Olivier Maillet

Xin Zhou

Rasul R Gazizulin

Rob Ilic

Jeevak M Parpia

Olivier Bourgeois

Andrew D Feerman

Eddy Collin
email: eddy.collin@neel.cnrs.fr

Measuring Frequency Fluctuations in Nonlinear Nanomechanical Resonators

Keywords: nanomechanics, nonlinearity, bifurcation, frequency uctuations, damping uctuations

Advances in nanomechanics within recent years have demonstrated an always expanding range of devices, from top-down structures to appealing bottom-up MoS 2 and graphene membranes, used for both sensing and component-oriented applications. One of the main concerns in all of these devices is frequency noise, which ultimately limits their applicability. This issue has attracted a lot of attention recently, and the origin of this noise remains elusive up to date. In this Letter we present a very simple technique to measure frequency noise in nonlinear mechanical devices, based on the presence of bistability. It is illustrated on silicon-nitride high-stress doubly-clamped 1 beams, in a cryogenic environment. We report on the same T /f dependence of the frequency noise power spectra as reported in the literature. But we also nd unexpected damping uctuations, amplied in the vicinity of the bifurcation points; this eect is clearly distinct from already reported nonlinear dephasing, and poses a fundamental limit on the measurement of bifurcation frequencies. The technique is further applied to the measurement of frequency noise as a function of mode number, within the same device. The relative frequency noise for the fundamental exure δf /f 0 lies in the range 0.5 -0.01 ppm (consistent with literature for cryogenic MHz devices), and decreases with mode number in the range studied. The technique can be applied to any types of nano-mechanical structures, enabling progresses towards the understanding of intrinsic sources of noise in these devices.

Within the past decade Nano-Electro-Mechanical-Systems (NEMS) have developed with a broad range of applications extending from physics to engineering. In the rst place, their size makes them very sensitive transducers of force. [START_REF] Moser | Ultrasensitive Force Detection with a Nanotube Mechanical Resonator[END_REF][START_REF] Li | Ultra-sensitive NEMS-based Cantilevers for Sensing, Scanned Probe and Very High-frequency Applications[END_REF] This had been demonstrated e.g. in the seminal work of D. Rugar et al. in which a cantilever loaded by a magnetic tip reached a detection sensitivity corresponding to the force exerted by a single electronic spin at a distance of about 100 nm. [START_REF] Rugar | Single Spin Detection by Magnetic Resonance Force Microscopy[END_REF] More recently, NEMS have been applied to the detection of small quantities of matter (mass spectroscopy), with precision reaching the single proton. [START_REF] Chaste | A Nanomechanical Mass Sensor with Yoctogram Resolution[END_REF] Nowadays, even the quantum nature of the mechanical degree of freedom is exploited for quantum information processing. [START_REF] Pirkkalainen | Hybrid Circuit Cavity Quantum Electrodynamics with a Micromechanical Resonator[END_REF] In all applications, the quality of the device is intrinsically linked to its level of displayed noise. [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF] Specically, frequency noise in NEMS appears to be a key limiting parameter whose physical origin is still unknown. [START_REF] Sansa | Frequency Fluctuations in Silicon Nanoresonators[END_REF] Besides, only few quantitative experimental studies are available in the literature, 710 especially at low temperatures. [START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] The nonlinear frequency noise reported for carbon-based systems [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF] is one of the most striking results, revealing the complex nature that the underlying mechanisms can possess.

Frequency noise (or phase noise [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF][START_REF] Greywall | Evading Amplier Noise in Nonlinear Oscillators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] ) can be understood in terms of pure dephasing, [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF] making an analogy with Nuclear Magnetic Resonance (NMR); and its impact on a mechanical resonance can be modeled experimentally by means of engineered frequency uctuations. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] The physical origin of intrinsic frequency noise is indeed still elusive, since all identied mechanisms studied explicitly display much weaker contributions than the reported experimental values: adsorption-desorption/mobility of surface atoms, [START_REF] Atalaya | Diusion-induced Dephasing in Nanomechanical Resonators[END_REF] experimentally modeled under a Xe ow, [START_REF] Yang | Surface Adsorbate Fluctuations and Noise in Nanoelectromechanical Systems[END_REF] or the nonlinear transduction of Brownian motion. [START_REF] Zhang | Spectral Eects of Dispersive Mode Coupling in Driven Mesoscopic Systems[END_REF][START_REF] Maillet | Nonlinear Frequency Transduction of Nanomechanical Brownian Motion[END_REF] These eorts in understanding the microscopic mechanisms at work in mechanical dephasing are accompanied by theoretical support. The nonlinear dephasing/damping has been proposed to originate in nonlinear phononic interactions between the low frequency mechanical modes and thermal phonons. [START_REF] Atalaya | Nonlinear Damping and Dephasing in Nanomechanical Systems[END_REF] Finally, a common speculation reported in the literature is that frequency noise is related to defects, [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF][START_REF] Dutta | Low-frequency Fluctuations in Solids: 1/f Noise[END_REF] which can be either extrinsic or constitutive of the material (like in a glass). The presence of these so-called Two-Level Systems (TLS) is also proposed to explain damping mechanisms in NEMS, [START_REF] Hoehne | Damping in High-frequency Metallic Nanomechanical Resonators[END_REF][START_REF] Venkatesan | Dissipation Due to Tunneling Two-level Systems in Gold Nanomechancial Resonators[END_REF] and have been shown recently to lead to peculiar features (especially in the noise) for mesoscopic systems such as quantum bits and NEMS. [START_REF] Behunin | Dimensional Transformation of Defect-induced Noise, Dissipation, and Nonlinearity[END_REF] Properly measuring frequency noise is not easy; a neat technique presented in the literature relies on cross-correlations present in the two signals of a dual-tone scheme. [START_REF] Sansa | Frequency Fluctuations in Silicon Nanoresonators[END_REF] Moreover, in the spectral domain dephasing and damping are mixed. [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF][START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF][START_REF] Miao | Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators[END_REF] In order to separate the contributions, one has to use both spectral-domain and time-domain measurements. [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF][START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] All of these techniques may not be well suited for large amplitude signals (especially when the system becomes bistable), preventing the exploration of the nonlinear range where nonlinear damping/dephasing may dominate.

In this Letter we present a method based on bifurcation enabling a very simple measurement of frequency noise in nonlinear bistable resonators. Building on this method, we characterize the intrinsic frequency noise of high-stress Silicon Nitride (SiN) doubly-clamped beams in cryogenic environment (form 1.4 K to 30 K). In particular, we study the three rst symmetric modes (n = 1, 3, 5) of one of our devices, and demonstrate the compatibility of our results with existing literature. The temperature-dependence is indeed similar to Ref., [START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] but we nd an unexpected damping noise which is amplied through the bifurcation measurement. This result is distinct from the reported nonlinear phase noise of Ref. [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF] in which the device was not bistable. The phenomenon seems to be generic, and we discuss it in the framework of the TLS model. Note that our results demonstrate the existence of an ultimate limit to the frequency resolution of bifurcation points in nonlinear mechanical systems.

Results and discussion

A typical doubly-clamped NEMS device used in our work is shown in Fig. 1. It consists of a 100 nm thick SiN device covered with 30 nm of Al. The width of the beam is 300 nm and the length L = 250 µm. Another similar sample of L = 15 µm has been characterized.

The beams store about 1 GPa of tensile stress, and we dene A to be their rectangular cross-section. For fabrication details see Methods below. The device is placed in a [START_REF] Chaste | A Nanomechanical Mass Sensor with Yoctogram Resolution[END_REF] He cryostat with temperature T 0 regulated between 1.4 K and 30 K, under cryogenic vacuum ≤ 10 -6 mbar. The motion of the beam is driven and detected by means of the magnetomotive scheme. [START_REF] Cleland | External Control of Dissipation in a Nanometer-scale Radiofrequency Mechanical Resonator[END_REF][START_REF] Collin | In situ Comprehensive Calibration of a Tri-port Nano-electro-mechanical Device[END_REF] For experimental details see Methods below. A Laplace force F (t) = F 0 cos(2πf t) with F 0 ∝ I 0 LB 0 is created with a static in-plane magnetic eld B 0 and an AC current I 0 fed into the metallic layer (Fig. 1). Fields B 0 of the order of 1 T, and currents I 0 up to 0.5 µA have been used. The detected signal is the induced voltage V (t) proportional to velocity. It is measured with a lock-in from which we can obtain the two quadratures X, Y of the motion. We call R = √ X 2 + Y 2 the amplitude of the motion (at a given frequency), dened in meters peak. For all the T 0 , B 0 settings used in the present work, the Al layer was not superconducting. A key feature of the magnetomotive scheme is that it enables the ability to tune the Q factor of the detected resonances: [START_REF] Cleland | External Control of Dissipation in a Nanometer-scale Radiofrequency Mechanical Resonator[END_REF] this is the so-called loading eect.

At low drives, in the linear regime, the quality factor of the resonance Q = f 0 /∆f is dened from the linewidth ∆f and the resonance frequency f 0 of the mode under study. We consider here only high-Q resonances Q 1. In this limit, the X peak is a simple Lorentzian, whose full-width-at-half-height gives ∆f . For large excitation forces, our doubly-clamped beams' mechanical modes behave as almost ideal Dung resonators. [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF][START_REF] Matheny | Nonlinear Mode-Coupling in Nanomechanical Systems[END_REF] A typical Dung resonance is shown in Fig. 2. The maximum of the resonance shifts with motion amplitude as f max = f 0 + βR 2 max . β is the so-called Dung parameter. We assume β > 0, but the case β < 0 is straightforward to adapt. R max is the maximum amplitude of motion; it always satises R max = F 0 Q/k 0 with very good accuracy. [START_REF] Collin | Addressing Geometric Nonlinearities with Cantilever Microelectromechanical Systems: Beyond the Dung Model[END_REF] k 0 is the mode's spring constant with f 0 = 1 2π k 0 /m 0 and m 0 the mode mass. In the nonlinear Dung regime, a damping parameter ∆f can still be dened from the Q factor deduced from the peak height R max .

When frequency noise is negligible, the so-called decoherence time T 2 = 1/(π∆f ) dened from such frequency-domain measurements is just equal to T 1 , the relaxation time of the amplitude R in time-domain. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] When R max ≥ 2 √ 3 R s a response hysteresis opens 29 (see Fig. 2). The point in the (R, f ) space at which this begins is called the spinodal point, with R s = 1 3 1/4 ∆f β . Beyond this point, two stable states coexist in a range of frequencies, with the system jumping from one to the other at f up (in an upward frequency sweep) and f down (for downward). These are the two bifurcation points. The maximum amplitude of the resonance is reached only on the upper branch. These frequencies write explicitly:

f up = f 0 + 2βR 2 up - βR 2 up 2 - 1 4 ∆f 2 for R up ≥ R s , (1)
and:

f down = f 0 + 2βR 2 down -(βR 2 down ) 2 - 1 4 ∆f 2 for 3 1/4 √ 2 R s < R down ≤ R s (Case 1)
,

f down = f 0 + 2βR 2 down + (βR 2 down ) 2 - 1 4 ∆f 2 for R down ≥ 3 1/4 √ 2 R s (Case 2). (2)
These functions are displayed in Fig. 3. As we increase the driving force F 0 , the maximum amplitude R max linearly increases while the peak position shifts quadratically. The upper bifurcation point f up then shifts monotonically towards higher frequencies, with a monotonically increasing amplitude R up when F 0 is increased. On the other hand, the lower bifurcation point f down has rst an amplitude R down that decreases (f down being given by Case 1 above), and then it increases again (f down being then dened through Case 2). At

the spinodal point R s , f up = f down = f s = f 0 + √ 3
2 ∆f . The method we present builds on the work by Aldridge and Cleland: 30 the bifurcation

σ f = 0 . 4 8 H z C o u n t f d o w n -< f d o w n > ( H z ) 0 8 0 0 0 1 6 0 0 0 -1 . -1 . -0 . 0 . 0 . 1 . 1 . f d o w n -< f d o w n > ( H z ) T i m e ( s ) 1 0 -4 1 0 -3 1 0 -2 1 0 -6 1 0 -4 1 0 -2 1 0 0 1 0 2 S f ( a . u . )
F r e q u e n c y ( H z )

A )

-1 . 0 -0 . 5 0 . 0 0 . 5 1 . 0 0 5 0

1 0 0 1 5 0 σ δf = 0 . 2 6 H z C o u n t δf ( H z ) 0 8 0 0 0 1 6 0 0 0 -0 . 8 -0 . 4 0 . 0 0 . 4 0 . 8 δf ( H z ) T i m e ( s ) 1 0 -4 1 0 -3 1 0 -2 1 0 -4 1 0 -2 1 0 0 B ) S δf ( a . u . )
F r e q u e n c y ( H z ) positions are essentially arbitrarily well dened (in the sense that in an ideal system only the thermal activation of the bifurcation process will limit the stability), and can be used for sensing/amplifying. We thus devise a technique enabling the characterization of frequency uctuations themselves; indeed, the imprint of frequency uctuations had been reported earlier in noise-induced bifurcation relaxation. [START_REF] Defoort | Scaling Laws for the Bifurcation Escape Rate in a Nanomechanical Resonator[END_REF] We show in Fig. 4 histograms obtained on the f down frequency position of the resonance of Fig. 2. They are measured by ramping the frequency down from above f up at constant speed, and measuring the switching to the higher branch through a threshold detection. We repeat this protocol typically 1000 times to acquire enough statistics (see Fig. 4 graph A left inset). The histogram obtained directly from the frequency time-trace is then t to a Gaussian (of standard deviation σ f , graph A in Fig. 4), while the power spectrum S f (f ) of the uctuations is also computed (dened as the FFT of the auto-correlation function). It displays a 1/f -type structure (see Fig. 4 A right inset).

Some precautions have to be taken in order to ensure that the acquired data is unbiased:

we rst make sure that the bifurcation jump occurs within a single point of the acquisition trace (we thus have to lower the ltering time constant of the lock-in compared to Fig. 2). Typically, we take one point every 40 ms with a frequency resolution typically 10 times smaller than the measured Gaussian spread. Second, we verify that we do not suer from Brownian-type motion amplitude noise (at the mode frequency) that would activate relaxation of states when we are close enough to the bifurcation points. [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] Such activated bifurcation generates non-Gaussian and asymmetric statistics, which is ramping-speed dependent. [START_REF] Aldridge | Noise-enabled Precision Measurements of a Dung Nanomechanical Resonator[END_REF] No such characteristics have been seen in our experiments: we rst check that the ramping speed (of order 0.1 -1 Hz/sec) does not change the measured histogram; and second, we add a controlled amount of force noise (at the mechanical resonance) in order to see when relaxation is indeed noise-activated. [START_REF] Defoort | Scaling Laws for the Bifurcation Escape Rate in a Nanomechanical Resonator[END_REF] We see that a force noise equivalent to a bath temperature of about 10 6 K has to be reached in order to aect the frequency statistics.

Note that 10 6 K is also the range of eective temperatures that are needed in order to see (asymmetric) frequency uctuations transduced from Brownian motion through the Dung nonlinearity. [START_REF] Maillet | Nonlinear Frequency Transduction of Nanomechanical Brownian Motion[END_REF] Clearly, at 4.2 K with no added noise no such phenomena can occur. In the following, we make sure that no extra force noise is injected in the setup while measuring frequency uctuations. Finally, the frequency drifts of our generators are characterized: we take two of them of the same brand, and measure the frequency stability of one against the clock of the other. Slow frequency uctuations occur at the level of 1 mHz for 1 MHz signals over minutes, and 10 mHz for 10 MHz. This is at least two orders of magnitudes smaller than what is seen here over the same periods of time, and can be safely discarded.

We see that frequency uctuations display a typical 1/f -type behaviour (right inset in Fig. 4 A), as reported by others. [START_REF] Sansa | Frequency Fluctuations in Silicon Nanoresonators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] Indeed, the time-trace has clearly some slowly drifting component (left inset in the same graph). This means that the statistics obtained depends on the acquisition bandwidth. For pure 1/f noise, the standard deviation σ f (which is the square root of the power spectrum integral) depends on ln (f high /f low ), with f high the fastest frequency probed (dened from the time needed to acquire 1 bifurcation trace ∆t min , about 10 seconds) and f low the lower frequency cuto (set by the total acquisition time, about 3 hours). In order to be as quantitative as possible, we look for an estimate of frequency noise which is as much independent from the protocol as possible. We therefore study the frequency jumps δf (t) = f down (t i+1 ) -f down (t i ) instead of f down (t), see Fig. 4 B left inset.

The variance of this quantity σ 2 δf is the well known Allan variance, [START_REF] Allan | Time And Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators[END_REF] computed for acquisition time ∆t min = t i+1 -t i . Note that this quantity safely suppresses equipment low frequency drifts (like, besides the one characterized for the generator, e.g. temperature drifts due to the 4 He bath).

For a perfectly 1/f noise, the Allan variance at ∆t min → 0 is independent 6 of f high , f low (see Methods). However, our power spectrum ts S 0 /f 1+ with ≈ 0.4 ± 0.2. We calculate that over the most extreme settings that have been used, our Allan variance should not have changed by more than 50 % (and in the data presented here by much less). Moreover, to prove that the ∆t min → 0 assumption can be applied we display in the right inset of Fig. 4 B the power spectrum S δf (f ) of δf uctuations: the data match the spectrum ∝ f directly computed from the t of S f (f ).

We can then compare the values of σ f and σ δf that have been obtained. According to theory (see Methods) the rst one should be about twice the second one in our conditions. This is indeed what we see in Fig. 4, conrming that σ δf is a good quantitative measurement of frequency uctuations. Besides, the frequency stability dened as σ δf /f 0 ≈ 0.3 ppm falls within the expected range according to reported measured devices. [START_REF] Sansa | Frequency Fluctuations in Silicon Nanoresonators[END_REF] We thus demonstrate that our simple technique is functional: the key being that since the spectrum is 1/f -type, we do not need to be especially fast to characterize frequency noise. With a Phase Locked Loop (PLL) setup one could measure uctuations on much shorter timescales, 10 but our 10 s repetition rate is perfectly well adapted.

We now build on our method to thoroughly characterize frequency noise in SiN string devices. Let us apply this technique to the upper bifurcation point f up . The method explained above is easily reversed in sweep direction and threshold detection. Similar time-trace, spectrum and histogram to Fig. 4 obtained this way are shown in Fig. 5. We see that the power spectrum displays the same 1/f 1+ law with ≈ 0.4. This is true for the complete F r e q u e n c y ( H z )

Figure 5: Large amplitude statistics. Similar graphs as in Fig. 4 obtained on the upper bifurcation point f up (frequency jumps time-trace in the left inset, power spectrum of frequency in the right inset and histogram of δf in the main of the graph). The line is a Gaussian t, while the power spectrum follows 1/f 1.4 (see text).

study we realized on the same device, and proves that dierent data sets can be consistently compared. The histogram is again Gaussian. But surprisingly, σ δf is now much bigger on the upper branch than on the lower one. We therefore make a complete study as a function of driving force (i.e. motion amplitude). We discover that the standard deviation σ δf depends quadratically on motion amplitude R. Measuring at another magnetic eld B 0 , we nd that it also depends linearly on the Q of the mechanical mode. However, measuring at dierent temperatures T 0 in the 1.4 K -30 K range, we realize that the small amplitude value obtained is temperature-dependent, while the large amplitude one is not.

This suggests the normalized plot of Fig. 6, where the increase σ δf (T, R up,down ) from the

extrapolated σ δf (T, R s ) is plotted against the normalized variable R 2 up,down /R 2
s . The notation R s ∼ 0 means that the value is obtained from the linear t, extrapolating at R → 0. In order to verify the robustness of the result, we realize the same analysis with a similar device of 15 µm length. Some typical data is displayed in the inset of Fig. 6. The noise properties obtained for this other device are very similar to the initial 250 µm long beam (but the quadratic dependence is dierent). However the spectra better t with ≈ 0.2 ± 0.2. We proceed with similar measurements performed on modes n = 3 and n = 5 of the 250 µm beam sample. All spectra display the same 1/f 1.4 dependence as the n = 1 mode.

The same normalized plots are displayed in Fig. 7. However, this time the inferred quadratic dependencies are much weaker.

The nonlinear dependence of the frequency noise is rather unexpected; indeed, the nonlinear dephasing features observed for carbon-based devices [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF] have not been reported for nitride structures. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] A possible source for such an eect could be a purely intrinsic property of the bifurcation eect itself. However, since our statistics could not be altered by reasonable changes in eective temperatures and frequency-sweep ramping speeds, such an explanation is improbable. If we then suppose the bifurcation process to be perfectly ideal, the nonlinear frequency noise observed should originate in one of the parameters dening the bifurcation frequencies. When the experiment is performed reasonably far from the spinodal point (which is always our case), we have:

f up ≈ f 0 + δφ(t) + βR 2 up , (3)
f down ≈ f 0 + δφ(t) + 3βR 2 down , (4)
R up ≈ R max , (5)
R down ≈ 1 2 2/3 R max ∆f β 1/3 , (6)
adapted from Eqs. (1-2), where we explicitly introduced the stochastic frequency variable δφ(t). For strings [START_REF] Lifshitz | Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators[END_REF]

β ∝ 1/(m 0 f 0 ) E Y A L 3
, the nonlinear frequency noise could be caused by (Gaussian) uctuations of the Young's modulus E Y = E 0 + δE(t). However, to have the measured characteristics, this noise would have to be δE/E 0 ∝ 1/∆f and mode-dependent, which is dicult to justify: this explanation seems again unphysical.

The only possibility left is an internal motional noise with R max = R max 0 + δR, leading to uctuations ∝ δR Rmax 0 βR 2 max 0 . The proper scalings, as reported in Figs. 6 and7, are thus only achieved by assuming damping noise δΓ(t) with δR Rmax 0 = -δΓ ∆f . As a result, it follows from Eqs. (3-6):

f up (t) ≈ f 0 + δφ(t) + βR 2 max 0 1 -2 δΓ(t) ∆f , f down (t) ≈ f 0 + δφ(t) + 3 2 4/3 (∆f 2 βR 2 max 0 ) 1/3 together with f s (t) = f 0 + δφ(t) + √ 3 2 ∆f 1 + δΓ(t)

∆f

. This means that both bifurcation frequencies suer from frequency noise δφ, while only the upper branch experiences damping uctuations δΓ: they are amplied by the measurement method through a factor βR 2 max 0 /∆f . Note that the frequency noise extrapolated at R → 0 on the upper branch matches the one of the lower branch, but does not equal the one obtained at the spinodal point, simply because the expressions Eqs. (3-6) do not apply near R s ; this is emphasized through the writing R s ∼ 0 in our graphs. The Allan deviation σ δf extrapolated to R → 0 is thus characteristic of the frequency noise δφ, while the slopes of the graphs in Figs. 6 and 7 are 2 √ 3 times the Allan deviation σ δ∆f of the damping uctuations. To our knowledge, the latter has not been reported in the literature so far. The mode parameters together with these 4.2 K frequency and damping noise gures are summarized in Table 1. σ δ∆f is temperature-independent in the range studied, while σ 2 δf is linear in T 0 ; this is displayed in Fig. 8. The same temperaturedependence of frequency noise has been reported in Ref. [START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] (within an overall scaling factor) for a very similar device. In order to compare the various results, values from the literature are presented in Table 2. We give the Allan deviation when it is reported, otherwise we list the direct frequency noise; the damping noise in the third line is recalculated from Fig.

IV.19 in Ref. [START_REF] Defoort | In Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF] We can only speculate on the microscopic mechanisms behind the reported features. The entities generating such noise are thought to be atomic-scale two level systems (TLS), which δf as a function of temperature (rst mode of 250 µm device), as obtained for small motion R → R s ∼ 0. The line is a linear t, with the T 0 = 0 K extrapolated value emphasized by the arrow (see text). The variance of damping uctuations σ 2 δ∆f is constant in the same range of temperatures. 36 Hz X could be defects or intrinsic to the materials in use. [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF][START_REF] Behunin | Dimensional Transformation of Defect-induced Noise, Dissipation, and Nonlinearity[END_REF][START_REF] Defoort | In Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF] A signature that also supports this view is the presence of telegraph frequency noise in many NEMS experiments (see e.g.

Ref. [START_REF] Defoort | In Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF] ). Ref. [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF] analyzed frequency noise in beams, i.e. structures with no built-in stress.

These Authors assumed that thermally activated motion of a defect in a double well potential from one minimum to the other caused a shift in the local Young's modulus. The motion of many such defects (following the mathematical arguments of Ref. [START_REF] Dutta | Low-frequency Fluctuations in Solids: 1/f Noise[END_REF] ) then causes a change in the average Young's modulus and consequently a change in the resonance frequency of the beam with power spectrum ∝ T /f . The same argument was applied in Ref. [START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] to analyze the frequency noise of string structures, where the built-in stress is large, even though the resonance frequency is nearly independent of the Young's modulus in the high-stress limit.

We thus believe that it is more appropriate, in the interpretation of the present measurements on strings, to consider frequency uctuations due to stress uctuations. Indeed, point defects in crystals are characterized as elastic dipoles that cause an orientation dependent change in the strain (and consequently the stress) of a crystal. [START_REF] Nowick | Anelastic Relaxation in Crystalline Solids[END_REF] In our highest Q device, for the rst exure n = 1 frequency uctuations at 4.2 K represent about 20 % of the linewidth, and damping uctuations about 5 %. These parameters fall with mode number n (see Tab. 1), while both frequency f 0 and linewidth ∆f increase.

This means that the eect of uctuations is the strongest on the rst mode, but is usually dicult to visualize on standard frequency-sweeps or time-decay data; it is for instance expected that for a device rather equivalent to our 250 µm beam, the measurements performed in Ref. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] did not report any such features (see Methods).

Conclusion

In this Letter we present a very simple and reliable method to measure and characterize frequency noise in bistable resonators. The technique has been employed to describe thoroughly the intrinsic frequency noise of high-stress silicon-nitride doubly-clamped beams. The measurements have been performed at low temperatures in cryogenic vacuum, on two devices of very dierent lengths/fundamental resonance frequencies. The 3 rst symmetric exural modes of the longest beam have been studied.

We report on the Allan deviations of the frequency noise, presenting the same basic features as in Refs.: [START_REF] Sansa | Frequency Fluctuations in Silicon Nanoresonators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] spectra of 1/f -type, scaling linearly with temperature. The reported magnitudes of the noise δf /f 0 fall in the 0.5 -0.01 ppm range, as expected for MHz devices.

We have also found unexpectedly damping uctuations, which are amplied in the vicinity of bifurcation points. Our technique seems the most adapted for the detection and the quantication of such a noise process to date. We nd that damping noise can be as large as about 5 % of the total width of the resonance peak in our highest Q devices. It also sets a nite resolution attainable for the measurement of the frequency position of bifurcation points.

These features seem ubiquitous to all NEMS devices, and we do believe that damping noise and frequency noise originate in the same microscopic mechanism. But the latter remains elusive, and the most discussed candidate is based on Two-Level Systems (TLS). [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF][START_REF] Fong | Frequency and Phase Noise of Ultrahigh Q Silicon Nitride Nanomechanical Resonators[END_REF] Because of the strength of the frequency noises reported here for high-stress devices, we propose that TLS are responsible for noise on the stored stress in the structure instead of the Young's modulus, as it was proposed in earlier papers discussing stress-free beams (e.g.

Ref. [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF] ).

Our technique can be easily adapted to any types of devices, including MoS 2 and carbonbased systems in which nonlinear frequency noise has been reported. [START_REF] Schneider | Observation of Decoherence in a Carbon Nanotube Mechanical Resonator[END_REF] We think that it should help advance the understanding of the underlying fundamental microscopic mechanisms that also signicantly degrade the properties of existing NEMS devices, and hinder their applicability.

Methods

Device Fabrication

The structure was fabricated using e-beam lithography on a silicon substrate covered with a 100 nm silicon nitride (SiN) layer. The stochiometric nitride was grown using low pressure chemical vapor

Measurements

The voltage drive was delivered by a Tektronix AFG3252 generator, through a 1 kΩ bias resistance which created the drive current. The motion was actuated with the magnetomotive technique through a force F (t) = I 0 ζLB 0 cos(2πf t), which also leads to the detection of the velocity ẋ(t)

of the oscillation through a voltage V (t) = ζLB 0 ẋ(t). ζ is a mode-dependent shape factor. [START_REF] Cleland | External Control of Dissipation in a Nanometer-scale Radiofrequency Mechanical Resonator[END_REF] In the high Q limit, the velocity in frequency domain is iω 0 x(ω) with ω 0 = 2πf 0 , hence an inverted denition for the signal quadratures X and Y with respect to displacement x(t). Due to the symmetry of the scheme, only symmetric modes (n = 1, 3, 5 • • • ) can be addressed (ζ = 0 otherwise).

The magnetic eld was generated with a small superconducting coil fed with a 10 A Kepco current source. The detected signal was processed with a Stanford SR 844 RF lock-in amplier. Due to the nite impedance of the electric circuit seen by the NEMS (whose own characteristic impedance varies as B 2 0 ), the mechanical resonances are loaded by an additional damping ∝ B 2 0 . This provides the ability to tune the quality factors in situ. [START_REF] Cleland | External Control of Dissipation in a Nanometer-scale Radiofrequency Mechanical Resonator[END_REF] Our calibration procedure is described in Ref. [START_REF] Collin | In situ Comprehensive Calibration of a Tri-port Nano-electro-mechanical Device[END_REF] It enables us to give all mechanical parameters in S.I. units (we thus quote X,Y in meters), while minimizing the loading eect. Loading is negligible for the 15 µm beam, but still large in the 250 µm device.

Spectra Mathematical Properties

Let us consider a frequency power spectrum for the stochastic resonance frequency ω 0 = 2πf 0 of type S ω 0 (ω) = A 0 / |ω| 1+ (dened from -∞ to +∞). The variance can be dened from the integral of the spectrum, leading to σ 2

ω 0 = 1 2π S ω 0 dω = 2A 0 2π ω - low -ω - high
with ω low and ω high the lower and higher frequency cutos imposed by the experiment. For → 0 we have σ ω 0 ∝ ln [ω high /ω low ]. Since δω = ω 0 (t i+1 ) -ω 0 (t i ) ≈ ∂ω 0 (t)/∂t × ∆t min when ∆t min → 0, we have S δω (ω) ≈ ∆t 2 min ω 2 S ω (ω). Thus

σ 2 δω ≈ 2A 0 2π (∆tmin ω high) 2 2 ω - high 1-(ω low /ω high) 2- 1-/2
. The Fourier Transform imposes ∆t min ω high ≈ π, and in the case → 0 we obtain σ δω ∝ 1 -(ω low /ω high ) 2 , which is essentially independent of the cutos. [START_REF] Cleland | Noise Processes in Nanomechanical Resonators[END_REF] For = 0, a small dependence to the bandwidth appears in the Allan variance σ 2 δω . For our acquisition bandwidths, this does not result in a too large scatter in data (within error bars).

Impact on Frequency-domain & Time-domain Measures

With a noise of type S ω 0 (ω) = A 0 / |ω| 1+ , we can take as an estimate of the relevant uctuations timescale τ -1 c ∼ ω low /π: the weight is at the lowest accessible frequencies. We thus always verify σ ω 0 τ c 1, which means that the phase diusion of the mechanical mode is in the Inhomogeneous Broadening limit (IB), in analogy with Nuclear Magnetic Resonance. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF][START_REF] Zhang | Spectral Eects of Dispersive Mode Coupling in Driven Mesoscopic Systems[END_REF][START_REF] Maillet | Nonlinear Frequency Transduction of Nanomechanical Brownian Motion[END_REF] In frequency-domain, the response χ meas (ω) is the convolution of the standard (complex-valued, dening the two quadratures) susceptibility χ(ω) with the (Gaussian) distribution of frequencies

p(δϕ) = 1/ 2πσ 2 ω 0 exp -1 2 δϕ 2 /σ 2
ω 0 , with δϕ = 2πδφ in Rad/s. This means that at each scanned frequency ω, the measurement is performed over a long enough timescale such that all uctuations are explored. On the other hand, the small damping uctuations are simply ltered out by the acquisition setup (here, a lock-in amplier): they have no relevant impact on the resonance peak measured, even at very large motion amplitudes. We conclude that only frequency noise will contribute to the denition of a T 2 , the decoherence time involving relaxation T 1 and dephasing σ ω 0 . [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] In time-domain, in the linear regime the complex susceptibility χmeas (t) (i.e. decay of the two quadratures) is simply the Fourier Transform (FT) of χ meas (ω). It can also be written χmeas (t) = exp (iδϕt) χ(t) with χ(t) the FT of χ(ω) and exp (iδϕt) = exp -1 2 σ 2 ω 0 t 2 the average over frequency uctuations. In the nonlinear regime, the averaged decay of the two quadratures writes χmeas (t) = exp (iδϕt) exp -i2

βR 2 max 0 ∆ω δγ t κ [t] χ(t) with κ [t] = 1-exp[-t∆ω]
t∆ω and χ(t) dened in Ref., [START_REF] Collin | Addressing Geometric Nonlinearities with Cantilever Microelectromechanical Systems: Beyond the Dung Model[END_REF] the second average being over damping uctuations δγ = 2πδΓ; we wrote ∆ω = 2π∆f in Rad/s. The function κ is characteristic of the decay of the nonlinear frequency pulling due to the Dung term, 28 ∝ R 2 max 0 . In practice, this assumes that both quadratures are measured independently, averaging many decay traces starting from the same (noisy) t = 0 amplitude R max , imprinted by the slow uctuations of damping δγ. The second average can be explicitly calculated: exp -i2 δ∆ω (t κ [t]) [START_REF] Li | Ultra-sensitive NEMS-based Cantilevers for Sensing, Scanned Probe and Very High-frequency Applications[END_REF] . In time-domain, the impact of damping uctuations (of variance σ 2 δ∆ω ) is thus amplied by the same term as in the bifurcation measurement: βR 2 max 0 ∆ω . However, to have a measurable eect (within experimental error bars) both amplitude R max 0 and uctuations σ δ∆ω have to be very large; in experiments of the type of Ref. [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] based on devices similar to our 250 µm, no such eect has been reported. Measuring the decay of the two quadratures leads then to the denition of T2 , roughly equivalent to T 2 . [START_REF] Maillet | Classical Decoherence in a Nanomechanical Resonator[END_REF] Note that the decay of the motion amplitude | χ(t)| 2 remains unaected by frequency and damping noise, leading to the proper T 1 denition. Graphical TOC Entry Some journals require a graphical entry for the Table of Contents. This should be laid out print ready so that the sizing of the text is correct.

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Figure 1 :

 1 Figure 1: Device and setup. SEM image of the 250 µm device measured in this work. The gate electrode (brown) is not used here. The actual NEMS device is the red-colored string in between the two (light blue) electrodes. The lock-in detector (violet), magnetic eld and drive generator (in green) are also depicted in a schematic fashion to illustrate the magnetomotive technique.

Figure 2 :X 2 +

 22 Figure 2: Nonlinear (Dung) resonance. A) Dung resonance line (X and Y quadratures) measured on the 250 µm device at 4.2 K in vacuum, for a drive force of 81 pN, in a 1 T eld. The directions of frequency sweeps are depicted by arrows. Vertical dashed lines indicate the two bifurcation points f up,down . B) Amplitude R = √ X 2 + Y 2 corresponding to A). Bifurcation points are indicated with their amplitudes R up,down .

  t e r e t i c r a n g e

Figure 3 :

 3 Figure 3: Bifurcation branches. Calculated bifurcation frequencies from Eqs. (1-2) for the 250 µm device and magnetomotively-loaded Q of 5 000 (1 T eld).

Figure 4 :

 4 Figure 4: Statistics on frequency at low amplitudes. A): histogram obtained on the f down relaxation point of Fig. 2 (left inset: actual frequency time-trace; right inset: power spectrum of the frequency uctuations, displaying 1/f -type structure). B): histogram obtained on the frequency jumps δf computed from the f down time-trace (left inset: δf time-trace; right inset: power spectrum). The lines are Gaussian ts, and the power-law dependencies of the spectra are discussed in the text.

Figure 6 :

 6 Figure6: Universal plot for mode 1. Frequency noise increase vs squared amplitude normalized to the spinodal value R s , for the rst mode n = 1 of the 250 µm device. Various T 0 and B 0 (hence Q) have been used (see legend). Squares stand for upper branch bifurcation, triangles for lower. Inset: same result obtained with a 15 µm beam having β = 1.1 × 10 19 Hz/m 2 (4.2 K and Q = 17 000 red squares; 800 mK and Q = 31 000 blue dots; the magnetomotive eld broadening was negligible). The green line is a linear t (see text).

5 Figure 7 :

 57 Figure 7: Universal plot for modes 3 and 5. Normalized frequency noise plot for the same 250 µm device, for modes n = 3 (left) and n = 5 (right). The lines are linear ts (see text).

Figure 8 :

 8 Figure 8: Temperature dependence of frequency uctuations. Allan variance σ 2

  deposition (LPCVD) at the Cornell NanoScale Science & Technology Facility (CNF). It stores a biaxial stress of about 1 GPa. A 30 nm Aluminum coating has been evaporated onto the sample in a Plassys e-gun machine. Its resistance at low temperature is about 1 kΩ for the 250 µm long device, and 100 Ω for the 15 µm one. It served as a mask for the structure during the SF 6 Reactive Ion Etching (RIE) step used to pattern the SiN. The structure was then released using a nal XeF 2 etching of the underlying silicon.

13

 13

Figure 9 :

 9 Figure 9: ForTable Of Contents Only.

Table 1 :

 1 Mode parameters, frequency and damping uctuations for modes n = 1, 3 and 5 (250 µm long beam, 4.2 K). Mode number n Freq. f 0 Unloaded Q Dung β σ δf at R s ∼ 0 σ δ∆f from R up R s MHz 600 000 ± 10 % 8.5 ± 0.5 × 10 15 Hz/m 2 0.28 ± 0.05 Hz 0.11 Hz ±10 %

	n = 1	0.905

n = 3 2.68 MHz 450 000 ± 10 % 1.25 ± 0.2 × 10 17 Hz/m 2 0.1 ± 0.02 Hz 0.005 Hz ±15 % n = 5 4.45 MHz 400 000 ± 20 % 5.7 ± 0.5 × 10 17 Hz/m 2 0.09 ± 0.02 Hz 0.0025 Hz ±30 %

Table 2 :

 2 Mode parameters, frequency and damping uctuations for dierent devices (fundamental exure n = 1). The damping noise gure in the third line is recalculated from Fig.IV.19 in Ref.[START_REF] Defoort | In Non-linear dynamics in nano-electromechanical systems at low temperatures[END_REF] SiN/Al d.c. beam 4.2 K (this work) 0.905 MHz 600 000 ± 10 % 0.28 ± 0.05 Hz 0.11 Hz ±10 % 15 µm SiN/Al d.c. beam 4.2 K(this work) 17.5 MHz 18 000 ± 10 % 1.45 ± 0.1 Hz 0.6 Hz ±10 % 2 × 3 µm Si/Al goalpost 4.2 K Refs. 31,33 7.1 MHz 4 700 ± 10 % 1 ± 0.5 Hz on f 0 0.35 Hz ±10 % 380 µm SiN d.c. beam 5 K Ref. 11 0.64 MHz

	Device	Freq. f 0	Q	σ δf or σ f 0	σ δ∆f
	250 µm 2 200 000	0.01 Hz on f 0	X
	3.2 µm Si cantilever Room Temp. Ref. 7	45.2 MHz	6 000

Acknowledgement

We thank J. Minet and C. Guttin for help in setting up the experiment, as well as J.-F. Motte, S. Dufresnes and T. Crozes from facility Nanofab for help in the device fabrication. We acknowledge support from the ANR grant MajoranaPRO No. ANR-13-BS04-0009-01 and the ERC CoG grant ULT-NEMS No. 647917. This work has been performed in the framework of the European Microkelvin Platform (EMP) collaboration. At Cornell we acknowledge support from the NSF under DMR 1708341.