence.

  l i t u d e ( n m ) D r i v e f r e q u e n c y ( H z )

Figure 4 . 9 :X 2 +

 492 Figure 4.9: frequency (left) and time (right) domain responses of the resonator for an excitation force F 0 = 1.1 pN bringing the resonator far into the bistable regime. The free decay of the steady-state resonant response is triggered by detuning the excitation force away from the resonance frequency by ∆ = -2π × 250 Hz at t = 0. Solid lines on the left panel are the application of the Dung model through Eq. (2.67) with β = 1.8 mHz/nm 2 , while those reproducing the experimentally observed non-linear oscillations on the right panel follow the expression obtained through Lindstedt-Poincaré method and κ = 2 mHz/nm 2 (see text). The magenta points represent the free decay of the amplitude R = √ X 2 + Y 2 , while the red solid line is an exponential t yielding the energy decay time T 1 = 57 ms.

  a y t i m e ( m s ) D i s p l a c e m e n t ( n m )

Figure 4 . 10 :

 410 Figure 4.10: T 1 and T 2 times obtained with the methods described in the main text, as a function of mechanical displacement. The dotted line is set at 58 ms.

  p l i t u d e ( n m ) D r i v e f r e q u e n c y ( H z )

Figure 4 .

 4 Figure 4.11: in-phase (left) and quadrature (right) responses in frequency domain with an applied force F 0 = 83 fN, for three levels of gate induced frequency noise. The extracted parameters (FWHM W , resonance frequency, maximum amplitude) are shown for the response undergoing a frequency noise magnitude σ = 2π × 33 Hz. Solid lines are the application of Eq. (3.34).

Figure 4 .

 4 Figure 4.13: left: measured decay times as a function of injected noise level σ. The blue solid line is obtained by numerically extracting the linewidth from the convolution. The dashed line is a guide for the eyes. Right: analogous pure dephasing rate as dened in the main text, extracted from T 2 and T 2 measurements, with a reference T 1 = 57 ms. The blue solid line is calculated from the T 2 , which is numerically extracted from the convolution formula.

Figure 4 .( 4 . 38 )

 4438 Figure 4.14: left: frequency domain response with a frequency noise magnitude σ = 2π × 5.1 Hz, and the Lorentzian t having the same FWHM. Right: time domain response of X 2 + Y 2 with the same frequency noise magnitude, and the exponential t which is the inverse Fourier transform of the Lorentzian t on the left panel.

Figure 4 .

 4 Figure 4.15: measured shape factor as a function of the noise magnitude. The red solid line is the shape factor with ∆ω calculated from Eq. (4.20).

5 . 1

 51 right panel). Thus a new process has been developed by our collaborator Thierry Crozes, which relies on two lithography steps. It led to develop a sample with a bottom trench located 40 µm below the string (see Fig. 5.1 left panel).

Table 5 . 2 :

 52 relevant parameters for the three samples measured. The damping rates indicated are obtained in vacuum (P < 10 -6 mbar) for a magnetic eld B = 100 or 200 mT, depending on the sample.

Figure 5 . 1 :

 51 Figure 5.1: left: colorized SEM picture of the 300 µm device, with g = 40 µm the gap between the string and the bottom trench. Right: zoom on the right clamp of the 150 µm device (g = 20 µm), which emphasizes the spongy bottom resulting from the XeF 2 etching. The shadowed region is a good marker of the undercut digging length.

Figure 5 . 2 :

 52 Figure 5.2: resonance lines measured at 5 Torr (orange dots) and 0.5 Torr (grey dots) with the 100 µm sample. Solid lines are Lorentzian ts, yielding resonance frequencies ω 0 (P = 5 Torr) = 2π × 2.2531 MHz and ω 0 (P = 0.5 Torr) = 2π × 2.2548 MHz, and damping rates Γ (P = 5 Torr) = 2π × 6 kHz and Γ (P = 0.5 Torr) = 2π × 1 kHz.

Figure 5 . 3 :

 53 Figure 5.3: SEM picture of the Al surface. The typical size of the grains is about 30 nm, with a size varying between 10 and 60 nm.

Figure 5 . 4 :

 54 Figure 5.4: gas damping contribution as a function of the cell pressure in the high pressure limit, i.e. from 1 to 100 Torr, measured with the 100 µm long NEMS at 4.2 K. For the rst mode, theoretical predictions with (black solid line) and without (orange solid line) slippage correction are represented.

5. 5 . 149 0Figure 5 . 5 :

 514955 Figure 5.5: gas damping contribution versus cell pressure for the 100 µm long NEMS at 4.2 K[START_REF] Defoort | Slippage and Boundary Layer Probed in an Almost Ideal Gas by a Nanomechanical Oscillator[END_REF]. The green dashed line is the molecular damping prediction, proportional to P . The green solid line interpolates the data in the transition range between molecular (P below 1 Torr) and viscous (above 10 Torr) regimes. The dashed line is a low pressure asymptote. The black solid line is a t from the Navier-Stokes regime for the rst mode, incorporating slippage corrections. The grey dashed line is an estimate of the acoustic radiation contribution, while the horizontal dashed lines mark the intrinsic linewidths of each mode for comparison. The vertical pink dashed lines are markers for the transition to the molecular regime (λ = w) and to the regime featuring an anomalous damping reduction (λ = g = 4 µm).

Figure 5 . 6 :

 56 Figure 5.6: left: gas damping as a function of pressure, for the 150 µm sample (g = 20 µm).The solid line is the molecular regime prediction, with corrections at high pressures[START_REF] Yamamoto | Flow of a rareed gas past a circular cylinder[END_REF]. The magenta dashed line marks the pressure for which λ = g. The rather important error bars come from the signicant intrinsic linewidth, limiting the resolution. Right: same graph for the 300 µm sample (g = 40 µm). Dots are data taken with the cryostat n • 1 (see Table5.1), while hollow squares are data taken with the cryostat n • 2. All the data are taken at

Figure 5 . 7 :

 57 Figure 5.7: left: gas damping measured at 4.2 K (red dots) and at 1.5 K (blue dots) with the 300 µm device. The temperature was homogeneous over the whole cell. The dashed lines mark, for the two temperatures, the condition λ = g. Right: gas damping measured with the 150 µm device. Red full dots: gas damping measured with a cryostat temperature 4.2 K, with no heating of the sample holder (data shown in Fig. 5.5). Blue empty dots: gas damping measured with the cryostat cooled down to 1.3 K while regulating the sample holder temperature at 4.2 K.

Fig. 5 .Figure 5 . 8 :

 558 Figure 5.8: gas contribution to the NEMS damping normalized to the molecular regime prediction Γ mol of Eq. (5.18), as a function of the mean free path normalized to the gap λ/g, for the three samples measured throughout the experiment, at 4.2 K. Solid lines are P(Kn) ts from Eq.(5.19).

Figure 5 . 9 :

 59 Figure 5.9: left: t parameters α and α as a function of temperature for the 150 µm sample with g = 20 µm. The dierent symbols account for the three tting routines explained in the text. Dashed lines are guides for the eyes. Right: c parameter rounding the P function between the two asymptotic regimes. The dashed line is set at c = 1.21.

Figure 5 .

 5 Figure 5.10: t parameter α as a function of the gap g between the NEMS and the bottom trench. The dierent symbols account for the three tting routines explained in the text. The red solid line is a t function α = 50/g.

5 . 11 ), using the bare 4

 5114 He atomic mass and the geometric NEMS surface 2l(w + e). Knowing the layer density for helium ρ at = 11.6 × 10 18 atoms/m 2 [129], we can compute how many layers are adsorbed on the NEMS.

2 )Figure 5 . 11 :

 2511 Figure 5.11: added mass on the 150 µm resonator probed by resonance shift measurements, in units of area density of adsorbed atoms, for dierent temperatures. Solid lines at low pressures follow the Dubinin-Astakhov isotherm model (Eq.(5.21), see text for the t parameters), while dashed lines at high pressures for temperatures below 2 K are the application of Eq. (5.25) for the unsaturated lm growth. Above 2 K, the high pressure growth is interpreted as the reactive component of the viscous ow taking place (see Section 5.3.1).

Figure 5 . 12 :

 512 Figure 5.12: adsorption distribution function g(ε), obtained from Eq. (5.22).

5. 5 .

 5 Unveiling the Knudsen layer at the lowest pressures 161 veloped in series of Kn = T /x: a i... = b Kn 2 + .... Here, coecients b (k) i...

Figure 5 .

 5 Figure 5.13: schematics of the physical situation probed by the NEMS when entering the Knudsen layer. The brown trench is the bottom of the sample, clamped and thermalized to the sample holder. The shaded region corresponds to the situation where our expansions fail, and more involved theory and/or simulations are required.

Figure 5 . 15 :

 515 Figure 5.15: eective density as a function of pressure for the 3 dierent devices, i.e. 3 dierent gaps, for the same temperature T 0 = 4.2 K. The solid line is the application of Eq. (5.35) for Kn dened with g = 40 µm, while the dashed line is the low (n ∝ P 2 ) pressure asymptote.

5. 7

 7 Résumé en françaisDans ce chapitre nous étudions l'interaction entre un résonateur nano-mécanique et un uide simple, ici l'hélium 4 gazeux, dans la perspective d'étudier les uides quantiques tels que l'4 He superuide et l' 3 He liquide et superuide, à des échelles de longeur jusqu'ici inexplorées. Nous mesurons le taux de dissipation mécanique dû à l'interaction avec le uide de 10 Torr à 5 mTorr, et nous développons diérents modèles qui capturent avec un minimum d'hypothèses les résultats sur 2 ordres de grandeur en pression: de 100 Torr où le uide est bien décrit par les équations de l'hydrodynamique classique, jusqu'à ∼ 1 Torr dans la limite du gaz raréé, où nous faisons appel à une description ballistique qui explique la dissipation par le transfert d'impulsion des atomes du gaz au résonateur lors de chocs aléatoires. Une correction au régime hydrodynamique prenant en compte le glissement du uide sur les parois du résonateur permet de décrire le début de la transition entre les deux régimes. Aux plus basses pressions, nous observons une déviation par rapport à la prédiction du régime moléculaire, reportée dans [35]. Cette déviation est observée pour des températures allant de 1.4 K à 15 K pour trois échantillons de diérentes tailles, et dans deux cryostats aux caractéristiques diérentes, ce qui conrme que le phénomène est reproductible. Nous remarquons que la déviation apparaît dans tous les cas pour un libre parcours moyen λ du gaz qui devient comparable à la distance g entre la sonde nanomécanique et le fond de l'échantillon. Le rapport entre la courbe expérimentale et la prédiction du régime moléculaire suit une loi d'échelle simple du nombre de Knudsen λ/g lorsque 0.1 < λ/g < 1, qui semble universelle. Nous développons un modèle phénoménologique qui explique la déviation par rapport au régime moléculaire par les processus d'interaction entre les particules du gaz et le mur du fond. La nature diusive de cette interaction, déjà soulignée aux plus hautes pressions pour expliquer le glissement, indique que les particules proches du fond vis-à-vis du libre parcours moyen sont en équilibre avec le mur plutôt qu'avec le gaz en volume. Ces processus diusifs sont qualitativement sondés pour plusieurs températures à travers le dépôt de couches adsorbées de gaz sur le NEMS, qui augmentent sa masse et donc diminuent sa fréquence. Nous mettons en lumière le rôle du potentiel d'adsorption, prédominant aux basses températures. La nature même du paysage énergétique au mur a donc pour conséquence une désadaptation des propriétés d'équilibre du gaz près du mur, donc une déviation à la statistique d'équilibre de Maxwell-Boltzmann caractéristique des propriétés en volume. Cette déviation s'établit sur une distance de l'ordre de λ en partant du mur, au-delà de laquelle les collisions sont plus fréquentes entre atomes issus du volume qu'avec des atomes issus de la zone proche du mur, dite couche de Knudsen. Dans la couche de Knudsen, en revanche, les collisions sont fréquentes entre particules du gaz hors équilibre et le NEMS voit donc une statistique du gaz modiée lorsque g ∼ λ: nous montrons par la hiérarchie des équations de conservation (densité, quan-Chapter 6

3 . 4

 34 Interaction between a mechanical mode and TLS: qualitative description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.3.5 Relaxational TLS-phonon interaction . . . . . . . . . . . . . . . . 192 6.3.6 Resonant TLS-phonon interaction . . . . . . . . . . . . . . . . . 202 6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 204 6.4.1 Measurement protocol . . . . . . . . . . . . . . . . . . . . . . . . 204 6.4.2 Results for the damping . . . . . . . . . . . . . . . . . . . . . . . 205 6.4.3 Results for the frequency shift . . . . . . . . . . . . . . . . . . . . 209 6.4.4 Discussion on the contribution of normal state electrons . . . . . 212 6.4.5 Discussion on the TLS location . . . . . . . . . . . . . . . . . . . 215 6.4.6 Link to frequency noise . . . . . . . . . . . . . . . . . . . . . . . 216 6.5 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . 218 6.6 Résumé en français . . . . . . . . . . . . . . . . . . . . . . . . . 220 Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems 6.1 Introduction

Chapter 6 .Figure 6 . 1 :

 661 Figure 6.1: Shift of the resonance frequency and diagram of the bifurcation frequencies as a function of the motion amplitude, with the parameters of the 250 µm long NEMS. The arrows indicate the direction of the evolution of bifurcation frequencies along with an increase in the applied force. The red solid line is the application of Eq. (6.1). The blue solid line is made of two parts. Beyond the turning point when following the arrow, the line is the application of Eq. (6.2). Between x s and the turning point, it is simply Eq. (6.1) applied to the lower branch bifurcation frequency. The green solid line is the Dung shift of the resonance frequency, following Eq. (2.70). The blue dotted line is the asymptotic behaviour of ω bif,↓ (see text).

Figure 6 . 2 :

 62 Figure 6.2: Bistable resonance line, with markers used in dening the procedure (threshold and jumps, see text). This resonance line corresponds to a force F 0 = 81 pN applied to our 250 µm device, measured at 4 K in a 1 T eld.

Chapter 6 .Figure 6 . 3 :

 663 Figure 6.3: typical frequency noise track, taken over about 4 hours, with a step 0.03 Hz. Down sweep measured on the 250 um device, for the setup of Fig. 6.2.

Figure 6 . 4 :

 64 Figure 6.4: frequency noise power spectral density S f (ω), obtained from FFT on the autocorrelation signal of the frequency noise track shown in Fig.6.3. The two cut-o frequencies correspond to the global measurement time (ω low ) and the acquisition time for one resonance frequency (ω high ). The red solid line is a 1/ω 1+µ -type t, here with µ = 0.4.

Figure 6 . 5 :

 65 Figure 6.5: left panel: histogram of the frequency noise track shown in Fig. 6.3. Right: histogram on jump events, computed from the same track. Red solid lines on both histograms are Gaussian ts. Insets: time track of jumps (left) and jumps spectrum (right), with the red curve being the application of Eq. (6.5).

Figure 6 . 6 :

 66 Figure 6.6: histogram of the frequency jumps taken on the upper branch of the bistable resonance shown in Fig. 6.2. Right: histogram on jump events. Insets: time track of jumps (left) and frequency spectrum (right), with the red curve being the 1/ω 1+µ t.

Figure 6 . 7 :

 67 Figure 6.7: Allan deviation as a function of displacement at the two bifucation points, as a function of squared amplitude of motion multiplied by the Q factor, for two Q factors. The t yields σ δf = 0.25 + 2.02 × 10 9 Qx 2 .

Chapter 6 .

 6 Internal energy relaxation and frequency noise in nano-mechanical systems

Figure 6 . 8 :

 68 Figure 6.8: Allan deviation as a function of temperature measured on the lower branch, or on the upper branch with the quadratic dependence in displacement (small here) removed.

Figure 6 . 9 :

 69 Figure 6.9: frequency noise, with the extrapolated value for the spinodal point contribution subtracted, as a function of the mechanical energy (∝ x 2 ), normalized to the spinodal point mechanical energy (∝ x 2 s ). The green solid line is a simple linear curve of slope 0.13 Hz. Inset: same graph for the 15 µm device, with a slope 0.7 Hz.

Chapter 6 . 5 Figure 6 . 10 :

 65610 Figure 6.10: frequency noise data of the third and fth mode of the 250 µm beam, rescaled with the same procedure as for the rst mode (see Fig.6.9).

(6. 26 ) 6 . 3 .Figure 6 .

 26636 Figure 6.12: schematic representation of the relaxational interaction, detailed for one TLS: as the beam is undergoing a deformation when resonantly excited, the resulting local strain eld changes the interatomic potential and modulates the TLS asymmetry. The dots represent defects randomly distributed in the beam, the dashed line on the beam is the neutral axis, where the applied strain eld is zero. The TLS equilibrium conguration is the dashed potential, while the full line represents the strain-modulated potential.

. 30 )

 30 Therefore, we have δp = -ω[δp c sin(ωt) + δp s cos(ωt)], and we can explicit the energy change (6.27): δU

6. 3 .Figure 6 .

 36 Figure 6.13: magnitude of the real and imaginary part of a TLS susceptibility normalized to γ • E/(2k B T ), as a function of the reduced parameters u = E/(k B T ) and v = ∆/ε, for two temperatures representative of the two limit behaviours xed by ωτ rel .

Figure 6 . 14 :

 614 Figure 6.14: average critical eld as a function of temperature. The intermediate region between superconducting and normal state due to the type II nature of the thin lm superconductor is not shown, but it explains the rather large error bars. The critical temperature is evaluated at T c = 1.4 ± 0.1 K.

Figure 6 . 16 :

 616 Figure 6.16: damping rate as a function of the cryostat base temperature with the small linear slope subtracted, with the same convention for the dots as in Fig.6.15. Added solid lines are various ts corresponding to dierent scenarios detailed in the text. Inset: same data for the 50 µm device embedded in a microwave cavity.

6. 4 .Figure 6 . 18 :

 4618 Figure 6.18: relative frequency shift δω/ω 0 as a function of temperature, with the dierential stress contribution (see Fig.6.17) removed. The black solid line is the application of Eq. (6.54), i.e. a logarithmic t yielding a slope C = 2.3 ± 0.3 × 10 -5 . Inset: same procedure applied to the 50 µm long NEMS embedded in a microwave cavity. The t slope is C = 0.6 ± 0.1 × 10 -5 .

(6. 62 ) 6 . 4 .

 6264 Results and discussion213where we have introduced fermionic creation/annihilation operators c ( †)

Figure 6 . 20 :

 620 Figure 6.20: Allan deviation of the frequency noise for the 15 µm device in the superconducting (red squares) and normal state (blue dots). The red solid line is a constant value set at 1.45 Hz. The blue solid line is a t function 1.1/ √ T . Inset: damping noise measured with our technique. The blue dashed line is a guide for the eyes.

6. 6 1

 61 Résumé en français Dans ce chapitre nous mesurons la dissipation intrinsèque du NEMS jusqu'à 30 mil-liKelvin. Nous mettons en évidence le rôle des excitations de basse énergie couplées à la déformation du NEMS dans la relaxation d'énergie mécanique. Ces excitations, permises par la structure désordonnée des matériaux constitutifs du NEMS, sont modélisés comme des atomes se déplaçant par eet tunnel entre deux positions équivalentes du réseau atomique (TLS). Nous obtenons également le bruit en fréquence intrinsèque en développant une nouvelle technique de mesure utilisant la non-linéarité du NEMS. L'étude poussée nous permet de lier phénoménologiquement les deux observations. Main achievements . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

  Chapter 2. Experimental and theoretical tools

		SF 6 plasma			XeF 2 gas
	Al	Al	Al	Al	Al	Al
	SiN		SiN	SiN		SiN
		Si
					Si

  Figure2.17: eect of the loading impedance on the measured resonance properties for a long (300 µm) nanobeam. Left: eect on the damping rate. The solid line is a quadratic law following Eq.(2.38). The zero eld value is the extrapolated intrinsic damping rate, left as a free parameter. Here we obtain Γ

m = 2.96 ± 0.5 Hz. The parameters used for the NEMS resistance (1.1 kΩ) and mass (2.5 × 10 -14 kg) are obtained by dierent procedures, which leaves no other free parameter. Right: eect on the resonance frequency. Note the small dynamic range. The solid line is a t from Eq. (2.38) with C l = 30 pF.

  To keep the modelling simple, we describe it as a discrete capacitance C, of about a few 100 pF,

	but which we keep as a t parameter, since electrical connections are likely to vary
	a lot in length depending on the experiment requirements. The canonical electrical
	circuit associated to a basic experiment, i.e. driving and detecting a NEMS around
	resonance, is displayed in Fig. 2.13, and its transfer function is:

  The measured shift of the resonance frequency δω 0 [I h (0)]

	due to Joule heating will then serve as a reference curve (see Fig. 2.19).

  Figure2.24: Main: shift of a 300 µm NEMS resonance frequency as a function of the bias voltage applied on the gate electrode. The red solid curve is a t from Eq. (2.61), from which we obtain a coupling strength ∂ 2 C(0)/∂x 2 = 4.3 × 10 -5 F/m 2 . Note the voltage oset V off = 315 mV added to consistently t the whole data range. Inset: Residual error on the t. All the error is contained into 10 % of the NEMS damping rate (see dashed lines).

  Figure 2.28: example of bistable response for a 300 µm string. The hysteretic region is located between the two bifurcation frequencies ω ↑,↓ . Solid lines are ts from the solutions of Eq. (2.69), with a Dung coecient β = 2.75 mHz/nm 2 .

	52	Chapter 2. Experimental and theoretical tools
	with γ m,n being proportional to the overlap of two dierent modes proles. It has a ω ↑ / ( 2 π) similar expression to the non-linear constant, which owes to their common origin:
	Amplitude			Amplitude
		Frequency
				Frequency
	Figure 2.29: left: bistable response, where the dashed line represents the unstable state.
	Right: zoom in the bifurcation region where the upper branch metastable state and the
	unstable state merge.
	2.9.3 Dispersive mode coupling
	It is straightforward to realize that the mechanism responsible for the Dung nonlin-
	earity will also break the eigenmodes orthogonality, since the geometric nonlinearity
	is a global tensioning eect. As such, it couples dierent eigenmodes, dispersively at
	rst order, i.e. there is no energy ow from one mode to another. If we assume that
	two modes m, n are excited, they are coupled through a biquadratic potential:
		V m,n =	3 4	γ m,n x 2 m x 2 n ,	(2.71)

  Figure2.30: left: third mode in-phase resonance lines of a 300 µm NEMS while the rst mode is resonantly excited, with a resonant amplitude x 1,res = 10 nm (black), 144 nm (red), 240 nm (green). Solid lines are Lorentzian ts, all having the same height and width. Right: shift of the resonance frequency as a function of the 1 st mode resonant amplitude. The red solid line is a quadratic t, yielding a mode coupling coecient β 1,3 = 2.75 mHz/nm 2 .

	1 s t m o d e r e s o n a n t a m p l i t u d e ( n m )

Table 3 .

 3

1: Measured mode parameters relevant to the data presented here, with loaded linewidths at a eld B = 0.84 T. All agree fairly well with analytic computation.

  where the average notation ... i ≡ N -1 i=0 /N , using cycling notations when i + j ≥ N . The SR844 lock-in has a lter with asymptotes falling at 24 dB/oct, and the equivalent measurement bandwidth BW is obtained from the lock-in time constant τ m through the relation BW × τ m = 5/64. By denition S Λ (ω) = πC Λ [0]/BW, where Λ = XX, YY or XY. Note that by construction, if the linearized description still holds, X and Y

are uncorrelated, i.e. S XY (ω) = 0, and S x (ω) = S XX (ω) = S Y Y (ω), where S x is the position noise spectrum dened through Wiener-Khinchin theorem (3.4).

  Figure3.12: left: raw measured power spectral density on X and Y quadratures at the level of the lock-in amplier, on sample 300 µm n • 1. The green solid line is the application of Eq. (3.41), with a t line capacitance C eq = 500 pF, while dashed lines show contributions of Eq. (3.42) (green), (3.43) (pink) and (3.44) (blue). Right: corrected power spectral density, where parasitic contributions S cross and S back have been substracted. The red solid line is a Lorentzian t yielding a linewidth Γ 1 = 140 Hz in agreement with the linewidth obtained through sine wave response measurements.

  The total power due to injected noise is recalculated by integrating the injected voltage noise spectrum over the lter's bandwidth, yielding values as high as 10 nW directly injected in the nanomechanical beam. This is taken into account in the t procedure following our thermal model (see Chapter 2, section 2.7). Examples for both limits are shown in Fig.3.16, with

theoretical lines obtained from Eq. (3.37) without free parameters. Again, the spectra observed for high eective temperatures appear strongly asymmetric, with a long tailed structure which is reminiscent of the exponential distribution

(3.16)

. In addition, no cross-correlation between the two quadratures was observed.

  Meanwhile, adiabatic dephasing is determined by the low-frequency spectrum of the environment variable coupled to the longitudinal spin component, and in the zero frequency limit the pure dephasing rate Γ φ ∝ S X (0).

Obviously the NEMS is a classical oscillator, not a quantum spin. Its dynamics is not described by Bloch-type equations. Yet, conceptually, some analogies are still possible, especially with respect to pure dephasing. Indeed the bare Lorentzian linewidth Γ accounts for dissipation and thus the usual relaxation time 2/Γ can be identied with a T 1 time. Meanwhile, as pointed out in Chapter 3, if the NEMS undergoes uctuations of its resonance frequency that are slow enough, it adiabatically follows them. Under sinusoidal excitation, the response accumulates dephasing, and for long enough measurement times, it is averaged over all realizations of the uctuations assuming ergodicity. This leads to inhomogeneous broadening, hence to a change in the linewidth. This change is not caused by additional dissipative processes and thus, one can dene by analogy a measurable (classical) decoherence time T 2 = 2/∆ω, where ∆ω is this inhomogeneous linewidth. As quadratures include the phase dependence, averaging over uctuations independently on both would lead to quadratures decaying with a time constant T 2 in free decay experiments. Building on these analogies, one can introduce a pure dephasing rate

  The gate electrode as a source of frequency uctuationsTo separate damping and dephasing contributions, we need to implement a measurement scheme that allows us to observe the resonator's decay, i.e. its return to equilibrium after a perturbation, in real time. Since the response is a small signal competing with instrumental noise, in time domain, one needs to average over many decays to obtain reasonable error bars on the exponential decay time constant. However, our best (i.e. longest) resonators display at best linewidths above 1 Hz, which translates as decay times T 2 ≤ 300 ms at best. In addition we need a magnetic eld high enough to address their motion with a good signal to noise ratio, but we know that long devices are more aected by external loading losses (see Chapter 2 section 2.6.3). The best compromise is thus to use long devices with small magnetic elds: with the de-

	110 Chapter 4. A model experimental approach to classical decoherence for mechanical systems
	4.1.3

vice used in this experiment for time-resolved measurements (see Chapter 3, Table

3

.1, sample 300 µm n •

  The typical oset voltage in Chapter 4. A model experimental approach to classical decoherence for mechanical systems this experiment was about 300 mV, which is rather large (more typical values are about 100 mV). The σ g values are given in r.m.s. units, that is, σ g = δV 2

	g	1/2	.
	The spectral lines observed should be, in this conguration, the Fourier transform
	of Eq. (4.16). In other words, the average susceptibility in frequency domain is the
	convolution of the bare susceptibility by the Fourier transform of the exponential term,
	called p(δΩ):

2

. Note that the DC voltage that is computed is the eective one, i.e. the one that takes into account the oset on the gate. The latter is measured during the coupling strength characterization, as shown in Chapter 2, section 2.8.3.

  The n th moment of this distribution, M n = δΩ n , writes:

			.33)
	with σ =	√	2ασ 2 g . The choice to leave a

√ 2 outside this denition is a matter of convention, such that σ is without numerical factors the frequency noise standard deviation. Again, due to the quadratic nature of the coupling, frequency uctuations 4.3. Quadratic coupling 123 take only positive values, which appears through the Heaviside function Θ(δΩ).

  , theory predicts that the coecient κ should coincide with the Dung coecient β. Over the whole dynamic range we veried experimentally that κ = β within 10 %, where β = 1.8 mHz/nm 2 . Moreover, we did not observe any discrepancy be-

	126 Chapter 4. A model experimental approach to classical decoherence for mechanical systems
	tween relaxation and decoherence times for our device, nor any non-linear damping, as
	shown in Fig. 4.10, within error bars over the range explored (see Chap. 6 for further
	discussion).
	/T 1
	2t/T 1	.	(4.37)

Besides

  Nous mesurons le temps de décohérence du mode fondamental de exion hors du plan d'une longue poutre nano-mécanique par la technique habituelle de balayage en fréquence de la raie de résonance mécanique. En eet, chaque point de la raie est moyenné sur des temps susamment longs pour intégrer statistiquement toutes les réalisations des uctuations de fréquence. En présence de déphasage adiabatique, la raie est élargie. La largeur de raie inhomogène (par analogie avec la RMN) est directement une mesure du temps de décoherence T 2 , tandis que la forme de la raie nous renseigne sur la statistique des uctuations: ceci est démontré en créant deux types de bruit en fréquence, l'un Gaussien, l'autre fortement asymétrique.An de mesurer le temps de relaxation T 1 en s'aranchissant de la contribution du déphasage adiabatique, nous développons une technique résolue en temps de mesure de

	132 Chapter 4. A model experimental approach to classical decoherence for mechanical systems
	chapitre est diérent: il s'agit simplement ici de séparer pertes d'énergie et perte de
	cohérence de phase.

l'oscillation mécanique, insensible à la phase de l'oscillation. En coupant l'excitation initialement à résonance, nous mesurons l'enveloppe des pseudo-oscillations du signal relaxant á l'équilibre sur le temps caractéristique T 1 . En l'absence de uctuations, nous montrons que le déphasage adiabatique est négligeable avec notre niveau de résolution, et qu'il n'y a pas de phénomène de relaxation ou déphasage non-linéaire visible comme cela peut être le cas pour des résonateurs à base de graphène ou de nanotubes de carbone. Lorsque le bruit en fréquence est articiellement ajouté, nous observons clairement une séparation des deux temps caractéristiques T 1 et T 2 , qui est capturée quantitativement par un modèle théorique simple. Nous pouvons en particulier extraire un taux de déphasage adiabatique pur, qui peut être quantitativement lié au spectre intégré des uctuations lentes de fréquence. Ces résultats, reportés dans

[START_REF] Maillet | Classical decoherence in a nanomechanical resonator[END_REF]

, reposent sur des techniques simples à mettre en ÷uvre, et invitent à une réexion sur le caractère fondamentalement quantique de certains phénomènes à l'÷uvre dans les systèmes mésoscopiques et/ou à la limite quantique.

  6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.2 Measuring intrinsic frequency noise . . . . . . . . . . . . . . . 172

6.2.1 Dynamical bifurcation as a tool to measure frequency noise . . . 172 6.2.2 Analysis procedure and spectral features . . . . . . . . . . . . . . 175 6.2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3 Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) . . . . . . 187 6.3.1 Structural two-level defects (TLS) in amorphous materials . . . . 187 6.3.2 Single TLS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.3 TLS energy distribution . . . . . . . . . . . . . . . . . . . . . . . 190 6.

  1/ω type spectrum leading to very slow characteristic timescales. Decoherence due to frequency noise has also a small eect in time domain for our devices having the best Q factors, seen as an extra Gaussian decay. The damping uctuations are again amplied by a term βx 2 max /Γ , but the decaying term (t) = (1 -e -Γ t/2 )/(Γ t/2) in the exponential is a rather weak dependence, and with our devices we see that extremely large motion (on the millimeter scale for our longest devices) is needed to observe a mesurable impact on the decay of the two quadratures, i.e. on T 2 (see Chapter 4. Such displacements are in practice never reached. This explains why no eect could be seen in Ref. [34], leading to T 1 ≈ T 2 ≈ T 2 , where T 1 = 2/Γ is the energy decay time. Eventually, the bifurcation method presented here appears to be a unique, simple and reliable way to access damping noise δΓ quantitatively.

	Mode	ω 1 /(2π)	Bare Q 1	β	σ δf	σ δΓ
	n=1	0.905 MHz 600 000 8.5 mHz/nm 2 0.28 ± 0.05 Hz	0.11 Hz ± 10%
	n=3	2.68 MHz	18 000 125 mHz/nm 2 0.1 ± 0.02 Hz	0.005 Hz ± 15%
	n=5	4.45 MHz	4 800	570 mHz/nm 2 0.09 ± 0.05 Hz 0.0025 Hz ± 30%

Table 6 .

 6 1: internal frequency noise of the three rst symmetric modes at T = 4.2 K for the 250 µm device.

Table 6 .

 6 ) Unloaded Q 1 σ δf or σ f 2: internal frequency noise reported in this work (two rst rows) and in some recent works (in order of appearance

	250 µm SiN/Al d.c. beam 4.2 K 0.59 MHz	600 000	0.25 ± 0.05 Hz
	15 µm SiN/Al d.c. beam 4.2 K 17.5 MHz	18 000	1.45 ± 0.1 Hz
	2 × 3 µm Si/Al goalpost 4.2 K 7.1 MHz	4 800		1 ± 0.5 Hz (σ f )
	380 µm SiN d.c. beam	5 K	0.64 MHz	2 200 000	0.01 ± 0.5 Hz (σ f )
	3.2 µm cantilever	300 K	45.2 MHz	6 000		36 Hz
	Device		σ δf or σ f			σ δΓ
	250 µm SiN/Al d.c. beam	0.28 ± 0.05 Hz	0.11 Hz ± 10 %
	15 µm SiN/Al d.c. beam	1.45 ± 0.1 Hz	0.6 Hz ± 10 %
	2 × 3 µm Si/Al goalpost	1 ± 0.5 Hz (σ f )	0.35 Hz ± 10 %
	380 µm SiN d.c. beam	0.01 ± 0.5 Hz (σ f )		X
	3.2 µm cantilever	36 Hz

X

  6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 195TLS over one cycle of the strain oscillation: P c = -ω

	2π	δU δt dt. It yields straightfor-
	wardly:

  at low temperatures. The energies of the two rst transverse-polarized bulk modes are of the order c t π/e and c t π/w where c t ∼ E/ρ is the transverse sound speed and E is the elastic Young's modulus. For 6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 199temperatures below 1 K such modes' thermal populations are vanishingly small, and thus these modes do not exchange energy with TLS. This is where dimensionality reduction comes into play, as it amounts to saying that the dominant phonon wavelength hc t /(2.82k B T ) of the bulk equilibrium phonons is bigger than transverse dimensions. It means that we have to consider four families of modes: one bulk longitudinal with sound speed c l = E/ρ, one torsional with c r = E/[2(1 + ν)] and two exural, with c f = σ 0 /ρ in the string limit. These four families of modes are all 1D channels into which phonons in thermal equilibrium propagate.A TLS can relax from the high to the low energy state by emitting a phonon, with a rate τ -1 e→g , or go from low to high energy by absorbing one, with a rate τ -1 g→e .

	Only the longitudinal and torsional polarizations survive, and new families of modes
	emerge, that are solutions of the 1D Euler-Bernoulli equation (2.2) [43]: our usual
	exural modes.

  energy eigenbasis and |N k,s refers to the number of electrons in a state |k, s in Fock space. Because of fermionic rules of occupation, for an electron of spin s going from state |k, s to state |q, s by TLS scattering there must be initially one Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems electron in |k, s and zero in |q, s . Assuming conduction electrons are thermalized at a temperature T , this appears by introducing the Fermi-Dirac distribution

A model experimental approach to classical decoherence for mechanical systems

An example is shown in Fig.2.19 with an almost DC heating current (50 kHz). The data are well reproduced by theory up to moderate heating currents, choosing a and b as in the example of Fig.2.18 which is taken from the same device and measurement campaign. For strong heating, which corresponds to temperatures of 15 K or more at

en unités réelles (force appliquée en Newton, amplitude détectée en mètres), nous décrivons une technique de calibration in-situ du résonateur, reposant sur le chauage local dû à la circulation d'un fort courant électrique dans la couche métallique déposée sur la poutre. La abilité de cette procédure est testée avec la caractérisation sans paramètre ajustable de phénomènes non-linéaires tels que le tirage de fréquence et le couplage de modes mécaniques de vibration, qui ont pour origine le raidissement d'une poutre doublement encastrée subissant une déformation, ou l'accordage de fréquence et l'amplication paramétrique rendus possibles par la modulation de fréquence obtenue par couplage électrostatique de la poutre à une électrode de grille. L'accord obtenu est excellent, ce qui nous permet entre autres d'invoquer ces phénomènes de façon quantitative dans les chapitres à venir. Les phénomènes de dissipation électro-mécanique sont brièvement introduits à travers l'étude de l'amortissement causé par les courants induits par la technique magnétomotive. Certains mécanismes mal connus à ce jour causant l'amortissement mécanique seront approfondis dans la dernière partie de ce manuscrit.

Dans ce chapitre, complémentaire du précédent, nous décrivons une expérience ayant pour but de modéliser la décohérence classique d'un système nano-mécanique par analogie avec la Résonance Magnétique Nucléaire (RMN) ou les systèmes d'information quantique. Des concepts issus de ces champs d'étude sont introduits, tels que le temps de relaxation T 1 qui est le temps caractéristique sur lequel s'eectue le transfert irréversible d'énergie du système à son bain, et le temps de décohérence, qui inclut les pertes d'énergie mais incorpore aussi la perte de cohérence (ici classique) de la phase associée à l'oscillation du système. La décohérence est articiellement créée par une source de bruit lent (spectre piqué en-dessous de 1 Hz) en tension couplée capacitivement à la vibration mécanique par une électrode de grille: le bruit est alors transduit en uctuations lentes de la fréquence de résonance du mode étudié, ce qui introduit un déphasage accumulé sur une échelle de temps comparable à T 1 , responsable d'une perte de cohérence de phase du signal bien que la relaxation d'énergie reste inchangée.La procédure est similaire à celle du chapitre précédent bien qu'ici le mécanisme à l'origine des uctuations de fréquence soit extérieur à la nano-poutre (nous utilisons une grille externe au lieu de la non-linéarité intrinsèque). En revanche, l'objet du

(5.9)

tité de mouvement, énergie) et une résolution approchée de l'équation de Boltzmann utilisant la méthode de Grad-Chapman-Enskog (développement des quantités thermodynamiques et de l'opérateur de collision en λ/x, x ≡ g et en les vitesses d'atomes), qu'une raréfaction du gaz a lieu dans la couche de Knudsen, avec une dépendance en pression de la densité eective du gaz asymptotiquement quadratique plutôt que linéaire comme c'est le cas en volume. Cette raréfaction a pour eet mesurable une réduction de l'amortissement mécanique par le gaz du NEMS : ce dernier agit comme une sonde non-invasive du milieu hors-équilibre du fait de ses très petites dimensions transverses.

Acknowledgements

Part II NEMS coupled to microscopic degrees of freedom

force per unit length in the frequency domain writes: ∂F g (z, ω) ∂z = ρ g A c ω 2 Λ(ω)Ψ(z)x(ω), (5.3) where ρ g is the gas mass density and A c = πw 2 /4 the eective cross-section of the beam experiencing friction. Λ(ω) is the friction coecient that is calculated from Eq. (5.2). By eective we mean that it is the cross-section for a cylinder of diameter w, and the correction to match the rectangular case will further be incorporated in the friction coecient. To detail physically the viscous force acting on the whole beam, one can write the instantaneous power injected into the uid due to the damping force integrated over the whole beam in time domain:

From then on the viscous force that appears into the brackets can be decomposed into two dierent components. The one proportional to ẍ is said to be reactive, adding inertia as if the eective mass of the NEMS were renormalized by the surrounding uid, dragged together by the motion. This contribution leads to a shift of the resonance frequency. On the other hand, the term proportional to ẋ acts as the dissipative component, leading to the linewidth broadening. What is left now to compute is the friction coecient Λ(ω). It can be explicited as Λ(ω) = Σ(ω)Ω(ω), where Σ(ω) is the Stokes solution for an innite cylinder [START_REF] Stokes | On the Composition and Resolution of Streams of Polarized Light from dierent Sources[END_REF]. Ω(ω) is a small correction used to match the rectangular cross-section term [START_REF] Elie | Frequency response of cantilever beams immersed in viscous uids with applications to the atomic force microscope[END_REF][START_REF] Collin | Silicon Vibrating Wires at Low Temperatures[END_REF], which renormalizes the damping by at most 15 %. The Stokes solution writes:

where u = √ 2δ/w and δ = 2 η/(ρ g ω). δ is called the viscous penetration depth.

It is the characteristic length over which the harmonic motion propagates in the uid, that is, over which the uid is perturbed by the oscillation. J and Y are the Bessel functions of rst and second kind.

We have written the power injected, and the dissipative part P d can be related to the linewidth broadening Γ v through P d = -mΓ v (ω) ẋ2 (t). In the end, the damping rate in the viscous regime depends on the oscillation frequency: Chapter 5. Energy relaxation in a surrounding uid this time, and the interaction potential V vw between a gas atom and the solid at a distance d is of Van der Waals type:

where n s = 60 atoms/nm 3 is the solid atomic density for aluminum (50 atoms/nm 3 for silicon) and a = 2.56 Å the van der Waals atomic radius (similar for silicon).

As one layer is about 0.36 nm thick (the helium spacing in bulk [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF]), we have

In this experiment we kept a pressure P < P s , and the high coverage situation only applies for temperatures below the superuid transition at 2.17 K. If the pressure is high enough, an unsaturated lm grows: it arises from the competition between the van der Waals force exerted by the surface on the lm and gravity. The thickness of this unsaturated lm depends on the gas pressure at the level h of the NEMS from the bottom of the cell which is at saturated vapour pressure P s :

where g 0 is the gravitational eld, labeled so to avoid any confusion with the gap g.

If one equates the gravitational potential (obtained by inverting the equation above)

with the van der Waals potential (5.23), one can obtain the number of adsorbed layers N = d/(0.36 nm):

.

(5.25)

Here N m = 3 is the value obtained from low coverage ts. The high coverage curves following Eq. (5.25) are represented in Fig. 5.21 with dashed lines, with no free parameters. Note that at these temperatures, the lm is superuid (the so-called Rollin lm) and thus creeps over surfaces [START_REF] Pobell | Matter and Methods at Low Temperatures[END_REF]. However, we do not probe its superuid properties in this experiment, and data focusing on superuid properties are not reported in this manuscript.

Interpretation in terms of density reduction near the wall

While signatures of the Knudsen layer manifest clearly and can be correlated to the adsorption dynamics at the wall, we have not explained so far why this dynamics should lead to an eective damping reduction on the NEMS. To get a deeper understanding, even at a qualitative level, it is useful to focus on the distribution of particles in Chapter 6. Internal energy relaxation and frequency noise in nano-mechanical systems

In the following we will use the energy basis to describe the interaction between phonons and TLS, which is thought to be the dominant channel of mechanical energy relaxation at low temperatures, where the bath ghts against coherent tunneling of the TLS. We later extend the formalism to the interaction between conduction electrons and TLS.

TLS energy distribution

Since there is no reason for a disordered material to host only one type of TLS, energywise, we can assume that it features instead a distribution over an energy range that should be limited by the glass transition temperature around 1000 K. Although there is no general consensus on what should be the density of energy states for two-level systems, two arguments lead to a rather simple description: rst, at low temperatures, the relevant TLS which are interacting with equilibrium phonons should have energies (and hence asymmetries) that are well below the glass transition temperature (∼ 1000 K). Therefore, considering that asymmetries range up to this energy, we can assume that the distribution on asymmetries is roughly constant in our range. Meanwhile, we can see through Eq. (6.16) that a small change in the barrier leads to a large variation of the tunneling matrix element ∆ 0 . Since at low temperatures, the range of ∆ 0 spanned is small, the distribution in the λ parameter is assumed to be roughly uniform. Then, the probability distribution writes:

P (∆, λ)d∆dλ = P 0 d∆dλ.

(6.20)

Hence P 0 is the number of TLS per unit energy per unit volume. Through a simple variable change we can obtain a distribution on asymmetries and tunneling matrix elements:

While this distribution explains well experimental ndings, it also raises the issue of an energy cut-o, as the distribution (6.21) clearly diverges for small tunneling amplitudes. We shall see in the following that the averaging over many TLS is not very sensitive to the low cut-o energy, provided that the latter is small compared to k B T . Ultimately, this cut-o is justied by the nite number of TLS in the structure.

We also introduce useful distributions on couples (ε, ∆), (ε, ∆ 0 ) and on reduced parameters (u, v) = (ε/(k B T ), ∆/ε): 6.3. Internal energy relaxation at milliKelvin temperatures: an overview of the Standard Tunneling Model (STM) 191

In particular, the distribution on dimensionless parameters (6.24) will allow to predict easily how the damping should behave at low temperature without explicit integration (see below).

Interaction between a mechanical mode and TLS: qualitative description

The interaction between TLS and phonons can be split into two parts, namely relaxational and resonant processes, which are predominant depending on the phonon energies at stake and lead to very dierent signatures. Though the two can be treated on an equal footing using Bloch equations for the pseudo-spin representing the TLS, here we describe them separately to clarify the physical picture and make the link with the NEMS macroscopic parameters clearer. One can quickly describe the two processes the following way:

• the relaxational interaction involves two parts: rst, the TLS energy levels are modulated by the oscillating strain eld imposed when the NEMS is driven.

Hence they are put out of their equilibrium, and relax energy towards their environment over a nite time delay after the strain modulation, which represents a loss of energy. We use the TLS susceptibility to a strain perturbation, which contains both reactive (frequency shift) and dissipative (damping) parts, as a framework in this case.

• the resonant interaction is easier to describe: when driven, a mechanical (phonon) mode ω is put out of equilibrium, and relaxes back by interacting with TLS whose energy splitting match the mechanical energy quantum ω relative to a mode: either the TLS absorbs or emits resonant phonons. Thus the mechanical damping rate is directly the phonon-TLS scattering rate averaged over the whole TLS distribution.

In the following, we will describe in detail these two processes, starting with the relaxational interaction, which is more subtle to address. Besides, we believe that the reduced dimensions and beam nature of our mechanical device require a specic