Christopher Bauërle

Kasper Grove-Rasmussen

Denis Fein

Matthieu Lauriane

Matthieu Dartiailh

Desjardins

Raphaël, Franois Zaki Federico

Merci

Michaël Rosticher

José Palomo

Aurélie Pierret

-butyl ketone NMR Nuclear magnetic resonance PCB

Keywords: Carbon nanotube quantum dots, Cavity quantum electrodynamics, Cooper pair splitter, strong coupling, spin-qubit

In this thesis, we study carbon nanotubes based quantum dot circuits embedded in a microwave cavity. This general architecture allows one to simultaneously probe the circuit via quantum transport measurements and using circuit quantum electrodynamics techniques. The two experiments realized in this thesis use metallic contacts of the circuit as a resource to engineer a spin sensitive spectrum in the quantum dots. The first one is a Cooper pair splitter which was originally proposed as a source of nonlocal entangled electrons. By using cavity photons as a probe of the circuit internal dynamics, we observed a charge transition dressed by coherent Cooper pair splitting.

Strong charge-photon coupling in a quantum dot circuit was demonstrated for the first time in such a circuit. A new fabrication technique has also been developed to integrate pristine carbon nanotubes inside quantum dot circuits. The purity and tunability of this new generation of devices has made possible the realization of the second experiment. In the latter, we uses two non-collinear spin-valves to create a coherent interface between an electronic spin in a double quantum dot and a photon in a cavity. Highly coherent spin transitions have been observed. We provide a model for the decoherence based on charge noise and nuclear spin fluctuations.

Mots clés : Boîtes quantiques en nanotubes de carbone; Electrodynamique quantique en cavité; Lame séparatrice à paires de Cooper; couplage fort; qubit de spin.

Introduction

Mesoscopic physics

Mesoscopic physics is the branch of condensed matter physics that focuses on systems of scale lying in between the micropscopic and the macroscopic world. Such systems typically contains millions of atoms, and are micrometers long. Building electronic circuits at this scale is particularly appealing since the quantum nature of electrons starts to emerge. Electrons in these circuits can be described as waves with a well defined phase relation over the whole circuit. In the ballistic regime, this was firstly demonstrated in nano-constriction called quantum point contact in 1988 [START_REF] Van Wees | Quantized conductance of point contacts in a two-dimensional electron gas[END_REF][START_REF] Wharam | One-dimensional transport and the quantisation of the ballistic resistance[END_REF], where the quantization of the conductance was measured for the first time.

Nevertheless, the coherence of these electronic waves can easily be lost due to collisions with other electrons or vibrations of the lattice. Besides the need to go to very small dimensions, one also need a clean material to limit these scattering events. This explains why the evolution of mesocopic physics is closely connected to the progress in materials discovery and synthesis. Nowadays, the variety of these high purity materials is impressive: 2D electron gas (GaAs/AlGaAs, Si/SiGe,...), semiconducting nanowire (InAs, InSb, SiGe,...), Graphene, Carbon nanotubes, Transition metal dichalcogenide monolayers (MoS 2 , WSe 2 , ...). Transport experiments usually require low temperature partly to limit inelastic scattering. Today, temperatures down to 20mK are routinely achieved in the lab using standard cryogenic refrigerators based on liquid 3 He and 4 He.

In condensed matter systems, one is often dealing with physical process involving a large number of particle. Interestingly, it is possible to identify elementary collective excitations, with defined quantum numbers. Such elementary excitations are sometimes called quasiparticles, and this approach of collective phenomena has been very prolific in the past century. In practice there is a large zoology of such quasiparticles, and among the most known, one can think of the Landau quasiparticles in Fermi liquid, the holes which are absence of electron in a Fermi sea, the vibrations in a crystal lattice called phonons, and many others. The family of quasiparticles also include non-collective excitations such as electron-hole bound state: the excitons, or the bogoliubons in superconductors, which are two of the many examples. Mesoscopic circuits open the possibility to isolate a single quasiparticle and make it interact with other kinds of quasiparticles. In that sense, mesoscopic physics appears as an extraordinary workbench for testing quantum theory.

Quantum dots circuits

In this thesis, we are interested in a special type of mesoscopic circuits: quantum dot circuits. When electrons are confined in a small region of a ballistic conductor they behave as standing waves, and display discrete energy levels, recalling the ones of an atom. A single quantum dot circuit has the same architecture of a transistor including a source and a drain electrode carrying a current and a gate electrode to change the state of the transistor. This is the reason why many of the nanofabrication techniques developed in the microelectronic industry can be transposed to quantum dot circuit fabrication.

Carbon nanotubes are excellent candidates for hosting such electron boxes [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]. Because of their small transverse dimension (diameter 1nm), one only needs to confine electrons in a short section of the nanotube to realize 3D confinement. This is generally done by applying an electrostatic potential on a nearby gate electrode. Circuits studied in this thesis are double quantum dots, but one can envision more complex ones such as the one illustrated in figure 1.

nm

Figure 1: multiple carbon nanotube quantum dot circuit: False-color scaning electron micrograph showing two carbon nanotubes (in white) suspended over gate arrays (in blue), and contacted by separated sources and drain contacts (in yellow).

Using multiple gate electrodes allows to shape in a controlled way the confinement potential of electrons in the carbon nanotube. Source [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF] Cavity and circuit quantum electrodynamics (C-QED)

In parallel to the progress made in mesoscopic physics, a strong effort in the AMO community (Atom, Molecule, Optics) was given to the study of light-matter interaction at its most fundamental level, by manipulating a single atom and a single mode of the electromagnetic field. Cavity QED is now a resource for entangling photons and atoms, quantum non-demolition read-out of states or the generation and stabilization of non-classical states of light [START_REF] Sayrin | Real-time quantum feedback prepares and stabilizes photon number states[END_REF]. This exquisite control over the atom-photon system is possible thanks to a strong decoupling to the outside world. Indeed, the very weak interaction with the environment allows the atom to have a fully quantum dynamics over long periods of time (the decoherence time of Rydberg atom is typically T atom ∼ 10ms [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]). While the realization of such closed systems in a condensed matter environment may seem unrealistic, the use of a non-dissipative circuit element, the Josephson junction, has made the transposition of such experiment in mesoscopic circuits possible. The Josephson junction inherits the quantum macroscopic coherence of superconductivity, and led to numerous architectures of superconducting circuits [START_REF] Mh Devoret | Experimental Aspects of Quantum Computing[END_REF]. The possibility to realize two-level systems, analogous to a single spin 1/2, with such circuits has been particularly investigated. Although, this artificial atom has significantly shorter coherence time than atoms (T 2 of several 100 µs have been reached with transmon qubit in a 3D-cavity [START_REF] Reagor | Quantum memory with millisecond coherence in circuit QED[END_REF]), they have large electric dipoles compensating the reduction of coherence times with respect to atoms. This places them very well in the race for quantum computing architectures.

Mesoscopic quantum electrodynamics

The idea of using a cavity has then been transposed to quantum dot circuits [START_REF] Childress | Mesoscopic cavity quantum electrodynamics with quantum dots[END_REF]. One can reproduce experiments achieved in Cavity and circuit-QED with new degrees of freedom. The most natural two-level system (TLS) one can think of is probably the spin 1 2 of an electron and an implementation of such a spin qubit is presented in chapter 5. The charge distribution in a double quantum dot can also encode a TLS. These two types of qubit have recently reached the strong coupling regime with a photon in a cavity in different platform (see figure 3 for the charge qubit) and the chapter 4 presents one of them. In this regime, the TLS and the cavity coherently exchange energy faster than any decoherence rate in the coupled system.

One of the resources in quantum dot circuit is the possibility to use specific contact reservoirs such as ferromagnet or superconductor to induce electronic correlation in the quantum dots. This idea is at the heart of the two experiments presented in chapter 4 and 5 and can be understood as the shaping of the electronic spectrum in quantum dots.

This thesis is organized as follows: the first chapter introduces double quantum dot circuits in carbon nanotubes, and emphasizes on two realizations that are the Cooper pair splitter and the ferromagnetic spin-qubit. In the second chapter a theoretical description of the cavity-circuit coupling is given. In particular, this chapter details the coupling of microwave cavity photons to double quantum dot circuits and how to improve the cavity-quantum dot circuit interface. Chapter three describes the nanofabrication of the mesoscopic-QED devices, and the measurement techniques used in this thesis work. Figure 3: strong charge-photon coupling in mesoscopic-QED: a. Cavity transmission measurement showing the avoiding crossing between a silicon charge qubit energy and the cavity resonance frequency [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF]. b. Similar measurement with a GaAs charge qubit coupled to a high impedance resonator [START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF]. A vertical cut of the left panel measurement display a splitting of the cavity resonance, called vacuum Rabi splitting. c. Vacuum Rabi splitting observed with a carbon nanotube charge-like qubit [START_REF] Bruhat | Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs[END_REF].

coupling in a superconducting double quantum dot for the chapter four, and the observation of highly coherent spin transition in a carbon nanotube based spin qubit.

Chapter 1

Hybrid double quantum dots in carbon nanotube

When an electron is confined to a small region of space, its spectrum will reveal discrete energy levels. Such systems are often called quantum dots or artificial atoms because of the analogy with an electron confined in the potential generated by a nucleus. In the same philosophy, one can also realize artificial molecules by bringing closely enough two quantum dots. Among the solid state platforms to implement such double quantum dots, one finds two-dimensional electron gases in semiconducting hetero-structures (GaAs/GaAlAs [START_REF] Oosterkamp | Photon Sidebands of the Ground State and First Excited State of a Quantum Dot[END_REF], Si/SiGe [START_REF] Nakul Shaji | Spin blockade and lifetimeenhanced transport in a few-electron Si/SiGe double quantum dot[END_REF]), nanowires (InAs [START_REF] Thelander | Single-electron transistors in heterostructure nanowires[END_REF], InSb [START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF],... ), graphene [START_REF] Güttinger | Spin States in Graphene Quantum Dots[END_REF],

and carbon nanotubes [START_REF] Mason | Local Gate Control of a Carbon Nanotube Double Quantum Dot[END_REF].

The double quantum dot (DQD) have more complex internal dynamics, and transition between the two dots can occur. Thanks to nanofabrication techniques, the tunneling rate between the two dots can easily be tuned in the microwave range, which make double quantum dot particularly attractive for quantum computing applications.

In this chapter, I will present several properties of carbon nanotubes (section 1.1), and introduce basic concepts about double quantum dots (see section 1.2). Then I will describe two specific implementations of hybrid double quantum dots (see sections 1.3 and 1.4) which experimental realization are detailed in chapters 4 and 5.

Carbon nanotube based quantum dot circuits

Carbon nanotubes possess several properties that make them very attractive for quantum dot circuit applications. In this section I review few of them.

Properties of carbon nanotubes

The atomic structure of carbon nanotubes is identical to the one of graphene, hence they share similar transport features. Their hexagonal atomic arrangement is built on covalent bonds between sp 2 orbitals with the three neighboring atoms. The forth valence electron occupy the p z orbital and is responsible for transport in carbon nanotubes.

Because p z orbitals are pointing out of plan, it is easy to realize good electrical contacts with almost every metal. Ferromagnetic and superconducting metals have thus been widely used to contact hexagonal carbon materials, with the idea of inducing specific The zone folding approximation: The rolling-up of a graphene sheet impose new boundary conditions to the transverse part of the wave vector. The Brillouin zone of a carbon nanotube is form of parallel lines. Whether or not one of the line is passing right through the middle of a Dirac cone will determine if the carbon nanotube is semiconducting or metallic. Source: [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF].

electronic spin correlations in it [START_REF] Jarillo-Herrero | Quantum supercurrent transistors in carbon nanotubes[END_REF], [START_REF] Cleuziou | Carbon nanotube superconducting quantum interference device[END_REF], [START_REF] Hofstetter | Cooper pair splitter realized in a two-quantum-dot Y-junction[END_REF], [START_REF] Herrmann | Carbon Nanotubes as Cooper-Pair Beam Splitters[END_REF], [START_REF] Tan | Cooper Pair Splitting by Means of Graphene Quantum Dots[END_REF], [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. This represents a big advantage compared to 2DEGs for which the choice of contacting metal is limited.

In return, this makes transport in graphene and carbon nanotubes very sensitive to their environment. This motivated the use of suspended carbon nanotubes to avoid interactions with the substrate [START_REF] Kuemmeth | Coupling of spin and orbital motion of electrons in carbon nanotubes[END_REF]. In the case of graphene, the encapsulation between two boron-nitride sheets has been found to be very efficient [START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF].

Many features of the band structure of carbon nanotube inherit from the graphene one. Indeed, carbon nanotubes can be seen as a rolled-up graphene sheet. A good approximation is to consider that curvature does not modify the bands dispersion, but only adds new boundary conditions to the transverse component of the wave vector k ⊥ . This is the zone-folding approximation, and holds as long as the overlap between neighboring p z orbital remains negligible. The quantization of k ⊥ limits the first Brillouin zone of carbon nanotube (CNT) to a set of parallel lines in the Brillouin zone of graphene with an angle determined by the rolling condition (see figure 1.1). Depending on whether or not one of the lines is passing by the center of one of the Dirac cones, the carbon nanotube can be semiconducting (a typical value of the semiconducting gap is E G = reciprocal space. Consequently, CNT initially predicted to be metallic (according to the zone folding approximation) can turn out to be semiconducting. Those CNTs are often referred to as "narrow gap" carbon nanotubes and have a semiconducting gap E G ranging between 10 meV and 100 meV. Strain in carbon nanotubes can produces similar effects [START_REF] Minot | Tuning Carbon Nanotube Band Gaps with Strain[END_REF], but in the end, metallic CNTs are fairly rare.

Two distinct low energy regions in the momentum space of electrons in CNT play a central role in transport experiments. They originate from K and K' Dirac cones of graphene, and are commonly referred to as K and K' valleys. This valley orbital degree of freedom can be pictured as whether the electron is rotating clockwise or counterclockwise around the nanotube. Because the transverse confinement in carbon nanotube is very strong, the first band above these K and K' valley is few hundreds of mV above, hence on can safely restrict himself to the K and K' bands to describe transport experiment. Since K and K' valleys are far apart from each other in momentum space, the valley is a robust quantum number to describe the state of an electron. K and K' states are degenerate in absence of magnetic field, hence each energy levels defined by longitudinal confinement is four-fold degenerate (2 × 2, for the electronic spin and the valley degree of freedom). However the presence of disorder can induce scattering with large momentum transfer, and mix K and K' states. This leads to the lifting of the valley degeneracy, which is defined by an energy coupling constant ∆ K,K . Spin-orbit coupling in carbon nanotube has also been reported, with values ranging from ∆ SO = 150 µeV [START_REF] Sand | Gate-dependent spinorbit coupling in multielectron carbon nanotubes[END_REF] (∆ SO = 300 µeV in [START_REF] Kuemmeth | Coupling of spin and orbital motion of electrons in carbon nanotubes[END_REF]), to ∆ SO = 3.4 meV [START_REF] Steele | Large spin-orbit coupling in carbon nanotubes[END_REF].

The Fermi velocity in carbon nanotube is v F ≈ 1.10 6 m.s -1 , and sets the minimal size of the dot L required to ensure well separated energy levels, since ∆E conf = hv F 2L (see the review paper [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]). Another interesting property of carbon nanotubes is the low density of nuclear spins, which make them particularly attractive for spin qubit experiments (see chapter 5). Indeed they are composed of 98.9% of 12 C which has no nuclear spin, and 1.1% of 13 C in normal synthesis conditions. With a diameter about 1 -2 nm, carbon nanotubes can be considered as 1D conductor which is highly beneficial for Cooper pair splitting for instance (see chapter 4).

Quantum dots in carbon nanotubes

The small transverse dimension of carbon nanotubes already provides a 1D confinement.

The longitudinal confinement in semiconducting CNT is commonly created by Schottky barriers, originating from a mismatch between the work functions of the contact metal and the CNT. Such Schottky barriers can be detrimental to obtain very transparent contact and reach the ballistic transport limit. Palladium have been found to produce low Schottky barriers [START_REF] Javey | Ballistic carbon nanotube field-effect transistors[END_REF], since its work function (φ P d = 5.1 eV) is very close to the one of carbon nanotubes. This is advantageous for reaching the Kondo or the Fabry-Perot regimes which demand very opened quantum dot. Low Schottky barriers are also beneficial for the closed quantum dot regime, since the height barrier can be subsequently tuned using a capacitively coupled gate electrode, in a controllable way.

When measuring the current through a quantum dot as a function of the gate voltage defining the dot, one sees peaks of current whenever an energy level in the dot is aligned with the chemical potential in the reservoirs (in the limit of small bias voltage, V SD = 1 e (µ S -µ d ) 0 with µ S (resp. µ D ) the chemical potential in the source (resp. drain) reservoir). The observed spectroscopic level spacing E add (N ) is the difference between the chemical potentials of the dot for two successive numbers of electron in the dot

(E add (N ) = µ N -µ N -1
), where the chemical potential of the dot is defined by: µ N = E(N ) -E(N -1), with E(N ) the total energy of the N electrons in the dot. In the constant interaction model, E(N ) reads:

E(N ) = N i=1 i + U (N ) (1.1)
The first term is the sum of the different orbital energies filled by electrons. The distribution of orbital levels i is determined by the longitudinal confinement potential in the carbon nanotube (∆E conf = hv F 2L ), and the eventual spin-orbit coupling ∆ SO , and valley splitting ∆ K,K . The second term is the electrostatic energy due to electron-electron interactions in the dot and capacitive energy between the dot and all neighboring conductors, and it reads:

U (N ) = e 2 N 2 2C Σ + eN M j=1 C j C Σ V j (1.2)
Where C Σ is the total capacitance of the dot, C j is the capacitance between the dot and the conductor j, and V j is the voltage of the conductor j. The spectroscopic level Chapter 1. Hybrid double quantum dots 14 spacing E add (N ) can then be rewritten as:

E add (N ) = e C G C Σ ∆V G = e 2 C Σ + N +1 -N (1.3) C G
C Σ is the conversion factor from gate voltage to energy shift of levels in the dot, and is called the lever arm of the gate G on the dot.

∆V G = V G (N + 1) -V G (N ) (with V G (N )
the gate voltage at which the number of charge in the dot oscillate between N and N-1) is the distance between two current peaks in the gate voltage space. e 2 C Σ is the charging energy of the dot. A standard transport measurement in a single quantum dot is to measure the current (and/or the conductance) through the dot as a function of the gate voltage V G and the bias voltage V SD . In transport experiment, charging energies and lever arms are the two quantities that electrostatically describe the circuit. In the following, I will focus on the case of a double quantum dot, which is the circuit that has been used in the two presented experiments (chapter 4, and 5).

Basics of double quantum dots

When two quantum dots are close enough for electrons to jump from one dot to the other, the system behave as an artificial molecule. Such circuits have a more complex internal dynamics and are at the heart of charge qubits [START_REF] Gorman | Charge-Qubit Operation of an Isolated Double Quantum Dot[END_REF], and several spin qubit architectures [START_REF] Petta | Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots[END_REF], [START_REF] Pioro-Ladrière | Electrically driven single-electron spin resonance in a slanting Zeeman field[END_REF], [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF].

Characteristic energy scales

A few characteristic energy scales are sufficient to define the operating regime of a double quantum dot.

Charging energy

The on-site charging energy originates from coulomb interaction between all the electrons in the quantum dot, and is expressed in terms of an effective capacitance C Σ :

E C = e 2
CΣ (see the review [START_REF] Van Der Wiel | Electron transport through double quantum dots[END_REF] for the full formula). In the quantum dot geometry considered in this thesis, the on-site charging energy is ranging from 1 meV to 10 meV, and is the dominant contribution to the spectroscopic level spacing E add (N ). There is also a mutual charging energy U m arising from the finite capacitance C m between the two dots, and typical mutual charging energy value is few 10 meV.

Orbital energy Like in an atom, a set of discrete orbital energy level originates from confinement in the CNT. the spacing between these levels depends on the shape of the confinement potential and on the energy dispersion of electron close to the Fermi energy [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]. For carbon nanotubes (linear dispersion: E(k ) ≈ v F k ) in the approximation of strong confinement (valid for large number of confined electrons), the spacing between orbital levels is ∆E conf ≈ hv f 2L . For a 500 nm long quantum dot, ∆E conf = 4 meV. As explained in the section 1.1.1, each orbital energy level is four-fold degenerated, nevertheless spin-orbit coupling and K-K' valley mixing (∆ SO and ∆ K-K respectively) can lift this degeneracy, giving rise to a more complex energy level ladder.

Dot-lead tunnel rate

The coupling to each dot to its adjacent lead is described by the tunneling rates Γ L and Γ R . For a double quantum dot well coupled to the leads (Γ k B T ), dot energy levels will be broadened by the Γ s. In a C-QED perspective, the regime with low tunneling rates (k B T Γ, often called "closed quantum regime") is favored since internal degrees of freedom such as the electron spin will not suffer from virtual exchange processes with electrons in the reservoirs [START_REF] Hanson | Spins in few-electron quantum dots[END_REF].

Interdot tunnel rate An other type of tunneling rate, is the one in between the two dots, labeled t. This tunneling rate plays a dominant role in the internal dynamic of the double quantum. Its value is typically between 1 GHz and 10 GHz, which make double quantum dot easy to probe with standard excitation and read-out techniques using basic RF equipment. When t k B T , coherent tunneling between dot orbitals occur, thus forming molecular orbitals. These regime is called coherent regime of a double quantum dot [START_REF] Van Der Wiel | Electron transport through double quantum dots[END_REF].

Bias voltage In all transport experiments, a voltage bias V SD is applied between the two electron reservoirs. This voltage drop along the circuit define what we call a "bias window": µ S -µ D = eV SD (see figure 1.2.b). All electronic states in the double quantum dot laying in this energetic window will contribute to the transport, hence measuring the current flowing through the circuit or its conductance provide a spectroscopy of the levels in the double quantum dot. A lot of attention has been given to such spectroscopy in the high bias regime where multiple levels participate to the transport [START_REF] Sapmaz | Excited State Spectroscopy in Carbon Nanotube Double Quantum Dots[END_REF], [START_REF] Molitor | Observation of excited states in a graphene double quantum dot[END_REF], [START_REF] Xing | Gate-defined graphene double quantum dot and excited state spectroscopy[END_REF].

Electronic temperature All those energies have to be compared with the thermal energy of the electrons in the circuit, k B T elec , which sets the smaller resolvable energy in transport measurement. This temperature can be quite different from the base temperature of the cryostat. All experiment in this thesis work have been carried at T elec ≤ 50 mK ≈ 4 µeV (with T cryostat 20 mK).

At the single dot level, three main transport regimes can be identified, each with a specific energy ladder: the Coulomb blockade regime

(Γ L , Γ R , k B T elec E C ), the Kondo regime (k B T Γ L , Γ R E C ),

Stability diagram

To fully characterize the transport in a double quantum one has to measure the current as a function of the five gate voltages and the bias voltage V SD . This would produce a six-dimensional array, which is not very handy. A first approach is to look at the current as a function of V L and V R tuning energy levels in the left and the right dot respectively, When one begins to increase the bias voltage V SD across the double quantum dot, the condition for current is now satisfied over two triangles at the triple point positions in the gate-gate plane. Electron can tunnel through multiple levels, and this allows to probe directly the energy spacing between the i which contains information about spinorbit coupling and valleys mixing [START_REF] Steele | Large spin-orbit coupling in carbon nanotubes[END_REF], [START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF], without paying the price of the charging energy price, since only single electron processes occur. Since the bias voltage provides an absolute energy scale, this regime is useful for calibrating the lever arms of the gates with respect to the two dots 1 , and the complete procedure is detailed in the review paper ref. [START_REF] Van Der Wiel | Electron transport through double quantum dots[END_REF]. In figure 1 showing the characteristic honeycomb pattern of double quantum dots (measurement taken on device SPN15R). The points with high current are called "triple points" and correspond to the situation where levels in each of the dots are in the bias window. The white lines describing the hexagons are cotunneling lines. In such second order process, one of the dot is detunned from the bias window, and electrons tunnel through this dots via a state only virtually accessible. b. Schematic of the region in between two triple points. The grey lines represent delimitation between lower-energy electronic configuration considering only capacitances between dots and conductors. The avoiding crossing in the current line (in blue) is due to the interdot tunneling. This region is of paramount importance since it is where (0,1) and (1,0) charge configurations are degenerate, thus an electron is allowed to jump between the two dots. the new eigenstates are the molecular bonding and anti-bonding state in which a charge qubit can be encoded (see section 1.2.3). In this area, no charge is exchanged with the reservoirs, hence the double quantum dot forms a closed system. The qubit frequency is Ω = E AB -

while keeping V Γ L , V Γ R , V t constant
E A = 2 δ + 4t 2
, with E AB and E B the energies of the anti-bonding and bonding states respectively, and the detuning δ = L -R , where i is orbital energy in the dot i. The qubit frequency can easily be tuned by acting on the detuning δ . In the perspective of quantum information processing [START_REF] Loss | Quantum computation with quantum dots[END_REF], one can perform detuning DC gate for the initialization [START_REF] Vandersypen | NMR techniques for quantum control and computation[END_REF], and modulating δ at the qubit frequency allows to perform a rotation of the qubit state around an axis (determined by the phase of the modulating signal) lying on the equator of the Bloch sphere (see figure 1.5-b). The read-out of the qubit state can be done through charge sensing [START_REF] Gorman | Charge-Qubit Operation of an Isolated Double Quantum Dot[END_REF], direct DC current measurement, or using c-QED like read-out scheme [START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF], [START_REF] Delbecq | Coupling a Quantum Dot, Fermionic Leads, and a Microwave Cavity on a Chip[END_REF]. Untill very recently, the various implementation of charge qubit had never shown good coherence properties. The reason for such short coherence times is the charge noise inherent to the environment. It originates from uncontrolled charge jumps resulting in random fluctuation of the electric field in the vicinity of the qubit. one strategy to circumvent this issue consists in placing the qubit in a regime (often called sweet spot) where its energy is insensitive to charge noise at first order. Nevertheless, even by taking care to this last point, it is difficult for the coherence times T 2 to exceeded the few ns [START_REF] Shi | Coherent quantum oscillations and echo measurements of a Si charge qubit[END_REF], [START_REF] Hayashi | Coherent Manipulation of Electronic States in a Double Quantum Dot[END_REF], [START_REF] Cao | Ultrafast universal quantum control of a quantum-dot charge qubit using LandauZenerStückelberg interference[END_REF], [START_REF] Petersson | Quantum Coherence in a One-Electron Semiconductor Charge Qubit[END_REF], [START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF] (more recently a coherence times T 2 = 50ns has been measured in GaAs [START_REF] Scarlino | All-Microwave Control and Dispersive Readout of Gate-Defined Quantum Dot Qubits in Circuit Quantum Electrodynamics[END_REF]). Such coherence times correspond to a 1/f noise density of a few 10 -4 e/ √ Hz at 1 Hz. To reach longer coherence time, one can also try to hybridize the orbital degree of freedom to the electron spin (see section 1.4

for more details), to benefit from its excellent coherence property (T 2 0.6s in a NV center using dynamical decoupling pulses [START_REF] Bar-Gill | Solid-state electronic spin coherence time approaching one second[END_REF]). The subspace of the charge qubit can be represented on a Bloch sphere. The two poles are the two logical states and can be reach only for δ = 0. Close to δ = 0 the energy transition is mainly set by the tunnel coupling t. Like in NMR experiment [START_REF] Vandersypen | NMR techniques for quantum control and computation[END_REF], small amplitude modulations of δ at the frequency of the transition between |B and |AB (originating from a cavity electric field for instance) will make the state vector rotate about the δ axis. Such a coupling scenario is called transverse coupling since the rotation axis is perpendicular to the quantization axis. c. Energy dispersion of the double quantum dot states. For δ = 0, the first derivative of Ω( δ , t) is zero, which means that the energy transition is insensitive at first order to δ -noise. This region is called a sweet spot. Placing the qubit in such region allows to limit the decoherence of the qubit due to charge noise.

B AB 1,0 Energy ε δ ε δ detuning, t 2t 0,1 0,1 1,0 1,0 AB B 0,1 (b) (c) 0 (0,0) (1,0) (0,1) (1,1) ε δ (a) ε δ 2 + 4t 2
in a c-QED experiment, an important step toward quantum computing is to reach the strong coupling, where the coupling strength between cavity photons and the two-level system overcomes the decoherence rates of both parts (see chapter 2). This regime reveals the coherent exchange of excitations between the two systems. Surprisingly, the strong coupling regime had first been reached with charge qubit in double quantum dot [START_REF] Bruhat | Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs[END_REF], [START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF], [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF], then followed closely by spin qubits [START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF], [START_REF] Landig | Coherent spinphoton coupling using a resonant exchange qubit[END_REF], [START_REF] Mi | A coherent spinphoton interface in silicon[END_REF].

The superconducting double quantum dot

Whereas superconductivity has been discovered more than a century ago, it is still an active domain of research. One direction is the research of exotic superconductivity, such as high-T C superconductor [START_REF] Foltyn | Materials science challenges for high-temperature superconducting wire[END_REF]. Another line of research consists in studying superconductivity at the µm-scale and is often referred to as mesoscopic superconductivity. In this section, I first briefly introduce few results of the BCS ( Bardeen Cooper Schrieffer) theory of conventional superconductor as well as concepts in mesoscopic superconductivity, which are necessary for the understanding of the experiment presented in chapter 4.

Mesoscopic superconductivity

Homogeneous superconductor

In 1950, H. Fröhlich [START_REF] Fröhlich | Theory of the Superconducting State. I. The Ground State at the Absolute Zero of Temperature[END_REF] (and subsequently J. Bardeen and D. Pines with a more complete approach [START_REF] Bardeen | Electron-Phonon Interaction in Metals[END_REF]) showed that phonons in solids can mediate an attractive interaction between electrons. Nevertheless, the resulting force can be very weak. The next step was carried out in 1956 by L.N. Cooper [START_REF] Cooper | Bound Electron Pairs in a Degenerate Fermi Gas[END_REF]. He noticed that such attractive interaction can create bound pairs of two electrons above the Fermi sea no matter how weak the interaction is. Cooper's argument explains qualitatively the formation of a condensate of such Cooper pairs around the Fermi sea. In 1957 J. Bardeen, L.N. Cooper, and J.R.

Schrieffer found out the resulting ground state of such a scenario [START_REF] Bardeen | Theory of Superconductivity[END_REF], and showed that it naturally provides an explanation for all the observed phenomenon about superconductivity at that time. The conventional superconductor are the one described by the BCS theory, where phonons are responsible for the formation of bound electron pairs. While the electron-phonon interaction is in principle present for all pairs of electron (k,k'), the most efficient mechanism occurs for pairs (k↑, -k↓), since a broader range of q phonons can be involved 2 . By treating these pairing correlations c -k↓ c k↑ with a mean-field approximation, one can write the following hamiltonian:

H BCS = k,σ ξ kσ c † kσ c kσ + k ∆ k c † k↑ c † -k↓ + ∆ * k c -k↓ c k↑ (1.4)
Where c kσ , and c † kσ are respectively the annihilation and creation operator of an electron in the plane wave state (k,σ) with kinetic energy ξ kσ (where k denotes the vector momentum, and σ ∈ {↑, ↓}). We also introduced

∆ k = - k V kk c -k ↓ c k ↑ with V kk
the pairing potential. This hamiltonian can be diagonalize using the Bogoliubov transformation [START_REF] Tinkham | Introduction to Superconductivity[END_REF]:

γ k,↑ = u k c k↑ -v k c † -k↓ and γ † -k,↓ = u * k c † -k↓ + v * k c k↑ (1.5)
with:

u k = 1 √ 2 1 + ξ k ξ 2 k + ∆ 2 and v k = e iθ √ 2 1 - ξ k ξ 2 k + ∆ 2 (1.6)
Here we have considered ∆ to be independent of k. the diagonalized hamiltonian takes the form:

H SC = k,σ E k γ † kσ γ kσ (1.7)
The operators γ k,↑ and γ -k,↓ correspond to elementary excitations of the Cooper pair condensate. From the Bogoliubov transformation, they appear as superposition of electron and hole excitations. They are often called Bogoliubons quasiparticles and their

energy dispersion is E k = ξ 2 k + |∆| 2 .
One can derive the corresponding density of state of the quasiparticles:

ρ BCS (E) = E √ E 2 -∆ 2 ρ N (ξ F ) characterized by the energy gap |∆|.
The form of the ground state describing the Cooper pair condensate has been postulated prior to the diagonolization of the pairing hamiltonian, and it can be checked after that it coincides with the vacuum of excitations: γ k,↑ |GS = γ -k,↓ |GS = 0. The ground 2 The general situation where a phonon mediates a transfers of momentum between two electrons can be depicted as: (k, k ) → (k -q, k +q). Knowing that the k -q and k +q have to remain out of the Fermi sea and assuming a spherical Fermi sea, one find that a possible solution is q = k -k . Nevertheless, for k = -k, there is no condition on q, hence the interaction is expected to be much larger. state reads:

|GS = k (u k + v k c † k,σ c † -k,-σ )|0 (1.8)
Where |0 is the Fermi sea. One can identify (|u

k |+e iφ |v k |c † k,σ c † -k,-σ )|0
as an elementary block of Cooper pair condensate. We will see in the next section that such a elementary block can be partially isolated in a quantum dot. Importantly, all the terms of this product are linked to the same phase factor e iφ . Hence it plays a macroscopic role in the wavefunction of the condensate. Besides, because each elementary block is a coherent superposition of zero electron and two electrons in the pairs, the number of electron is undefined in the condensate. This is reminiscent of the uncertainty principle between phase and number ∆N ∆φ ≥ 1, where the BCS ground state is a well-defined phase state, hence realizing an example of macroscopic quantum coherence [START_REF] Annett | Superconductivity, Superfluids and Condensates[END_REF]. This result could be anticipated since the mean-field hamiltonian does not conserve the number, but only the parity.

Cooper pairs

In conventional superconductors, the Cooper pair wavefunction is an S-orbital and the spins are in the singlet state. However, triplet states are also possible (as long as the total wavefunction remain antisymmetric) and attracted a lot of attention in the perspective to create Majorana modes [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. Besides, Cooper pairs do not verify bosons statistic ([S p , S † q ] = δ pq (1-n -p↓ -n p↑ ) with S q = c -q↓ c q↑ ), hence they do not forms a Bose-Einstein condensate. An important length scale is ξ 0 = v F π∆ which correspond to the minimal length over which superconductivity can be established (for instance, it sets the diameter of superconducting vortices, or the depth of induced superconductivity in a normal metal). It can be interpreted as the spatial extension of a Cooper pair, and is an important parameter to determine the Cooper pair splitting efficiency (see chapter 4).

Excitation and semiconducting pictures

In the previous description of superconductivity, we are only dealing with positive excitation energies. Hence the most straightforward way to represent the state of the system is to fix the energy of the condensate at zero and then represent the density of excitation states as previously introduced above the condensate energy. This is called the excitation picture and is illustrated in figure 1.6-a.

Nevertheless, when one want to describe tunneling experiments, one is interested in transfers of charges and not of quasi-particles, which are two distinct things in superconductivity. For instance, when a normal metal and a superconducting metal are separated from each other by a tunnel barrier, the excitation picture is not the most handy. To illustrate this statement, lets consider the situation in which the normal metal is negatively biased with respect the superconductor such that eV normal < -∆.

In this situation one can inject a hole in the superconductor which decompose on the Bogoliubon states as:

c k↑ |GS = (u * k γ k↑ + v k γ † -k↓ )|GS = -v k γ † -k↓ )|GS (1.9)
with k > k F . This can seem a priori surprising and is due to the fact that a bogoliubon quasiparticles are a superposition of an electron and a hole. To clarify this point, one can introduce two hole-like and two electron-like quasi-particles excitations, connected by the fact that creating a hole-like excitation is equivalent to annihilate a pair and create an electron-like excitation. The new set of four excitation operators is:

γ † e,k↑ = u * k c † k↑ -v * k S † c -k↓ γ † h,k↑ = u * k Sc † k↑ -v * k c -k↓ γ † e,-k↓ = u * k c † -k↓ + v * k S † c k↑ γ † h,-k↓ = u * k Sc † -k↓ + v * k c k↑ (1.10)
which are related by γ h,kσ = Sγ e,kσ with S = c k↑ c -k↓ . One can notice that the two hole-like excitations γ † h,k↑ and γ † h,-k↓ lower the number of electron by one. Hence the same process can be rewritten as:

c k↑ |GS = u * k γ e,k↑ + v k γ † -k,↓ |GS = v k γ † -k,↓ |GS (1.11)
Now the annihilation of an electron in the superconductor corresponds to the creation of a hole-like excitation in the superconductor. The use of these four operators correspond to the semiconducting picture, and the previous situation is depicted in figure

γ h,k γ h,-k Δ γ k γ -k Δ ρ BCS ρ N ρ BCS (a) (b)
N S An electron incoming from a normal metal with a wave-vector k has a finite probability to be reflected on a normalsuperconductor interface as a hole of opposite wave-vector and spin. b. Mechanism responsible for the emergence of Andreev bound states in a superconductor -coherent conductor -superconductor junction. c. First observation of individually resolved Andreev bound states in a tunnelling spectroscopy experiment. In this experiment, a carbon nanotube plays the role of the coherent conductor. source: [START_REF] Pillet | Andreev bound states in supercurrent-carrying carbon nanotubes revealed[END_REF] interfaces are distant than less that the superconducting coherence length ξ 0 . This is the mechanism one want to exploit to split Cooper pair [START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF] (see chapter 4).

Proximity effect in quantum dots

When the superconductor is well coupled to a coherent conductor (Γ ∆), reflecting the fact that the time spent in the dot ( /Γ) is small compare to the correlation time of a Cooper pair ( /∆), multiple Andreev reflection occurs before the electron and the hole loose their relative phase. It gives rise to standing wave, forming Andreev bound states. Because the reflected hole takes a phase factor: -φ -arcos( E ∆ ) with φ being the macroscopic phase of the superconductor, the energy spectrum of such Andreev bound states is E A = ∆ 1 -τ sin 2 ( φ 2 ). These states carry supercurrent , as they depend on the superconducting phase(see schematic in figure 1.7-b,c), hence they can be detected via transport spectroscopy experiment [START_REF] Pillet | Andreev bound states in supercurrent-carrying carbon nanotubes revealed[END_REF].

When Γ, d ∆, one can treat Andreev reflection in a perturbative manner, and derive a low energy effective hamiltonian. Consider only one superconductor connected to a quantum dot with one orbital level d . The hamiltonian of such system reads:

H QD-SC = σ d d † σ d σ + t k,σ (d † σ c kσ + h.c) + k,σ E k,σ γ † k,σ γ k,σ (1.12)
Where d σ , and d † σ are respectively the annihilation and creation operator of an electron in the dot with a spin σ. At low enough temperature (T T C ), there are no quasiparticles in the superconductor, nevertheless the existence of these empty states can have a renormalization effect on the spectrum of the dot via second-order virtual processes involving them. A Schrieffer-Wolff transformation accounts for such second-order processes, by eliminating first order terms in γ kσ , and then subsequently preserving only second-order terms in γ kσ . This procedure is similar to the adiabatic elimination in atomic physics [START_REF] Walls | Quantum Optics[END_REF]. The Schrieffer-Wolff transformation reads: H SW = e iS H QD-SC e iS with S = i kσ X kσ γ kσ -h.c.. Noting H QD-SC = H 0 + H t with H t being the linear part in t, and using the Baker-Hausdorff-Campbell formula 3 , one has to find the X kσ such that H t + i[S, H 0 ] = 0. The X kσ that verify such a condition are:

X k↑ = tu * k E k -d d † ↑ + t * v * k E k + d d ↓ X k↓ = tu * k E k -d d † ↓ - t * v * k E k + d d ↑ (1.13)
After keeping only up to the 2 nd order terms of H SW in γ kσ , on obtain:

H ef f = H 0 + i 2 [S, H t ] = ˜ d n d + (Γ C d † ↑ d † ↓ + h.c.) (1.14)
with:

˜ d = d -π|t| 2 ρ N (E F ) d ∆ F ( d ∆ ) and Γ C = e iφ π|t| 2 ρ N (E F )F ( d ∆ ) (1.15) noting F (x) = 1/ √ 1 -x 2 . Since the term Γ C d † ↑ d † ↓ +h.c
. only couples states with the same parity, we can fully capture its effect by restricting ourselves to the subspace spanned by |0 , and | ↑↓ . The two eigenstates reads:

|Ψ ± = ( Γ * C |Γ C | 2 + E ± + E ± |Γ C | 2 + E ± d † ↑ d † ↓ )|0 (1.16)
with eigenenergies:

E ± = ˜ d ± ˜ d 2 + |Γ C | 2 (1.17)
interestingly, the form of the eigenstates strongly recall the expression of the elementary block of the BCS ground state. In this calculation we have considered the on-site

Coulomb repulsion energy U = 0.

Cooper pair splitting

Originally proposed as a source of electronic Einstein-Podolski-Rosen pairs [START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF], the [START_REF] Hofstetter | Cooper pair splitter realized in a two-quantum-dot Y-junction[END_REF] [67], current noise [START_REF] Das | High-efficiency Cooper pair splitting demonstrated by twoparticle conductance resonance and positive noise cross-correlation[END_REF], [START_REF] Herrmann | Carbon Nanotubes as Cooper-Pair Beam Splitters[END_REF]) and in multiple host materials (carbon nanotube [START_REF] Herrmann | Carbon Nanotubes as Cooper-Pair Beam Splitters[END_REF] [69], graphene [START_REF] Borzenets | High Efficiency CVD Graphene-lead (Pb) Cooper Pair Bibliography 182 Splitter[END_REF], InAs nanowires [23] [68]. Few of these experiment shown high splitting efficiency [START_REF] Das | High-efficiency Cooper pair splitting demonstrated by twoparticle conductance resonance and positive noise cross-correlation[END_REF], [START_REF] Schindele | Near-Unity Cooper Pair Splitting Efficiency[END_REF], [START_REF] Borzenets | High Efficiency CVD Graphene-lead (Pb) Cooper Pair Bibliography 182 Splitter[END_REF]. However, they do not conclude on the coherence of the splitting. One experiment have demonstrated coherent Detecting entanglement in a splitter geometry using only transport spectroscopy is indeed very challenging, since one ask to the system to be closed (weak coupling to the leads) to preserve coherence of quantum states, and at the same time, the DQD has to be sufficiently coupled to the leads to ensure measurable current. Besides, light-matter interaction has proven to be a powerful probe for closed system [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF], and has been widely applied to probe internal dynamics of quantum dot circuits [START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF], while weakly affecting its coherence [START_REF] Cottet | Probing coherent Cooper pair splitting with cavity photons[END_REF], [START_REF] Cottet | Subradiant Split Cooper Pairs[END_REF].

Consider a simplified geometry of a Cooper pair splitter, where a superconductor is connected to two quantum dots as depicted in figure 1.8-a. Similarly to the previous section one can derive a low energy hamiltonian (more details about this derivation are given in chapter 4): .18) with:

H ef f = i∈L,R i n i + t eh (d † L↑ d † R↓ -d † L↓ d † R↑ ) + t ind ee σ d † Lσ d Rσ + h.c. ( 1
t eh = πt L t R e iφ ρ BCS (E F )f (δx) i∈L,R 1 2 F ( i ∆ ) (1.19) t ind ee = -πt L t * R ρ BCS (E F )f (δx) L + R ∆ i∈L,R 1 2 F ( i ∆ ) (1.20) f 1D (δx) = cos(k F δx) i F ( i ∆ )e -|δx|/F ( i ∆ )ξ 0 i F ( i ∆ ) (1.
|V 1(2) = √ 2t * eh 2|t eh | 2 + E 1(2) |0 + E 1(2) 2|t eh | 2 + E 1(2) |S (1.22)
with eigenenergies (shown in figure 1.8):

E V 1(2) = 1 2 Σ ± 2 Σ + 8|t eh | 2 (1.23)
The last term in H ef f is the interdot tunneling assisted by Cooper pairs which is distinct from the direct tunneling term in between the two dots. Importantly, t ind ee depends on the energy sum Σ = L + R which can be used to engineer a specific coupling scheme between a double quantum dot and a microwave cavity (see chapter 4).

Geometrical factor

The width of the superconducting contact plays a crucial role in the splitting efficiency since, if the two QD-superconductor interfaces are too far apart from each other no splitting is expected. Thus, it is necessary to write the hamiltonian in real space and not in reciprocal space. When one is evaluating the Cooper pair splitting rate, terms such like:

k<k F e k.(r L -r R ) ∝ sin(k F (r L -r R ))/k F (r L -r R ) (in 3D
) appears, with r L and r R being the position of the QD-superconductor interfaces.

Similarly to Friedel oscillations, such terms are damped oscillations in real space whose damping depends on the dimension of the system (the sharp Fermi-Dirac distribution in reciprocal space give rise to damped oscillations in real space).

The geometrical factor presented here correspond to a one-dimensional geometry (meaning a point contact), which is relevant for carbon nanotube with a diameter about 1 nm. This factor sets a constraint on the distance δx = x L -x R . One notices that using a superconducting material with a large ξ 0 loosen this constraint (for instance ξ 0 100 nm for Aluminum in the dirty limit [START_REF] Hofstetter | Cooper pair splitter realized in a two-quantum-dot Y-junction[END_REF]). Going to a two-dimensional system (meaning a 1D contact, which is the case of nanowire with diameters above 50 nm) leads to a geometrical factor: f 2D (δr) ∝ cos(k F δr) √ k F δr e -δr/πξ for k F δr 1 [START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF], which is more restrictive than in the 1D scenario. Hence, carbon nanotubes appears as good candidate to host such a circuit.

Spin qubits in quantum dots

The founding paper of Divincenzo and Loss [START_REF] Loss | Quantum computation with quantum dots[END_REF] sets the different requirements of a qubit for quantum computing, comprising, initialization, coherent manipulation, and read-out.

The most natural candidate for a qubit and the one envision in this early proposal is the spin of an electron. Various platforms other than quantum dot have been study to realize such a spin qubit, and few of the more promising are [START_REF] Falk | Polytype control of spin qubits in silicon carbide[END_REF] (Silicon carbide), [START_REF] Jarryd | A single-atom electron spin qubit in silicon[END_REF] (P donor in Si), [START_REF] Childress | Coherent Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond[END_REF] (NV centers), [START_REF] Maurand | A CMOS silicon spin qubit[END_REF] (CMOS technology). In this section, we only focus on the different realizations of spin qubit in quantum dots circuits.

The electron spin is a very promising platform for quantum computing because of its extremely long coherence time (T 1 ≈ 1s [START_REF] Amasha | Electrical Control of Spin Relaxation in a Quantum Dot[END_REF] and T 2 = 0.87ms using appropriate dynamical decoupling sequences [START_REF] Malinowski | Notch filtering the nuclear environment of a spin qubit[END_REF]). Such coherence properties reflect the fact that the electron spin is well decoupled from its environment. The drawback is that this decoupling from the environment makes it hard to manipulate the state of spin. Indeed, the magnetic coupling strength between an electronic spin and a photon confined in a coplanar waveguide resonator with a standard geometry is very weak (g spin ≈ 1OHz [START_REF] Kubo | Strong Coupling of a Spin Ensemble to a Superconducting Resonator[END_REF]).

Strategies to circumvent this issue, are either to manipulate two-electron spin states by acting on the exchange energy (the case of triple quantum dot spin qubit) or to implement a coherent spin-charge interface allowing to control spin with electrical means which can be realized by using spin-orbit coupling [START_REF] Nadj-Perge | Spinorbit qubit in a semiconductor nanowire[END_REF], inhomogeneous Overhauser field [START_REF] Petta | Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots[END_REF], or a micromagnet [START_REF] Pioro-Ladrière | Electrically driven single-electron spin resonance in a slanting Zeeman field[END_REF], [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF], [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. At the end of this section another spin-qubit geometry is introduced [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF].

1.4.1 The big family of spin qubits

1.4.1.1 Singlet-triplet qubit
The singlet-triplet qubit is one of the first implementations of a spin qubit in quantum dot circuit [START_REF] Petta | Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots[END_REF]. It is based on a double quantum dot architecture and the information is encoded in the two-electron spin states T 0,(1,1) and a superposition of S (0,2) and S (1,1) .

Here the label (i, j) describe the charge occupation of the dots (i, j ∈ {0, 1, 2}) and

|S = 1/ √ 2(| ↑, ↓ -| ↓, ↑ ), |T + = | ↑↑ , |T -= | ↓↓ , |T 0 = 1/ √ 2(| ↑, ↓ + | ↓, ↑ )
are respectively the singlet and the three triplet states, describing the spin part of the state. For triplet states, the orbital part of the wavefunction has to be antisymmetric to satisfy the Pauli principle, thus one of the two electron of the states T 0,(0,2) , T +,(0,2) , T -,(0,2) has to sit on an higher energy orbital. This is the reason why these three states have a much higher energy and can be ignore in the rest of the discussion.

This qubit is operated close the degeneracy line between the charge filling states [START_REF] Van Wees | Quantized conductance of point contacts in a two-dimensional electron gas[END_REF][START_REF] Van Wees | Quantized conductance of point contacts in a two-dimensional electron gas[END_REF] and (0, 2), then one can safely ignore other charge distribution in the double quantum dot. For δ > 0 (see figure 1.9), the ground state is the singlet state S (0,2) . For δ < 0, in absence of electron tunneling and magnetic field, the three triplets and the singlet are degenerate. Applying a magnetic field allows to push the triplets T +,(1,1) and T -,(1,1)

away from S (= αS (1,1) + β (0,2) , with |α| 2 + |β| 2 = 1) and T 0,(1,1) which form the logical subspace of the qubit. Interestingly, both S (1,1) and T 0,(1,1) have S Z = 0, which make them insensitive to noise in B Z , hence the S-T 0,(1,1) qubit is a simple example of a decoherence free subspace. Since electron tunneling between the two dots conserves spin, close to δ = 0, S (1,1) and S (0,2) hybridize resulting in a avoiding crossing. The tunneling t and the detuning δ set the energy difference ∆E = E T 0,(1,1) -E S , often refereed to as the exchange energy. This exchange interaction originate from the sign inversion under exchange of fermionic particles and can be electrically controlled via electrostatic gate voltages. The exchange interaction can be recast from the hamiltonian of a double quantum dot:

H hub = i=L,R U i ni (n i -1) + U m nL nR + L nL + R nR + σ=↑,↓ (tĉ † L,σ ĉR,σ + h.c.) (1.24)
Where U i is the charging energy on the i dot, U m is the mutual charging energy between the two dots, i is the orbital dot energy occupied by the electron in the dot i, and t is the tunnel coupling between the two dots. Here we focus on the (1,1) charge configuration, where tunneling t is small compare to the other energy scales (Zeeman splitting and charging energies) and by considering only virtual tunneling to the doubly occupied singlet S (0,2) one can perform a Schrieffer-Wolff transformation and obtain a low-energy hamiltonian of the double dots [START_REF] Russ | Three-electron spin qubits[END_REF]:

H heis = J( δ ) ŜL ŜR (1.25)
In this Heisenberg hamiltonian, the exchange interaction reads [START_REF] Russ | Three-electron spin qubits[END_REF]:

J( δ , t) ≈ 4t 2 U/(U 2 -2 δ ) (1.26) With U = U L = U R .
By acting on the detuning δ it is then possible to modulate J( δ ).

If one look at the Bloch sphere of the S-T 0 subspace (figure 1.9), such modulation represent a rotation of the state vector around the z-axis. The full control of the qubit state requires a second rotation axis which can be provided by an inhomogeneous magnetic field: H S-T = J( δ )σ z + ∆B inh σx , where ∆B inh is the magnetic field difference between the right and the left quantum dot. Experimentally, this inhomogeneous magnetic field ∆B inh is created either by random nuclear magnetic field (as in GaAs [START_REF] Petta | Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots[END_REF]) or by embedding a micromagnet close to the double quantum dot in case of material with a low density of nuclear spins (as in Si [START_REF] Wu | Two-axis control of a singlet-triplet qubit with an integrated micromagnet[END_REF]). The latter allows a better control over the field gradient and going to materials with a low number of nuclear spins, such as silicon or carbon nanotube, improves the coherence of the spin qubit.

To initialize a singlet-triplet qubit in S (1,1) , first, one has to go to positive detuning to set the qubit in the S (0,2) state, and then go to large negative detuning in an adiabatic way with respect to the tunnel splitting. The read-out is commonly done using Pauli spin blockade. It consists in a spin-charge conversion mechanism exploiting the fact that state S (1,1) can tunnel toward S (0,2) whereas T 0,(1,1) cannot tunnel in the (0,2) region, due to Pauli principle. The spin part of the wavefunction (singlet or triplet) is then mapped-out on the charge states (1,1) or (0,2), which can be easily distinguished by measuring the conductance in a nearby quantum point contact. Such a measurement technique is projective (no-QND measurement) and does not allow to coherently couple qubits to each other. Coherence time T * 2 ≈ 240ns [START_REF] Wu | Two-axis control of a singlet-triplet qubit with an integrated micromagnet[END_REF] have been measured and its limitation is attributed to charge noise (local charge fluctuator that feed through to

T - T T + 0 S (1,1) S (0,2) ε L - detuning, δ ε ε R = energy, E J(ε) gμ B B J(ε) ΔB T 0 S (a) (b) 0 Figure 1
.9: Singlet-triplet qubit in a double quantum dot: a. Energy dispersion of the two-electron states in a double quantum dot. A external magnetic field B lift the degeneracy between the three triplet states by gµ B B. b. Bloch sphere illustrating the subspace of the singlet-triplet qubit. The exchange interaction J sets the transition energy. When J is small compare to ∆B, the quantization axis of the qubit aligns with the | ↑↓ -| ↓↑ axis of the Bloch sphere. In this situation, coherent oscillations between | ↑↓ and | ↓↑ can be driven via an RF modulation of J( ) [START_REF] Petta | Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots[END_REF].

noise in gate voltages) and fluctuations in magnetic field, since relative phase evolution of the two electron spins results in mixing of singlet and triplet.

Three-electron spin qubits

The original motivation to use triple quantum dots to encode a spin qubit is the possiblity to fully control the state of the qubit using exchange interaction only. Furthermore, such geometry allows one to reduce sensitivity to charge noise by operating the qubit at highly symmetric point in the stability diagram.

Exchange-only qubit

The spectrum of a triple quantum dots is much more complex than the one of a double quantum dots. To simplify the description we restrict ourself to 3 electrons and to one orbital per dot. With these constraints, the Hilbert space still contains 20 states. Applying a magnetic field allows to separate in energy the different subspaces (S = 3/2 and the two subspaces with S = 1/2 and S Z = ±1/2). In the following we focus on the subspace S = 1/2 and S Z = +1/2 which contains eight states [START_REF] Russ | Three-electron spin qubits[END_REF]. Two of these eight states lie in the (1,1,1) charge stability region (see figure 1.10) and are the logical states of the exchange-only qubit [START_REF] Russ | Three-electron spin qubits[END_REF] :

|0 = 1 √ 6 (| ↑↑↓ + | ↓↑↑ -2| ↑↓↑ ) = | ↑ 1 |S 23 + |S 12 | ↑ 3 (1.27) |1 = 1 √ 2 (| ↑↑↓ -| ↓↑↑ ) = |S 13 | ↑ 2 (1.28)
The dispersion of the two states is shown in figure 1.10. Similarly to the singlet-triplet qubit an effective low energy hamiltonian can be obtained:

H ef f = J L ( δ )σ L + J R ( δ )σ R (1.29)
Where σL = (1/4)( √ 3σ X -σZ ), and σR = (1/4)(-√ 3σ X -σZ ). As illustrated in Similarly to the singlet-triplet qubit the read-out can be performed via spin-to-charge conversion [START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF] or c-QED technique [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. it worth noticing that |0 and |1 both belong to the subspace S = 1/2 and S Z = +1/2, hence like for the singlet-triplet qubit, the logical subspace is a decoherence free subspace for noise in B Z .

Resonant exchange qubit (RX)

The resonant exchange qubit is very similar to the exchange only qubit, and it mainly differs in the way it is operated. Instead of applying only DC-pulses via or t, manipulation of the RX qubit state is performed via resonant modulation of the exchange interaction via . In addition, the triple quantum dot is tuned in the charge stability diagram such that the exchange interactions J L and J R are always ON. This provides a large modulation of the coupling strength [START_REF] Russ | Three-electron spin qubits[END_REF] η = 1 2ω RX (J ∂J ∂ + 3j ∂j ∂ ) with ω RX = J 2 + 3j 2 and J = (J L + J R )/2 and j = J L -J R . Such an operation principle has led to π-pulse time as low as 5ns [START_REF] Medford | Quantum-Dot-Based Resonant Exchange Qubit[END_REF]. Moreover, since Energy J (ε) no DC pulses are required, one can use only filtered gate line to limit low-frequency charge noise in the vicinity of the qubit. Recently, the realization of a RX qubit in a circuit QED architecture in GaAs have demonstrated strong coupling regime [START_REF] Landig | Coherent spinphoton coupling using a resonant exchange qubit[END_REF].

1 L J (ε) R 120°( a) (a) (b) (c) 1 S L S R T L TR 201 111 102 J (ε) L J (ε) R ε / U ε / U M ε 0 0

Spin-orbit qubit

An alternative method to the control of the spin via exchange interaction, is to slightly hybridize the spin degree of freedom to the orbital degree of freedom of an electronic system, in order to open a fast manipulation channel of the qubit transition. Qubits based on this idea are called spin-orbit qubit, and it gathers a large variety of physical implementations. The first realization was in a InAs nanowire [START_REF] Nadj-Perge | Spinorbit qubit in a semiconductor nanowire[END_REF] where the intrinsic spin-orbit coupling is large enough to realize this spin-charge hybridization. An alternative wording to describe the spin-orbit qubits, is to say that its operation principle rely on the implementation of an AC Zeeman splitting (gµ B B). Such AC Zeeman splitting can be engineered either through an oscillating motion of the electron in an inhomogeneous magnetic field [START_REF] Pioro-Ladrière | Electrically driven single-electron spin resonance in a slanting Zeeman field[END_REF] [91], or via a modulation of the Landé factor g [START_REF] Pei | A valleyspin qubit in a carbon nanotube[END_REF].

To further improve coherence properties of spin qubits, the general trend is to use nuclear-spin free host material (Si/SiGe, isotopically purified Si, carbon nanotube) to limit decoherence due to hyperfine interaction with nuclear spin which is the main limitation on coherence time in GaAs based spin-qubit. Nevertheless a possible strategy that has not been implemented yet to circumvent this issue is to use hole spin instead of electron spin. Because valence band is made out of p orbitals which has zero amplitude at the nuclei location, The Fermi contact hyperfine interaction between holes spin and nuclear spin is expected to be zero [START_REF] Prechtel | Decoupling a hole spin qubit from the nuclear spins[END_REF].

The ferromagnetic spin qubit

The goal of this section is to give a brief description of a peculiar type of spin-orbit qubit: the ferromagnetic qubit. The idea stemming from the proposal [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF], is to use non collinear ferromagnetic contacts in a double quantum dot geometry to induce spincharge hybridization. This implementation displays the same energy spectrum than spin qubit realizations using a micromagnet [START_REF] Pioro-Ladrière | Electrically driven single-electron spin resonance in a slanting Zeeman field[END_REF], [START_REF] Kawakami | Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot[END_REF]. Experimental results on such a spin qubit are presented in chapter 5.

Principle of the ferromagnetic qubit

The ferromagnetic qubit has a double quantum dot architecture where the two source and drain contacts are ferromagnet with non-collinear magnetizations (see figure 1.11).

the different directions of the magnetization induce different quantization axis on the two dots. Hence when an electron tunnels from one dot to the other, it experiences a modification of the orientation the magnetic field. Such a mechanism can be viewed as an artificial spin-orbit like interaction limited to two sites. This qubit has been design to be coupled to photons in a microwave coplanar waveguide resonator, via the charge

B ext (a) hυ c B eff, DC B eff, AC B eff, DC B ext B tot = + (b) M M M M ψ 2 ψ 2 hυ c ( ,0) ( ,0) ( ,0) ( ,0) (0, ) (0, ) (0, ) (0, ) (c)
V SD

.11: Working principle of the ferromagnetic spin qubit: a. Each ferromagnetic contact has a different magnetization represented by the large red arrows, which induced different quantization axis in each dot. Thus by driving tunneling between the two dots with the microwave electric field, the electron spin experiences a modification of its quantization axis, which is equivalent to an orthogonal AC magnetic field B AC ef f,⊥ . b. Bloch sphere summarizing the qubit-photon coupling mechanism. The overall quantization axis is set by an externally applied magnetic field B ext and the DC part of the induce magnetic field B DC ef f . c. Schematic representation of the ferromagnetic spin qubit with the two non-colinear ferromagnetic contacts. dipole in the double quantum dot, to enhance the coupling strength up to a few MHz [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF]. However, hybridizing spin with charge will also affect the coherence, since the qubit will start to be sensitive to charge noise. A more detailed discussion about the most favourable degree of hybridization is given in chapter 5. The energy of an electron confined in a coherent conductor is set by the constructive interference condition. If the coherent conductor is connected to a ferromagnet via a tunnel barrier, when the electron reflect on this interface, it will acquire a spindependent phase φ σ that adds to the phase acquired during the propagation into the conductor η. This results in a Zeeman splitting of the spectrum in the conductor.

Ferromagnetic interface-induced exchange fields

Consider a coherent conductor in contact with a ferromagnet via a tunnel barrier, as depicted in the figure 1.12. The ferromagnet can polarize electron spins in the coherent conductor via three different mechanism. The most obvious one is the straight magnetic field generated by the ferromagnet. This mechanism is not expected to be the dominant one.

A spin dependent phase shift acquired by the electron when reflecting on the barrier can also induce a polarization of the spectrum. This phase shift does not depends on the quality of the electrical contact since it also happens for ferromagnetic insulator [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF]. This mechanism requires a coherent conductor with a finite length (typically a quantum dot) for constructive interferences to occur. In the single electron picture, this phase shift yields to an effective Zeeman splitting (sometimes called confinement-induced exchange field ) in the spectrum of the coherent conductor [START_REF] Cottet | Controlling spin in an electronic interferometer with spin-active interfaces[END_REF]:

2δ = hv F 2L φ ↑ -φ ↓ 2π (1.30)
where v F is the Fermi velocity, L is the size of the coherent conductor (size of one of the two dots, for the ferromagnetic spin qubit), and φ ↑(↓) is phase shift for a spin-up(down) electron.

The last polarization mechanism is a kind of proximity effect, where the electronic wavefunction spreads over the ferromagnet and hybridize with first atomic layers, hence resulting in a spin polarization of the total quantum state. Its strength depends on the amplitude of the transmission (t lead ) and vanishes for opaque barrier (t lead = 0).

With the idea of implementing a long-lived spin qubit, the last mechanism is not very interesting since a finite Γ lead ∝ |t lead | 2 means that the qubit subspace is coupled to other states in the leads through which it can loose its coherence. Therefore, the ability to tune in-situ the coupling Γ lead is highly desirable for a ferromagnetic spin qubit.

Chapter 2

Mesoscopic -QED

The idea of coupling photons in a cavity to atomic system originates from cavity quantum electrodynamics (C-QED) where a lot of effort has been given to study light-matter interaction at the most fundamental level. The first experiments have been done with Rydbergs atoms [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] and trapped ions [START_REF] Leibfried | Quantum dynamics of single trapped ions[END_REF]. Such architectures provide a controlled environment for the atoms, but allows also to manipulate their state. The idea had then been transposed on chips forming the circuit-QED community [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF], [START_REF] Chiorescu | Coherent dynamics of a flux qubit coupled to a harmonic oscillator[END_REF], where superconducting circuit are coupled to microwave resonator. Such systems exploit the rigidity of the superconducting phase to engineer macroscopic coherent few levels systems. In these experiment, the properties of the artificial atoms can be easily tuned, whereas in cavity-QED they are set by natural constant. The cavity has then been used as a quantum bus to couple several artificial atoms to each other [START_REF] Majer | Coupling superconducting qubits via a cavity bus[END_REF], [START_REF] Sillanpää | Coherent quantum state storage and transfer between two phase qubits via a resonant cavity[END_REF], [START_REF] Fedorov | Implementation of a Toffoli gate with superconducting circuits[END_REF] . Once again, this idea can be extended to quantum-dot circuits. While atoms and superconducting circuits are intrinsically closed systems, quantum dot circuits are able to be well coupled to their condensed matter surrounding. In this regime, the cavity endorses a new role since it is used as a powerful probe for condensed matter [START_REF] Desjardins | Observation of the frozen charge of a Kondo resonance[END_REF]. The aim of this chapter is to introduce a theoretical description of the coupling between the microwave cavity and the quantum dot circuit, as well as the relevant experimental parameters for designing the circuit-cavity interface. The section 2.1 focus on theoretical description of the coupling.

The section 2.2 introduces formalisms on the dynamics of the full coupled system. The last section 2.3 describes few strategies to improve the quality of the cavity-quantum dot interface.

Quantum dot circuit -Cavity coupling description 2.1.1 Microscopic description of electron-photon coupling

In Cavity-QED, the cavity electric field is approximated to be constant in the vicinity of the atom (dipolar approximation). It enables to write the light-matter interaction as

H int = - → Ê . - → d where - →
Ê is a the uniform cavity electric field felt by the atom, and -→ d is the atomic electrical dipole. Such a description is not necessarily true for circuit-QED experiments. Indeed, superconducting circuits strongly influence the distribution of the electric field. Moreover the degrees of freedom of interest are macroscopic, hence the dipolar approximation is not valid. Circuit description of the full system with capacitive couplings between the different conductors is more appropriate. Although the microscopic description of Cavity-QED and circuit-QED can be different, the physics of both systems is described by the same Jaynes-Cummings hamiltonian (see section 2.1.3).

Mesoscopic-QED can be thought of as an intermediate between Cavity and circuit-QED [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF]. The discrete levels in quantum dot are reminiscent of cavity-QED atomic degrees of freedom. Besides, because of the high density of state in the fermionic reservoir, field screening occurs, resulting in a strong inhomogeneity of the electric field, similarly to circuit-QED (see figure 2.1-a). These screening effects are due to plasmons living in cavity conductors but also in the fermionic reservoirs of the nano-circuit. Tunneling events also happen inside the nano-conductor, and between fermionic contact and the nano-conductor. These tunneling events are the focus of most mesoscopic experiments.

Noticing that the dynamics of plasmons is much faster than the tunneling events, one can consider plasmons only via their average contribution, and introduce a photonic pseudo-potential V ⊥ ( -→ r ) containing the strong inhomogeneities [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF].

(a) (b) One can choose which dipole in the circuit to address with light by designing the shape of these protrusion.

g lead L g L g R g lead R γ lead, L γ L, R γ R, lead φ L -φ R ,
In this picture, the interaction between cavity photons and discrete electronic orbitals in the nano-conductor is described by the following hamiltonian:

H e-ph = -e dr 3 Ψ † ( - → r ) Ψ( - → r )V ⊥ ( - → r )(â † + â) (2.1)
Where Ψ( -→ r ) is the field operator of conduction electrons in the nano-conductor. Interestingly, the photonic pseudo-potential can account both for cavity and circuit-QED

experiments. Indeed, V ⊥ ( - → r ) = - → E . - → r corresponds to cavity-QED boundary conditions, whereas in circuit-QED, V ⊥ ( - → r ) is a constant different in each conductors.
To identify the different coupling mechanism, the interaction hamiltonian can be expressed 1 as

H e-ph = ĥint (â † + â), with: ĥint = n g n ĉ † n ĉn + n =n γ n,n ĉ † n ĉn + h.c. (2.2)
and:

g n = -e dr 3 |φ n ( - → r )| 2 V ⊥ ( - → r ) (2.3) γ n ,n = -e dr 3 φ n ( - → r )φ n ( - → r )V ⊥ ( - → r ) (2.4)
The first term in the expression of ĥint corresponds to the modulation of quantum dot orbitals by the AC-field of the microwave cavity, where g n is the spatial overlap between the orbital wavefunction and the photonic pseudo-potential. The second term describes the modulation of the hopping amplitude t n,n , and is proportional to the overlap of the two involved orbital wavefunctions and the photonic pseudo-potential. Although this last mechanism explains light-matter coupling in Cavity-QED, it is generally very small in mesoscopic-QED because of the larger spatial separation of the relevant orbitals. In the following only the coupling mechanism due to g n term will be considered. According to the physics one want explore, a specific electrical dipole in the circuit can be addressed (for instance the dot-dot dipole or the dot-lead dipole) via the shaping of V ⊥ ( -→ r ).

Coupling to a double quantum dot

Consider a double quantum dot, with only one orbital on each dot as illustrated in figure 2.1-b. The interaction hamiltonian of the double quantum dot reads:

H e-ph,DQD = (g L nL + g R nR )(â + â † ) (2.5) Where ni = ĉ † i ĉi with i ∈ (L, R). Close to zero detuning ( δ = L -R = 0),
g DQD (â + â † ) = +|H e-ph,DQD |- = +|(g L nL + g R nR )(â + â † )|- g DQD = t 2 δ + 4t 2 (-e dr 3 (|φ L ( - → r )| 2 -|φ R ( - → r )| 2 )V ⊥ ( - → r )) (2.6)
Where φ L ( -→ r ) and φ R ( -→ r ) are the wavefunctions of one electron on the left and on the right dot respectively. It appears that a constant photonic potential V ⊥ ( -→ r ) results in no coupling, and one have to engineer a gradients of potential to realize a DQD-cavity coupling (see section 2.3.2 for more details).

Input-Output formalism

In this section we describe the dynamic of a bare microwave cavity with an high quality factor Q. The so-called input-output formalism is well adapted to the description of a resonator coupled to its electromagnetic environment. It establishes a relation between the drive tone, the cavity features, and the outgoing signals, which are experimentally accessible quantities. In this section we follow the presentation given in appendix E of [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF], and only provide a brief review of the main results by trying to emphasize the physics behind the equations. The microwave cavities used in this thesis work are connected to two transmission lines, which carry the input and output signals. Consider a single mode of such a two-sided cavity. The Hamiltonian of the cavity mode coupled to its environment reads:

H = H cav + H bath + H int (2.7)
with:

H cav = ω c â † â H baths = ω q i=1,2 q
b † q,i bq,i

H int = -i i=1,2 q (f q bq,i â † -f * q b † q,i â)
bq,1 and bq,2 are the annihilation operator of modes on both sides of the cavity (see figure 2.2). H cav and H baths are the hamiltonians for the isolated cavity and the external electromagnetic baths respectively. H int describes the exchange of photons between the cavity mode and the outsides modes. It is expressed within the rotating wave approximation (RWA), which is valid only for high quality factor cavity (|f q |/ω c 1).

Then one can write the three Heisenberg equations of motion for the operators â, bq,1 , and bq,2 . The evolution of â is given by:

dâ dt = i [ Ĥ, â] = -iω c â - κ 2 â - √ κ 1 bin,1 (t) (2.8)
The first term of equation ( 2.8) describe the free evolution of the cavity mode. The second term regroup all dissipation processes, with κ = κ 1 +κ 2 . The last term represents the incoming field from the driving port 1. Since no drive tone is applied on the port 2, one consider bin,2 = 0. Besides, It has been assumed that the bath modes are Markovian which justify κ i (ω) = κ i = 2π|f | 2 ρ i , where ρ i is the constant density of state in the surrounding electromagnetic field. The newly introduced operator bin,1 reads: bin,1 (t) = 1 √ 2πρ 1 q e -iωq(t-t 0 ) bq,1 (t 0 ) (2.9) Thus, bin,1 is constructed as the sum of all incoming modes from a reference time t 0 < t normalized by the density of modes ρ 1 . Such a normalization give the dimension of √ ω to bin,1 which can be misleading since a and b q have no dimension. Similarly, one can define an outgoing field bout,2 (which is the one of interest in transmission measurement).

With these new operators the system of three linear equations is simplified to the two following expression (and the expression of bin

): bout,2 (t) = √ κ 2 â(t) (2.10) bout,1 (t) = √ κ 1 â(t) + bin,1 (t) (2.11)
From an experimental perspective, the two quantities of interest are the transmission of the cavity and the average photon number in it. By definition the cavity transmission T reads:

T ≡ bout,2 bin,1 = - √ κ 1 κ 2 -i(ω -ω c ) + κ 2 (2.12)
In practice, the cavity is not perfect and possess internal dissipation channels. These internal losses are artificially introduced by adding a third port without source term ( bin,L = 0) and then only influence the total decay rate κ = κ 1 + κ 2 + κ L . The mean photon number in the cavity n 0 ≡ â † â ω=ωc can be obtain using the two relations 2.11 and 2.10, and by introducing the two powers P in (t) = ω b † in,1 (t) bin,1 (t) , and

P out (t) = ω b † out,t (t) bout,2 (t)
:

n 0 = 2 √ AP in P out ω c ∆f -3dB (2.13)
A is the asymmetric ratio A = κ 1 κ 2 (equal to the ratio of coupling capacitances).

Jaynes-Cummings hamiltonian

When cavity photons are coupled to an internal dipole of the quantum dot circuit (ie:

no transition toward the leads), a realistic first approximation is to ignore dissipation in the system, and describe the full system with the Rabi hamiltonian:

H Rabi = ω c â † â + ω q 2 σz + g(â + â † )σ x (2.14)
where ω c /2π is the cavity resonance frequency, ω q /2π is the qubit frequency (having in mind that any transition in the quantum dots can form a qubit), and g is the coupling strength whose exact expression greatly depends on the physical nature of the quantum dot transition (for a DQD:

g = 2(g L -g R )t/ 2 δ + 4t 2 )
. This kind of photon-qubit interaction is often referred as transverse coupling since it involve a coupling axis (here σx ) transverse to the quantization axis σz .

By assuming ω c ω q and g ω c , one can further simplify the interaction term, and neglect the effect of counter rotating terms2 (Rotating Wave Approximation). This results in the Jaynes-Cummings hamiltonian:

H JC = ω c â † â + ω q 2 σz + g(âσ + + â † σ-) (2.15)
Although dissipation is not negligible in most mesoscopic-QED devices (Γ 2 is ranging from a few M Hz [START_REF] Bruhat | Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs[END_REF], [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF] to hundreds of M Hz [START_REF] Deng | Charge Number Dependence of the Dephasing Rates of a Graphene Double Quantum Dot in a Circuit QED Architecture[END_REF], [START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF], [START_REF] Stockklauser | Microwave Emission from Hybridized States in a Semiconductor Charge Qubit[END_REF]), the Jaynes-Cummings hamiltonian still provides qualitative understanding of the energy spectrum.

Dispersive regime When the qubit frequency is far detuned from the cavity resonance frequency (|∆ qc | = |ω q -ω c | g), the two sub-systems cannot exchange energy.

Nevertheless, virtual exchange of photon via second order processes can still occur, and change the cavity frequency as well as the qubit frequency. These effects can be captured by going into the dispersive picture associated with the unitary transformation:

U = e g ∆qc (σ + â-σ -â † ) (2.16)
and the effective hamiltonian up to the second order in g/∆ qc reads:

H disp = U H JC U † = ω c + χ disp σz â † â + 2 ω q + χ disp σz (2.17)
where χ disp = g 2 ∆ is the dispersive charge susceptibility (it is a susceptibility in the sense of the linear response theory). Interestingly the term χ disp σz â † â can either be interpreted as a shift of the cavity frequency ∆f c = χ disp σ z or as a change of the qubit frequency χ disp â † â (AC-stark shift [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF]). The Lamb shift χ disp /2 is also present in H disp .

The dispersive regime is widely used in cavity and circuit-QED [START_REF] Wallraff | Approaching Unit Visibility for Control of a Superconducting Qubit with Dispersive Readout[END_REF] for the read-out of the qubit state since the measurement of f c directly inform about the state of the qubit σ z . Such a read-out technique is almost QND (Quantum Non Demolition) because [H disp , σz â † â] = 0. Nevertheless, it is not purely QND since H disp is only an effective hamiltonian, and one would need to go towards real longitudinal coupling to ensure a QND-measurement of the qubit [START_REF] Didier | Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction[END_REF].

Resonant regime When the qubit frequency is close to the cavity resonance frequency

(|∆ qc | = |ω q -ω c | g)
, the spectrum of the full system is drastically modified. Photons and electronic excitations cannot be distinguished anymore, and the new eigenstates are light-matter hybrid states. For instance in the subspace with n excitation (|e, n -1 , |g, n ), the two eigenstates are:

|n, -= -sin(θ n )|e, n + cos(θ n )|g, n + 1 (2.18) |n, + = cos(θ n )|e, n + sin(θ n )|g, n + 1 (2.19)
θ n is the mixing angle and verify: tan(2θ n ) = (2g √ n + 1)/∆ qc [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF]. These two states are called the n-th order dressed states. An illustration of this so-called Jaynes-Cummings ladder is given in figure 2.3-c. Nevertheless, the Jaynes-Cummings hamiltonian is a nondissipative description and observing this dressed states in experiments is a challenging task. The broadening of the spectroscopic peaks due to dissipation ( κ/2 and Γ 2 ) has to be small enough to resolve the 1-st order doublet of the Jaynes-Cummings ladder.

When such condition is meet, one reach the strong coupling regime (g > κ/2, Γ 2 ). In the literature, one refer to this splitting of the cavity resonance as the vacuum Rabi splitting [START_REF] Brune | Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity[END_REF], [START_REF] Chiorescu | Coherent dynamics of a flux qubit coupled to a harmonic oscillator[END_REF](the hallmark of the strong coupling regime). Resolving higher order doublet

Uncoupled regime

Resonant regime ( ) Dispersive regime ( )

= = -

Master equation of the coupled system 2.2.1 Semi classical description

The master equation approach is very efficient to describe the dynamics of a mesoscopic-QED system. Here, we will focus on a single transition in the circuit between two states |0 and |1 , called the qubit in the following. The full hamiltonian reads:

H = ω q 2 σz + + g(â † σ-+ âσ + ) + ω c â † â Jaynes-Cummings Hamiltonian + iâ † in e -iω RF t -iâ * in e iω RF t
d dt â = 1 i [â, H] = -(i∆ cd + κ/2)â -igσ -+ in e -iω RF t (2.21) d dt σ-= 1 i [σ -, H] = -(i∆ qd + Γ 2 )σ -+ igσ z â (2.22)
With κ = 2πη|β| 2 is twice the decay rate and correspond to the spectroscopic linewidth of the bare cavity. The term Γ 2 σ-has been added manually to describe decoherence of the qubit. We also define two frequency detunings: ∆ cd = ω c -ω RF and ∆ qd = ω q -ω RF .

When the photon number in the cavity is high enough ( â † â 10), the semiclassical approximation is valid, meaning one can treat â as a classical quantity â â , and in the resonant case (ω c ω RF ) one can use: â ae -iω RF t . In the stationary regime, a reads:

a = in ∆ cd + iκ/2 -χ(ω RF ) σz (2.23)
With the charge susceptibility of the transition χ:

χ(ω RF ) = g 2 ∆ qd + iΓ 2 (2.24)
The quantity χ(ω RF ) characterizes the change of the qubit population due to a small amplitude modulation at the frequency ω RF /2π of the potential, hence it is the charge susceptibility in the linear response theory. Note that for Γ 2 = 0 we recover χ(ω RF ) = χ disp , obtained in the dispersive regime (|ω q -ω c | g) without dissipation. To quantify the coherence of a qubit-cavity interface, a key parameter is the ratio: C = g 2 (κ/2)γ called cooperativity. For C > 1, a coherent regime can be defined. In the resonant regime (|∆ qc | g), it manifest itself as the strong coupling regime, and the presence of the vacuum Rabi splitting.

In the dispersive regime, the relevant quantity that have to be compared to decoherence rates is the dispersive shift χn ph . This shift being dependent on the photon number Quantum dots The Γ normalizing the two axis contains all sources of decoherence, arising both from the cavity and the qubit. Adapted from: [START_REF] Schuster | Resolving photon number states in a superconducting circuit[END_REF] in the cavity, one can in principle resolve two successive photon numbers as long as χ > (κ/2), Γ 2 [START_REF] Schuster | Resolving photon number states in a superconducting circuit[END_REF]. Such a condition defines the strong dispersive regime (see figure 2.4).

This regime provides an absolute calibration of the photon number in the microwave cavity [START_REF] Brune | From Lamb shift to light shifts: Vacuum and subphoton cavity fields measured by atomic phase sensitive detection[END_REF], and allows to distinguish between a coherent state, a thermal state, or a quantum state such as a Fock state.

Lindbladian formalism

When the number of photon in the cavity is close to one, the semiclassical approximation is not valid anymore. A more general method to account for dissipation in quantum systems is to describe the evolution in terms of a Lindblad master equation. This fully quantum description has been used to model experimental results presented chapter 4 using the Qutip python package. The Lindblad master equation reads (a detailed derivation can be found in [START_REF] Blum | Density Matrix Theory and Applications[END_REF]):

d dt ρ S (t) = 1 i [H, ρ S (0)] + α xD x(ρ S (t)) (2.25)
with:

D x(ρ S (t)) = xρ(t)x † - 1 2 ρ(t)x † x - 1 2 x † xρ(t) (2.26)
the operators x are called collapse operator and each of them describe a different decoherence process. Hence, together with their corresponding rates α x, they have a direct physical interpretation. This is one of the reason why Lindbladian formalism is very handy for describing experimental results. In our case the different process taken into account are the cavity relaxation (x = â and α â = κ 2 ), the qubit relaxation (x = σ-and α σ-= γ), and the pure phase decoherence of the qubit (x = σ+ σ-and α σ+ σ-= Γ φ ).

When the temperature is non negligible compared to cavity photon energy one has to include thermal population of the cavity (x = â † and α â † = √ n th κ/2), likewise for electronic temperature and the electronic transition energy (x = σ+ and α σ+ = √ n th γ).

Because the numerical resolution time increases with the size of the Hilbert space, this approach requires to limit oneself to a finite photon number. Therefore, this technique is complementary to the semiclassical approach described earlier.

Cavity read-out engineering 2.3.1 Phase and Transmission read-out

In presence of a quantum dot circuit the expression of the transmission T reads:

T (ω RF ) ≡ bout,2 bin,1 = - √ κ 1 κ 2 -i(ω RF -ω c ) + κ 2 + iχ(ω RF ) σz (2.27)
In the presented experiments (chapter 4, and 5), the two measured quantities are the phase shift ∆φ and the amplitude shift ∆A of the transmitted signal, and they are defined by:

T = (A 0 + ∆A)e i(φ 0 +∆φ) (2.

28)

With (A 0 the reference amplitude and φ 0 the reference phase in absence of coupling to the circuit. Knowing that the cavity response is measured at ω RF = ω c , the phase shift ∆φ and the amplitude shift ∆A read:

∆φ = ∆ω c κ/2 = Re(χ(ω c )) κ/2) σz (2.29) ∆A = ∆(κ/2)A 0 κ/2 = Im(χ(ω c )) A 0 κ/2 σz (2.

Improving the cooperativity

There are two strategies to improve the cooperativity of the system 2g 2 /κΓ; either increase the coupling strength g or reduce the decoherence rates of the coupled system.

Increasing of g:

We focus here on the coupling strength between cavity photons and the dipole associated to one electron in a double quantum dot. From the expression of g DQD (equation 2.6) one can identify the different strategy to improve the DQD-cavity coupling. For a given photonic potential gradient, one can fabricate the two dots far apart from each other while preserving tunneling between the dots, thus increasing the potential difference felt by the electron whether it is in the left or in the right dot. Other than increasing the double quantum dot dipole, the coupling strength can be enhanced by engineering strong spatial variation of the potential. The shaping of the photonic potential is done by acting mainly on its screening and one can use high frequency electromagnetic simulation (such as the HFSS software) to finely tune the geometry of the circuit.

One can also enhance the overall value of potential V ⊥ ( -→ r ). Recalling that for an LCresonator, the operator for the potential inside the capacitance reads:

V = q C = V ZP F (â + â † ) (2.31)
Phase Shi�

ε δ ω D Q D ω RF = ω cavity 2t Energy Amplitude ε δ ε δ ε δ ε δ (b) (c) (d) (a) (0,1) (1,0) (0,1)
(1,0)

V ZP F = ω 2 c Z c π (2.32)
V ZP F is the RMS zero-point uncertainty of V when the LC-resonator is in its ground state, and

Z c = π 2 L
C is the impedance of the resonator. Notice that the larger the impedance Z c the larger the potential, hence going to highly inductive resonator enhance the coupling strength. The few implementation of this approach has been done using, josephson junction SQUID array (Z c = 1.8kΩ in [START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF]), compact resonator (Z c = few 100Ω in [START_REF] Geerlings | Improving the quality factor of microwave compact resonators by optimizing their geometrical parameters[END_REF]), modified CPW geometry (Z c = 200Ω in [START_REF] Mi | A coherent spinphoton interface in silicon[END_REF]), or high kinetic inductance material such as NbTi (Z c = 1kΩ in [START_REF] Samkharadze | High Kinetic Inductance Superconducting Nanowire Resonators for Circuit QED in a Magnetic Field[END_REF], [START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF]).

In general, the increase of the electrical dipole of the double quantum makes the qubit also more sensitive to surrounding charge noise. In the same way, going to higher impedance resonator is accompanied by an increase of the photon loss rate κ. It is then important to simultaneously work on improving the cleanliness of the environment in order to limit Γ 2 and κ to low values.

Reduction of Γ 2 and κ: The two main sources of qubit loss of coherence are electrical noise caused by charge fluctuators mostly located at the interfaces in the device and in oxide layers [START_REF] Gao | Experimental evidence for a surface distribution of two-level systems in superconducting lithographed microwave resonators[END_REF], and magnetic noise due to random fluctuations of the surrounding nuclear spins [START_REF] Khaetskii | Electron Spin Decoherence in Quantum Dots due to Interaction with Nuclei[END_REF]. Improvement of the fabrication process (removing oxide layers [START_REF] Barends | Minimal resonator loss for circuit quantum electrodynamics[END_REF]), and of the material purity (using spin nuclear low density material, as isotopically purified Si), have shown to be very efficient for limiting the decoherence of the qubit as well as photon loss rate of the cavity. To limit the impact of these electromagnetic noises, another approach is to operate the qubit at an optimal working point, called sweet-spot, where the qubit is insensitive to noise at first order. This means to find an extrema of the qubit frequency as a function of gate voltages (on which charge noise is acting). In the same spirit, flattening the dispersion of the qubit frequency with respect to gate voltages limits the second order noise dependence. This idea originates from the superconducting qubit community and has lead to the transmon design [START_REF] Koch | Chargeinsensitive qubit design derived from the Cooper pair box[END_REF].

As previously seen in the section 2.1.2, the photon loss rate has an internal contribution κ int and an external contribution κ ext due to the coupling to input and output transmission lines. While lowering κ int is beneficial for any experiment, a low κ ext is not always wanted in C-QED experiments. Indeed, the time for manipulating and reading the state of a qubit using a dispersively coupled cavity is set by 1/κ ext . As a result, the target value is κ ext = κ int , allowing a manipulation and read-out time of few 100ns as a typical value; without degrading too much the total photon loss rate κ.

Chapter 3

Experimental techniques

This chapter begins by giving a motivation for a new fabrication technique of carbon nanotubes circuits (section 3.1). Then, most of the technical aspects involved in this thesis are introduced. It goes from the nanofrabrication of the devices carried out in the clean room facility of ENS (section 3.2), to low temperature measurements (section 3.3). A particular emphasis is given to the integration of the carbon nanotube into the quantum dot circuit, whose development represent a large part of this thesis work.

3.1 Toward more tunable carbon nanotube quantum-dot circuits

Various Nanofabrication approaches

Carbon nanotubes appear as excellent candidates for hosting quantum dot circuits (cf section 1.1). Nevertheless, it remains a challenge to experimentally access these properties, and obtain a defect free 1D conductor over several µm. The method to integrate the nanotube in the quantum dot circuit is to a large extent responsible for his damaging.

One of the earliest and easiest way to fabricate carbon nanotube devices is by postgrowth processing [START_REF] Bockrath | Single-Electron Transport in Ropes of Carbon Nanotubes[END_REF]. In this technique, carbon nanotube are grown all over the substrate, then they are imaged by atomic force microscopy or scanning electron microscopy. The circuit is subsequently patterned over the choosen carbon nanotube. But post-growth nanofabrication of the circuit electrodes contaminates the carbon nanotube with organic residues. Indeed, carbon nanotube easily absorbs polymer contained in cleanroom resist, and knowing that transport in CNT occurs through p Z -orbitals which are pointing outwards, it will drastically affect the transport spectrum. In addition, carbon nanotube growth produces amorphous carbon on the substrate which is an highly dissipative medium for microwave, hence making this technique incompatible with c-QED application.

A way to circumvent this last issue is to grow the carbon nanotubes on a separate chip with a predefined mesa structured, and stamp it in a dedicaded area of the microwave resonator. This local transfer technique has been developped by J. Viennot [START_REF] Viennot | Stamping single wall nanotubes for circuit quantum electrodynamics[END_REF], and allows to combine high quality factor microwave cavity and carbon nanotube quantum dot circuits on the same chip. But such sample still suffer from post growth contamination, since all the electrode of the circuit are fabricated after the integration of the CNT.

In case of transport-only devices, one strategy to obtain ultraclean carbon nanotube consist in growing carbon nanoutube on a fully-patterned device requiring no further nanofabrication. This technique enabled to produce suspended CNT and provided one of the cleanest transport spectrum in a CNT [START_REF] Kuemmeth | Coupling of spin and orbital motion of electrons in carbon nanotubes[END_REF]. Its main drawback is the very low yield of electrically connected nanotubes, which begin to be a problem for devices with increasing complexity. Moreover the high temperature and the chemically aggressive gas mixture needed for the growth of CNT (900 • C in the used recipe) can make metallic thin film to melt and strongly deform. In particular, the superconducting metal realizing the microwave cavity can display a strong decrease of its critical temperature. Recent efforts have been made to realize microwave cavity based on MoRe superconductor which exhibit high quality factor even after exposure to CNT growth [START_REF] Blien | Towards carbon nanotube growth into superconducting microwave resonator geometries[END_REF].

The most recent technique [START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF], [START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF], [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF], [START_REF] Gramich | Fork stamping of pristine carbon nanotubes onto ferromagnetic contacts for spin-valve devices[END_REF], [START_REF] Blien | Quartz tuning-fork based carbon nanotube transfer into quantum device geometries[END_REF] is similar to the previous local transfer technique, only differing in the fact that the circuit is patterned before the transfer step (see figure 3.1). This last point requires a deterministic transfer which is the main difference with previous transfert technique where few carbon nanotubes were transfered to the circuit chip and a post selection is needed. This technique allows to combine pristine carbon nanotube based quantum dot circuits with high quality factor microwave cavity but has remained unused in this perspective. In the following this new technique will be called stapling.

Stapling in a meso-QED device

As discussed in the previous section, one of the advantages of this technique is the possiblity to produce suspended carbon nanotubes, which make it attractive for realizing mechanical oscillator with carbon nanotube [START_REF] Steele | Strong Coupling Between Single-Electron Tunneling and Nanomechanical Motion[END_REF]. The suspended nature is also expected to reduce charge noise in carbon nanotubes (see figure 3.2). Indeed, substrate is believed to host numerous charge oscillators, hence putting some distance between the CNT and the substrate is expected to result in a less electrically noisy environment for the CNT. This is particularly interesting when one is willing to embedded an electricallyaddressable qubit in a CNT. Up to now, the confinement in most of the quantum dot Zhong [START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF], and inspired a few other groups. Each of them have different technical approach but the principle remains the same. Such a method allows to envision arbitrarily complex carbon nanotube circuit. Up to now the most successful one has been developed in the group of S. Ilani [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF], where circuits with up to 16 gates, and multiple carbon nanotubes have been realized as shown in the top left box. Sources: [START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF], [START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF], [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF], [START_REF] Gramich | Fork stamping of pristine carbon nanotubes onto ferromagnetic contacts for spin-valve devices[END_REF], [START_REF] Blien | Quartz tuning-fork based carbon nanotube transfer into quantum device geometries[END_REF] circuit made in the group was believed to be dominated by disorder originating from interaction with the substrate and nano-fabrication residues. Such a disorder yields a low tunability of the circuit. The stapling technique reduces the contamination of the CNT to a minimum. Thus, one can fully controlled the confinement potential in the CNT by means of electrostatic gates. One can start to envision more complex circuit and to modified independently the different properties of CNT based circuit. For instance, control over the lead-dot coupling rate can be instrumental for switching between a closed system regime to an open one ( chapter 4), or to modulate the induced Zeeman splitting in the Ferromagnetic spin qubit (chapter 5).

To summarize, the increasing complexity of quantum dot circuits, the low yield of nanofabrication, and the weak tunability are the reasons why going towards a new generation of sample is essential.

Device nanofabrication

Sample overview

The chips fabricated in this thesis consist in a carbon nanotube-based quantum dot circuit coupled to a microwave cavity (see figure 3.3). This microwave cavity is a centimeterlong superconducting coplanar waveguide resonator (CPW) with a fundamental mode around 6.5GHz. The quantum dot circuit is embedded at an anti-node of the cavity electric field and its characteristic length is of the order of the micrometer. All the clean-room process is performed on a 10mm × 10mm chip containing two devices. By doing so, it is easier to handle the chip, and it allows to be faster on all nanofabrication steps. The substrate used in all experiments shown in this thesis, is a standard high resistivity (10kΩ.cm) silicon wafer covered with 500nm-thick layer of silicon oxide. The high resistivity silicon allows for low microwave dissipation, and the 500nm of thermal oxide prevents DC gates from leaking between each other. A widely used instrument in the fabrication of these chips is the Scanning Electron Microscope (SEM). It can be used as a microscope with a resolution down to 10nm, but the SEM is mainly used to draw pattern in electrosensitive resist with a similar resolution. The basic idea is to use the electrosensitive resist as a stencil on which the desired metal is evaporated and the resist layer can subsequently be removed during the lift-off step (see figure 3.4 for more details).

Resist SiO2

eor UV

Resist deposi�on The lithography technique is used whenever one want to pattern a structure on the device. It can be applied for metal deposition as well as for etching a defined area. The fabrication of a our samples typically contains around five different lithography.

1. A few drops of resist are deposited on the substrate.

2. Spin-coating of the resist in order to obtain an homogenous layer of resist.

3. The resist layer is exposed to an electron or photon beam depending on the sensitivity of the resist. The nature of the beam is chosen according to the smaller dimension of the pattern one want to draw.

4. In case of a positive resist (as illustrated) the exposed areas are removed using the appropriate chemical. For negative resist only the exposed areas remain on the chip.

5. The remaining resist then act as a mask either for metal deposition or etching of an underlying layer.

6. Finally the resist mask is removed in a acetone bath.

Chapter 3. Experimental techniques 68

Nanofabrication steps

Throughout this thesis, two strategies have been adopted for the fabrication process.

They differ in the way the carbon nanotube is integrated into the circuit. The first one is a stamping technique [START_REF] Viennot | Stamping single wall nanotubes for circuit quantum electrodynamics[END_REF] in which the carbon nanotube is transferred to the circuit chip after the Nb resonator fabrication, and then contacts and gates are deposited on top of it. This technique has been developed before the beginning of this thesis work, and has revealed to be very efficient for combining high quality factor cavity (Q 10 4 ) and carbon nanotube based quantum dot circuit. But in such devices, the carbon nanotube undergoes chemical processes which can results in resist residue in the environment of the CNT or some adsorbat on the CNT. Consequently, the electronic confinement potential in the quantum dots will suffer from this disordered environment, and the resulting devices will be weakly tunable by gates voltages. The second strategy consists in integrating the carbon nanotube at the very end of the nanofabrication and this allow to produce pristine carbon nanotube based quantum dot circuit. The main steps of those two processes are briefly described in the following:

Stamping process 1. Alignment crosses and pre-contact The alignment crosses are references for all the following steps. In particular, they allow for a 50 nm-accurate alignment between the electronic lithography of contacts and gates of the quantum dot circuit.

The pre-contacts are made of gold and are the bridge between the fine electrodes of the quantum dot circuit and the larger niobium lines. Because niobium oxidizes and not gold this pre-contact are essential for ensuring a electrical contact between niobium lines and circuits electrodes. This step is done by electronic lithography (see figure 3.4).

2. Niobium resonator It contains the coplanar waveguide (CPW) resonator [START_REF] Goppl | Coplanar waveguide resonators for circuit quantum electrodynamics[END_REF] and the large part of the circuit line and bonding pad. Because the niobium is not compatible with lift-off techniques on large scales, it is first deposited on all the chip and then one selectively etch it using electronic lithography and a dry etching process based on SF 6 .

Growth of carbon nanotubes on a comb of cantilever and localization

This step is similar to the one in the stamping process except to the growth chip which is now a cantilever comb. This new design enables the growth of suspended carbon nanotubes. Then we identify good candidates carbon nanotubes for stapling by observing the cantilever comb using a scanning electron microscope (SEM). Because the development of this stapling technique represent an important part of this thesis, in the following, we focus on the different steps of the stapling process.

Niobium cavity fabrication

The fabrication of the niobium coplanar waveguide resonator is done by an etching process. 100 nm of Niobium is evaporated on the whole surface of the substrate in a ultra high vacuum (UHV) electron gun evaporator reaching pressures below 5.10 -10 . In such low pressure, evaporated atoms follow ballistic trajectories over few meters ensuring high purity metallic films. Because Niobium is a refractory metal, the heating necessary for its evaporation is usually a limitation for reaching a good vacuum quality during the evaporation, and it may prevents from getting a good quality thin film. Nevertheless, this issue can be circumvent by a powerful cooling system of the deposition chamber.

Indeed, the chamber is fitted with cryogenic panels which can be cooled down with liquid nitrogen allowing to stay below 1.10 -9 mbar during the niobium evaporation. It is worth noting that Niobium is more commonly deposited by a sputtering technique, which have the merit to produce more sticking thin film, nevertheless in the ENS clean room facility, such sputtering equipment was not available. After the niobium deposition, all the steps of the electronic lithography technique are followed (cf. figure 3.4), with an extra cleaning of the substrate in a 5 minutes ultrasonic bath in acetone followed by a 3 minutes soft oxygen plasma in the RIE chamber to completely remove resist residues.

Because all the dimensions in the CPW design are larger than 1 µm, this step can in principle be done by optical lithography which is much faster than the electronic one.

However, optical lithography requires to fabricate a mask for a specific design, which is not convenient when one is often introducing small changes in the cavity design. For this reason we favored electronic lithography even for the fabrication of the microwave cavity.

Quantum dot circuit contacts and gates

To define the contact and the gate electrodes, the standard lithography technique is used (cf. figure 3.4). These steps of the process only act in the 100 µm × 100 µm area where the quantum dot circuit is located. This allows us to precisely align a lithography with respect to the previous ones. Indeed, the SEM can sweep a 100 µm × 100 µm area only by deflecting the e-beam and without any mechanical displacement, which allows to easily reach an alignment accuracy as low as 50 nm.

For the lithography of the gate electrodes which is the thinner structure of the device, a stack of 100 nm of MMA resist (Methyl methacrylate) and 50 nm of PMMA resist has been used. This stack is useful for two reasons. First, for the same dose factor, the MMA resist is developped faster than PMMA, this way the opening in the MMA will end-up larger than the PMMA one, which trully defines the width of metallic electrode (see figure 3.5). This way the deposited metal is not in contact with resist on its edges, which can be a problem at the lift-off step for a single layer of PMMA. Second, The total thickness of the resist layer is lower than the standard layer of 500 nm of PMMA, allowing for a better writing resolution. Details about each metal depositions are given in appendix A.

Etching side trenches

To ensure the contact between the nanotube and the circuit electrodes, two trenches on both side of the quantum dot circuit have to be etched to host the comb of cantilevers (in blue in figure 3.3). Since these trenches are 2 mm long for 200 µm wide, they can be defined by optical lithography. Moreover, the used optical resist (AZ5214) is 1.5 µm thick and is more resistant to SF 6 plasma etching than PMMA resist. Thereby it provides a resistant enough mask to withstand reactive ion etching of 15 µm of silicon.

Once the two adjacent trenches have been etched, we can cleave the sample in two chip of 5 mm × 10 mm, and then staple a carbon nanotube to complete the circuit.

Growth of carbon nanotube

The carbon nanotubes are grown by Chemical Vapor Deposition (CVD) in a home made furnace (see figure 3.6-a). In the CVD process, catalytic metallic nanoparticules are used as a precursor for the growth. The used recipe has been developed in Basel by J. Furer [START_REF] Furer | Growth of Single-Wall Carbon Nanotubes by Chemical Vapor Deposition for Electrical Devices[END_REF] and is optimized to obtain single walled carbon nanotubes. The catalyst solution is composed of 39 mg of F e(N O 3 ) 3 -9H 2 O, 7.9 mg of M oO 2 and 39 mg of Al The distribution density is a key parameter: a high density favors the formation of CNT bundles (as the chance of two CNTs meeting increases), while a too low density results

Chapter 3. Experimental techniques 74 in a small number of CNTs. The catalyst density can be estimated by using optical microscope in dark field mode (using a low-angle light). Additional catalyst can be deposited as long as the satisfying distribution density is not reached. The growth chip can then be inserted in the quartz tube of the CVD furnace and the growth can start (further details on the growth recipe are given in appendix A).

During the development of the stapling process other options have been explored such as using suspended CNTs over pillars (in the spirit of [START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF], [START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF]). In this perspective, the strategy is to initiate the growth at one location of the chip and then obtain long enough CNTs to reach the pillars few hundred of micrometers away. This strategy is highly beneficial for limiting the number of CNT bundles, but it requires a directional growth. Guided by this idea, we followed the growth recipe of [START_REF] Zhong | Ultralow Feeding Gas Flow Guiding Growth of Large-Scale Horizontally Aligned Single-Walled Carbon Nanotube Arrays[END_REF], reported to produce long CNTs and aligned along the gas flow. The growth results are presented in figure 3.7. Nevertheless, because of a low success on the transfer of CNT, (most likely due to the uncontrolled angle between the growth chip and the circuit chip) we decided to use cantilevers which have a much better tolerance on the approach angle. The catalyst deposition is critical for getting long CNTs. It has been found that depositing the catalyst on the edge is a efficient way to obtain a descent number of long CNTs (longer than 500 µm). In this low flow growth recipe, the kite-mechanism [START_REF] Huang | Growth Mechanism of Oriented Long Single Walled Carbon Nanotubes Using Fast-Heating Chemical Vapor Deposition Process[END_REF] is taking place, where the catalyst nanoparticles are floating above the substrate and dragging the growing CNTs. The edge is believed to lift the nanoparticles because of local turbulent flow, then they are maintained thanks to the buoyant force. In such situation, CNTs can easily reach few millimeters. b. Similar growth with pillars prepatterned on the substrate resulting in suspended CNTS. The fact that the trajectories of the long CNTs are not modified by the pillars is consistent with the kite-mechanism.

Scale bar: 5 µm.

Embedding of the carbon nanotube

The Stapler The integration of the carbon nanotube is accomplished inside a vacuum chamber dedicated for this task (see figure 3.8). This chamber contains a micromanipulator stage responsible for coarse motion and a piezo-motor stage allowing a fine positioning of the cantilever chip with respect to the circuit chip. The system is also DC wired for transport measurements during the stapling. An argon gun is integrated to clean the circuit chip surface prior to stapling a carbon nanotube, or to remove the stapled CNT if this one does not meet the desired criteria. In this scenario the cantilever chip can be isolated in a buffer chamber while the circuit chip is being cleaned. A window is also encrusted in the cover of the main chamber to optically align the two chips with respect to each other. This equipment has been design to fit with 4K-measurements by adding a cold finger through the bottom flange.

Stapling principle After the carbon nanotube growth on the cantilever comb, the CNTs are localized by SEM. The goal of this SEM observation is to sort the pairs of cantilever depending on wether they contain a CNT suitable for stapling. A suitable nanotube for stapling has to be as close as possible to the tip of the cantilever (in the first 10 µm), it has to be isolated enough from other possible suspended CNTs (at least 5 µm), and in the ideal scenario, it should be a unique single walled carbon nanotube.

Although the SEM is not suited to characterize carbon nanotubes, it allows at least to distinguish between a bundle composed of plenty of CNTs and one which contains only 2 or 3 CNTs, by looking at the contrast and at the Y shape characteristic of the braiding of two CNTs (see figure 3.6). More details are given in appendix C. Once this localization step is finished, the cantilever chip is ready to be glued with PMMA to the piezo-motor stage inside the Stapler.

A very close attention is given to the time spent by the cantilever chip in ambient atmosphere. CNT are well known to have a high adsorption capacity, which is exploited to design gas sensor based on carbon nanotubes [START_REF] Wang | A Review of Carbon Nanotubes-Based Gas Sensors[END_REF]. For our use, this property is rather a drawback and we want to limit as much as possible the exposition time to the atmosphere, in order to avoid such contamination. For that reason the cantilever chips are stored in a primary vacuum between the carbon nanotube growth and the stapling step.

Once the circuit chip and the cantilever comb are inserted in the stapler, we can start the stapling procedure. The global strategy for stapling is the following:

-Slowly lowering the cantilevers using the nano-positionner stage until a current is detected in the circuit, ensuring that a nanotube has been contacted.

Driving a high current in the external sections of the CNT to cut it by Joule heating.

--Rising the cantilevers enough to avoid any dramatic collision during the extraction of the circuit chip from the stapler.

-Measuring the gate dependence of the current through the central section of the CNT.

-Depending on the resistance of the central section, it might be needed to improve the contact resistance by driving a high current through the central section.

-Transferring the sample holder from the stapler to the cryostat, while keeping all electrodes grounded and the circuit chip under Nitrogen atmosphere to limit the reactivity of the CNT to the oxygen.

Contact detection

During the approach, a bias voltage (typically around 500 mV) is applied to the outermost contact electrodes while keeping the other electrodes grounded (see figure 3.9). Keeping the gate electrodes grounded during the approach is a important point to prevent the CNT to snap to the gates. The current is monitored by measuring the voltage drop across a 100 kΩ resistor, hence any current between the biased contacts and the ground will be detected, in particular current in the two cutting sections. This setup provides large chance to detect a CNT regardless of the approach angle.

Cutting and room temperature characterization After mating the CNT to the circuit, the next step is to separate it from the cantilevers, to avoid detaching the CNT from the circuit when raising the cantilevers. This is done by driving a large current between two adjacent contact electrodes up to a threshold value after what the CNT burns [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF]. Figure 3.9: DC detection circuit: This is the circuit used to detect the contact between the CNT and the circuit chip while the cantilever chip is lowered. A typical value for V app is 500 mV, which allows for easy detection of the contact. V meas range from 10 µV to 2 mV , which correspond to contact resistance of few hundreds of MΩ.

current carrying capacity of the CNT which vary a lot from single walled CNT to multiwalled CNT or bundles [START_REF] Collins | Engineering Carbon Nanotubes and Nanotube Circuits Using Electrical Breakdown[END_REF]. Another signature of multi-walled CNTs or bundles is the presence of multiple current drops in the cutting I-V curve (see figure 3.10 (b)), which is consistent with sequential cuts of the different shells of multi-walled CNT or the different CNTs in case of a bundle.

The cutting mechanism is believed to be due to Joule heating rather than electromigration [START_REF] Collins | Current Saturation and Electrical Breakdown in Multiwalled Carbon Nanotubes[END_REF]. Indeed the highest temperature in the CNT is reached at the center of the suspended segment since it is the farthest point from the contact electrodes that act as thermal sinks. In that respect, the fact that the cut of the CNT always occurs at the center of the suspended segment is consistent with a Joule heating mechanism.

These cutting curves also provide information about the cleanliness of the CNT. In This operation is also highly beneficial for the total resistance of the resulting device.

Indeed, the cut of the CNT acts as a local annealing. This drastically improves the contact resistance between the CNT and the two contacts of the resulting circuit. The contact resistance after mating is typically between a few tens of MΩ and hundreds of MΩ, which is too high for any transport measurements at millikelvins temperature. V SD (V) V SD (V) I (µA) I (µA)

(a) (b)

Figure 3.10: I-V curve during the cutting of a CNT (a) Three distinct regimes can be identified in the cutting I-V curve. First, between 2 V and 3.5 V the current increases step-wisely due to a sequential improvement of the contact. Then the current reaches its threshold value (between 3.5 V and 4.5 V for this CNT). For some of the contacted CNTs, a decrease of the current is observed before the CNT is cut. This reflects the suspended behavior of the CNT [START_REF] Ioan M Pop | Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles[END_REF], and is due to electron-optical phonon scattering. (b) I-V curve for a multi-walled CNT or a bundle displaying characteristic multiple current drops. On this I-V curve, the current is not reaching zero after the cut of the CNT. This residual current, which can be large, is attributed to a contact between the CNT and the silicon at the bottom of the trenches which is grounded. No current is measured when the source and drain are exchanged which confirms this scenario. After the cutting of the CNT, the total resistance of the device lay in between 100 kΩ and few MΩ. If the circuit is still too resistive after this step, a post annealing is worth considering (more details in the next paragraph).

µm

To discriminate between a semiconducting or a metallic CNT, one can measure the current in the central section as a function of gate voltage V G , at a fixed bias voltage.

The gate voltage shifts the chemical potential in the CNT with respect to the Fermi energy E F of the leads. Thus one can tune E F into the band gap of the CNT, leading to a current suppression. For carbon nanotubes, three different behaviors can be identified:

semiconducting, narrow-gap, or metallic (see figure 3.12). According to the zone folding approximation, only semiconducting and metallic CNTs are expected [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]. Nevertheless a narrow gap behavior can be explained by transverse curvature or strain in CNTs predicted to be metallic by the zone folding approximation [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF].

Post annealing After cutting the external section of the CNT, the resistance of the central portion can remains too high for low temperature transport measurement (above 3 MΩ no current was detected below V SD = 50 mV, due to a too high contact barrier).

A solution is to drive a high current (typically a few µA) through the central section to enhance the wetting between the metal and the CNT, therefore improving the contact resistance. Similarly to the cutting step this can be understood as a local annealing of the contact, with the only difference, here V app do not exceed V threshold .

Transfer of the chip carrier Once the stapled carbon nanotube has a good enough resistance, one have to transfer the chip carrier inside the cryostat. This is one of the tricky step in the process since carbon nanotubes are very sensitive to any electrical discharges originating from triboelectric effect or transient current during connections.

Such sensitivity is believed to be due to a self heating of the CNT, and its reaction with oxygen, causing the breakdown of the CNT [START_REF] Collins | Current Saturation and Electrical Breakdown in Multiwalled Carbon Nanotubes[END_REF]. The main precautions consist in always keeping the DC lines of the circuit grounded and maintain the chip carrier in a nitrogen atmosphere to avoid the CNT to be in contact with The length of the cutting section (see figure 3.14) determines the current I threshold needed to cut the CNT. Indeed, by increasing the distance between the two adjacent contacts the center of the suspended segment will be more thermally decoupled from them. On one hand, it is preferable to keep V threshold high enough to avoid accidental cutting of the CNT. On the other hand, going to high bias voltage (typically above 10 V) might break the gates oxide if the CNT is not suspended. This can causes the burning of the CNT due to a high transient current and also the melting of the contacted gates.

Typical spacing between the contact electrode and the cutting contact is 1 µm, which correspond to a V threshold lying in between 4 V and 6 V.

The width of the cutting contact defines the contact resistance with the CNT, which affects the detectability of the CNT during the lowering of the cantilevers. It will also influence V threshold : a wider cutting contacts result in a lower V threshold .

Development in progress focus on the characterization of the CNT when it is still suspended on the cantilever comb, in order to post select the single walled carbon nanotubes.

Raman spectroscopy and Rayleigh spectroscopy are the two techniques considered.

Measurement techniques

In this section we describe the setup for transport measurement as well as for RF measurement of the cavity. hence when no microwave tone is applied, the cavity is in its ground state. In order to preserve this very low thermal population, one need also to cool down the incoming drive field. This is done by thermalizing the input lines at each stage and by attenuating the 50 Ω thermal noise of each stage as illustrated in the figure 3.15. For the output line, one require as less attenuation as possible since it carries the signal of interest. Therefore a NbTi microwave cable with low attenuation is used up to the 4 K stage where a HEMT cryogenic amplifier is inserted. On the 20 mK stage, two cryogenic circulators are placed on the output line to protect the microwave cavity from 4 K thermal noise originating from reflections on the HEMT amplifier (see figure 3.15).

Isolating the device from environment

Transport measurements

All the transport measurements performed in this thesis work are either differential conductance, or current measurements. To measure the current, a bias voltage is applied on one of the contact electrode of the carbon nanotube circuit with a Y okogawa GS200.

A voltage divider and a RC filter (f cut-of f = 1.6 kHz) at room temperature are used to minimized noise in addition to the second order RC filter at 20 mK. At the output of the CNT the current is of the order of 100 pA, thus we use a home-made current amplifier with a gain of 10 7 (Electrical schematic available in the PhD thesis of Jeremie Viennot [START_REF] Viennot | Charge and spin dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF]), then the voltage is read by a Keithley 2000. For the gate voltages, an Itest rack (µBILT BN 103) with five modules (BE2141) providing 20 output voltages is necessary. Indeed the number of gate electrodes can go up to 18 for two circuits on the same chip.

The whole measurement setup is presented in figure 3.16 and a typical measurement with this setup is shown in figure 3.17

Cavity transmission measurement

Measuring the cavity transmission provides information about the internal dynamic of the quantum dot circuit. In that sense, it is complementary to transport measurements which probes the circuit only via its coupling to the leads.

RF setup

During this thesis multiple RF measurement setup have been used. The initial one is based on a Lock-In detection scheme and is presented in [START_REF] Laure | Microwave as a probe of quantum dot circuits: From Kondo dynamics to mesoscopic quantum electrodynamics[END_REF]. This setup benefit from the ability of Lock-In amplifier to extract signal at a given frequency in a noisy environment. Nevertheless such a setup is not fitted for time resolved measurement since it is unable to do pulse sequences. A simpler setup consists in directly connecting a VNA

A = I 2 + Q 2 ϕ = -arctan Q I
Depending on where the coherent state of the cavity field is displaced in the I/Q space by the circuit state, it can be relevant to look at other quadratures than the amplitude and the phase of the coherent state of the cavity field. Nevertheless the phase and the amplitude have the advantage of being directly link to physical quantities: Dissipation or photon emission for amplitude and dispersive shift of the cavity resonance for the phase (see section 2.3).

Qubit spectroscopy In a qubit spectroscopy measurement, the qubit state is brought out of σ z = -1 (north pole of the Blcoh sphere) by applying an additional tone resonant with the qubit frequency. According to the formula 2.27, this change of state is visible in the cavity transmission signal. Therefore, such measurement allows to access the qubit energy dispersion with respect to any parameter of the system (typically a gate voltage or the magnetic field). In this procedure the qubit is manipulated while the cavity is filled with photons, which could be detrimental for the qubit coherence. Indeed, photon shot noise in the cavity gives rise to random fluctuations of the qubit frequency because of the AC-Stark shift (see formula 2.17), resulting in a broadening of the qubit linewidth, hence a lower visibility in the measurement. This phenomenon is often referred to as measurement-induced-dephasing [START_REF] Schuster | Ac Stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field[END_REF].

A cleaner way to perform the spectroscopy of the qubit is to separate in time the manipulation and the read-out. This can be done by performing a π-pulse (a π 2 -pulse also works but results in lower contrast) on the qubit and subsequently filling the cavity with photons (see figure 3.20). If the calibration of π-pulse is not possible, the poor The FWHM (Full Width Half Maximum) of the resonance is ∆f -3dB = 530 kHz corresponding to a quality factor Q = 12500. The measurement is performed at a input power such that the average photon number at the resonance is about n ph 10. The estimation of the photon number requires to know the input and output powers at the cavity ports, which are known with a 3 dB precision.

Qubit drive

Cavity filling

Cavity read-out satisfying σ z = 0. This sequential procedure impose to have a qubit relaxation time T 1 longer than the time needed for filling the cavity with photons (namely its relaxation time 1/κ). If this condition is not fulfilled, one can also start to fill the cavity before the end of the qubit manipulation.

Time domain measurement

This type of measurement are more demanding in terms of calibrations than continuous ones, since the duration and the power of the applied tone can be very critical. During those measurements, the AWG is repeatedly providing pulse sequence and the ADC is waiting for a trigger signal as soon as it is not acquiring-processing data anymore. Only a small selection of the possible pulse sequences is presented here, namely T 1 -measurement and Rabi oscillations.

T 1 -measurement The longitudinal relaxation of the qubit is described by the characteristic time T 1 . To measure it, the standard way is to do a π-pulse on the qubit (placing it in its excited state: σ z = 1), then read the cavity after a variable time τ allowing to reconstruct the full exponential relaxation (see figure 3.21-a). π 2 -pulse and saturation pulse also work. Indeed, all pulses which drive the qubit out of σ z = -1 are valid, even though driving the qubit in σ z = 1 leads to the highest contrast. An other possible strategy is to drive the qubit, then directly look at the time traces after the qubit drive tone has been turned off [START_REF] Bianchetti | Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics[END_REF], [START_REF] Scarlino | All-Microwave Control and Dispersive Readout of Gate-Defined Quantum Dot Qubits in Circuit Quantum Electrodynamics[END_REF].

Rabi oscillations In Rabi oscillations, the qubit is rotating in a perpendicular plane to the equator of the Bloch sphere, hence oscillating between its ground and excited states with frequency Ω R 2π , and finally ends up at the center of the Bloch sphere. Such oscillations are performed by applying a drive pulse of variable duration (see figure 3 Pulse sequences for T 1 -measurement and Rabi oscillations The delay between the filling and the read-out of the cavity is set to 1/κ. In all the presented pulse sequences the re-initialization of the system is done by waiting long enough for the cavity and the qubit to relax in their ground states.

frequency reads (see supplements of ref. [START_REF] Murch | Cavity-Assisted Quantum Bath Engineering[END_REF]):

Ω R 2π = g d ∆ qubit-cavity (3.1)
Where d is the amplitude of the drive tone, g is the qubit-photon coupling, and ∆ qubit-drive = f qubit -f cavity is the frequency detuning between the qubit and the cavity. Measurements of the dependence of Ω R 2π according to d (Power dependence), and ∆ qubit-drive (Chevron pattern) are the two standard sanity checks.

Chapter 4

The superconducting double quantum dot

Coupling double quantum dot circuits to microwave cavities provides a new powerful means to control, couple and manipulate qubits based on the charge or spin of individual electrons. In this chapter, we revisit this standard configuration by adding superconductivity to the circuit. We combine theory and experiment to study a superconductordouble quantum dot circuit coupled to microwave cavity photons. First, we use the cavity as a spectroscopic probe. This allows us to determine the low energy spectrum of the device and to reveal directly Cooper pair assisted tunneling between the two dots. Second, we observe a vacuum Rabi splitting which is a signature of strong chargephoton coupling and a premiere with carbon nanotube based quantum dot circuits. We show that our circuit design intrinsically combines a novel set of key features to achieve the strong coupling regime to the cavity. A low charging energy reduces the device sensitivity to charge noise, while sufficient coupling is provided by the shaping of the spectrum of the double quantum dot by the superconducting reservoir.

Introduction

Circuit quantum electrodynamics (cQED) allows one to probe, manipulate and couple superconducting quantum bits at an exquisite level using cavity photons. Transferring the methods of cavity quantum electrodynamics to quantum dots circuits is appealing for multiple reasons [START_REF] Cottet | Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena[END_REF]. Electrons confined in quantum dots can be used as quantum bits based on their spin or charge degree of freedom. In the context of quantum information processing, cavity photons were first envisionned as a powerful way to manipulate such quantum bits. Therefore, most experimental efforts [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF][START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF][START_REF] Basset | Single-electron double quantum dot dipolecoupled to a single photonic mode[END_REF][START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Bibliography 193 Impedance Resonator[END_REF] have been directed towards achieving the strong coupling regime, which allows one to Chapter 4. The superconducting double quantum dot 97 hybridize coherently light and matter [START_REF] Chiorescu | Coherent dynamics of a flux qubit coupled to a harmonic oscillator[END_REF]. To that purpose, tunneling between the double quantum dot and metallic contacts is usually considered as an undesired dissipation channel and minimized.

However, metallic reservoirs can also be seen as a resource, as quantum dot circuits can include a variety of normal metal, ferromagnetic or superconducting electrodes. The engineering of electronic states in devices combining materials with different electronic properties is at the heart of many recent methods put forward for quantum information processing. One particularly promising venue is the coupling of superconductors to nanoconductors. For exemple, semiconducting nanowires proximized by superconductors are under active investigation because of the possibility to induce non-local superconducting correlations in the topological regime [START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF][START_REF] Albrecht | Exponential protection of zero modes in Majorana islands[END_REF]. Double quantum dots with a central superconducting contact are sought for creating distant entangled spins by the splitting of Cooper pairs [START_REF] Hofstetter | Cooper pair splitter realized in a two-quantum-dot Y-junction[END_REF][START_REF] Herrmann | Carbon Nanotubes as Cooper-Pair Beam Splitters[END_REF][START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF][START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF].

Combining cQED architectures with hybrid superconducting mesoscopic circuits is only at its premises. This has been successful in the case of superconducting quantum point contacts, as epitomized by the recent manipulation of an Andreev qubit by a microwave resonator [START_REF] Janvier | Coherent manipulation of Andreev states in superconducting atomic contacts[END_REF]. Looking at hybrid superconductor quantum dot circuits, a single experiment has been reported so far, with a single dot [START_REF] Bruhat | Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua[END_REF].

Here, we present the first implementation of a hybrid superconductor-double quantum dot circuit coupled to a microwave cavity. Such an experiment was theoretically proposed to test the coherence of Cooper pair splitting between the two dots [START_REF] Cottet | Probing coherent Cooper pair splitting with cavity photons[END_REF][START_REF] Cottet | Subradiant Split Cooper Pairs[END_REF]. As we will show below, our circuit operates in a parameter regime where the coherent injection of Cooper pair appears in a different way as was theoretically considered in those references.

Nevertheless, our results confirm the idea that cQED tools are a powerful method to probe the spectrum of hybrid superconductor-quantum dot circuits. Surprisingly, our work also demonstrates the reciprocal, namely that adding a superconducting electrode to a double quantum dot circuit can be instrumental in building a strongly coupled mesoscopic cQED system.

As observed in previous experiments, the cavity transmission shows a resonance between the cavity mode and a circuit transition. We find that the behavior of our hybrid double dot is dominated by tunnel coupling between the left and the right dot, which results in a "charge qubit"-like transition. However the cavity transmission reveals the shaping of the Chapter 4. The superconducting double quantum dot 98 spectrum of the double quantum dot by the superconducting reservoir. This represents the first spectroscopic observation of Cooper pair assisted cotunnelling between the left and the right dot [START_REF] Sauret | Quantum master equations for the superconductorquantum dot entangler[END_REF][START_REF] Sauret | Spin-current noise and Bell inequalities in a realistic superconductor-quantum dot entangler[END_REF][START_REF] Chevallier | Current and noise correlations in a double-dot Cooper-pair beam splitter[END_REF], in equilibrium conditions. So far this effect had been observed only indirectly through out-of equilibrium transport measurements [START_REF] Tan | Cooper Pair Splitting by Means of Graphene Quantum Dots[END_REF][START_REF] Hofstetter | Finite-Bias Cooper Pair Splitting[END_REF].

We present a theoretical description of the new eigenenergies and eigenmodes of the dressed superconductor-charge qubit. The interplay between bare interdot tunneling and the superconducting proximity effect was not considered in previous derivations of the spectrum of superconductor-double quantum dots. Our theoretical results are validated by their ability to reproduce the very peculiar features of our measurements.

Importantly, the superconductor does not only modify the spectrum but also the electronphoton coupling. In addition to the usual coupling mechanism where cavity photons modulate the energy difference between the two dots, our theoretical model clearly highlights the possibility of coupling via a symmetric modulation of the dot energies.

In presence of a superconducting central electrode, such a common mode excitation effectively results in a strong sensitivity of the double dot tunnel barrier to the cavity electric field. This corresponds to the original driving mechanism proposed by DiVincenzo [START_REF] Loss | Quantum computation with quantum dots[END_REF][START_REF] Bertrand | Quantum Manipulation of Two-Electron Spin States in Isolated Double Quantum Dots[END_REF][START_REF] Reed | Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation[END_REF][START_REF] Martins | Noise Suppression Using Symmetric Exchange Gates in Spin Qubits[END_REF], which is implemented here for the first time in a cavity.

Our new coupling mechanism is more than a simple curiosity as it actually provides a way to reach the strong coupling regime between the cavity and our modified charge qubit. Such a regime was obtained recently using two different approaches [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF][START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF].

Mi et al. built their double quantum dot in a low charge noise material, namely SiGe.

Stockklauser et al. used a squid array resonator to boost the value of the electron-photon coupling. Here we demonstrate a third approach: the common mode coupling scheme allows us to decrease the charging energy of our device and correspondingly decrease the device sensitivity to charge noise, while keeping a sufficient coupling strength. It is worth noting that the strong electron-photon coupling was also obtained with the spin degree of freedom very recently [START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Mi | A coherent spinphoton interface in silicon[END_REF][START_REF] Landig | Coherent spinphoton coupling using a resonant exchange qubit[END_REF]. This chapter is organized as follows. Section 4.2 presents our experimental setup. Section

Sample and Measurement setup

We use carbon nanotube based double quantum dot circuits embedded in a high finesse superconducting microwave cavity [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Delbecq | Coupling a Quantum Dot, Fermionic Leads, and a Microwave Cavity on a Chip[END_REF][START_REF] Viennot | Charge and spin dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF]. The microwave cavity is a Nb coplanar From the transport measurement shown in figure 4.2-a we are able to assess the electrical contact between the double dot and a superconducting reservoir. We measure a superconducting gap ∆ of about 150µeV. The fact that we do not observe any measurable subgap currents indicate that our tunnel barriers are rather opaque. In the following, transport data will be omitted, as it would be blank at most working points (see appendix B.2). However, we will show that cavity signals can be very strong and contain signatures of tunneling processes between the dots and the superconducting lead. This is mainly because the charge qubit transition of a double dot is modified by the superconducting reservoir, as we explain now.

H DQD = L nL + R nR + t b (c † L↑ c R↑ + c † L↓ c R↓ ) + h.c. (4.1)
+ U m nL nR + 1 2 U L nL (n L -1) + 1 2 U R nR (n R -1)
where, for each dot i ∈ {L, R}, i denotes the orbital energy, U i the charging energy, c † iσ the electron creation operator with spin σ ∈ {↓, ↑}, and ni the electron number operator.

The mutual charging energy between the two dots is U m and t b is the direct hopping strength between the L and the R dot. Close to the (0,1)/(1,0) degeneracy line, the relevant eigenstates are the antibonding and bonding states:

|+ = u |L + v |R , |-= -v |L + u |R (4.2)
with the eigenenergies:

E ± = -U m + Σ ± 2 δ + 4t 2 b 2 (4.3)
Where

δ = L -R , Σ = L + R + U m , u = 1 2 -1 2 δ √ 2 δ +4t 2 b and v = 1 2 + 1 2 δ √ 2 δ +4t 2 b .
This gives a transition energy of the double quantum dot:

ω DQD = E + -E -= 2 δ + 4t 2 b (4.4)
We will see below that this quantity is deeply modified by the presence of the superconducting lead. We now outline the formal derivation of the spectrum of the double quantum dot in presence of a superconductor (see Appendix B.1 for details). We start by considering the hamiltonian of our double quantum dot-superconductor device:

H = H DQD + H S + H S-DQD , (4.5)
which naturally contains the double quantum dot hamiltonian H DQD discussed above.

Additionally, there is a term describing the quasiparticles in the superconductor:

H S = kσ E k γ † kσ γ kσ (4.6)
where γ † kσ (γ kσ ) are the creation (annihilation) Bogoliubov quasiparticle operators. Finally, H S-DQD accounts for electron tunneling between the superconductor and the two dots and can be written:

H S-DQD = kσ,i∈{L,R} t * i A i kσ c † jσ + h.c (4.7)
where t i ≡ √ Γ Si is the hopping strength between the superconductor and dot i and A i kσ is a linear combination of Bogoliubov operators (formula given in Appendix B.1). At second order in t i , the states are |+ and |-become coupled to the singlet and triplets states: |S , |T 0 , |T + , |T -, whose energies are, close to the degeneracy line:

E S = E T 0 = E T + = E T - L + R + U m ≡ Σ (4.8)
The low energy spectrum of the system can be obtained by a Schrieffer-Wolf transformation Ĥ = e -S Ĥe S corresponding to tracing out the superconducting quasiparticles.

Taking the same path as previous theoretical work on Cooper pair splitters [START_REF] Eldridge | Superconducting proximity effect in interacting double-dot systems[END_REF][START_REF] Simon E Nigg | Detecting nonlocal Cooper pair entanglement by optical Bell inequality violation[END_REF] we look for the appropriate S operator which eliminates H S-DQD to first order, resulting in an effective hamiltonian to second order in the tunnel couplings t i of the superconductor to the two dots. While this method is well known, it had so far always been applied

to the case of two completely decoupled dots, namely t b = 0. By including a finite hopping t b between the left and the right dot, our derivation yields novel results, which are crucial to interpret our experimental findings. Projecting the effective hamiltonian on the subspace {|+ , |-}, we get:

Ĥeff = (E + + δE + ) |+ + |+ (E -+ δE -) |--| + δt b |+ -| + δt b |-+| (4.9)
Below we focus close to the degeneracy line between (0,1)/(1,0) charge states but the (1,1)/(0,2) lines give rise to similar expressions. After tedious but straightforward calculations, the perturbative elements have the following expressions:

δt b = (Γ SR -Γ SL ) t b 2 δ + 4t 2 b 4Ln 2 ω D ∆ + π U m ∆ + U m 2 + Σ 2 + δ 2 + 4t b 2 ∆ 2 (4.10) -πt 0 eh Σ ∆ δ 2 δ + 4t 2 b 1 - 2 π U m ∆ δE -= -(Γ SR -Γ SL ) δ 2 δ + 4t 2 b Ln 2 ω D ∆ - π 2
U + ∆ + U 2 + + Σ 2 ∆ 2 (4.11) - π 2 Σ ∆ (Γ SR + Γ SL ) 1 + 1 π U + ∆ -πt 0 eh Σ ∆ t b 2 δ + 4t 2 b 1 + 2 |δr| πξ 0 U + ∆ δE + = (Γ SR -Γ SL ) δ 2 δ + 4t 2 b Ln 2 ω D ∆ - π 2 U - ∆ + U 2 -+ 2 Σ ∆ 2 (4.12) - π 2 Σ ∆ (Γ SR + Γ SL ) 1 - 1 π U - ∆ + πt 0 eh Σ ∆ t b 2 δ + 4t 2 b 1 - 2 |δr| πξ 0 U - ∆
where t 0 eh is the Cooper pair splitting amplitude (see Eq. B.9 in Appendix B.1), δr is the distance between tunnel contacts from the superconductor to each dot, ξ 0 is the superconducting coherence length and ω D is the Debye frequency, used as a cut-off.

The following notation was introduced:

U ± = ±U m + 2 δ + 4t 2 b (4.13)
The new eigenenergies are:

Ẽ± = E + + δE + +E -+ δE -± (E + + δE + -E --δE -) 2 + δt 2 b 2 (4.14)
which leads to a new transition energy of the form:

ω S-DQD = ( 2 δ + 4t 2 b )Z ( δ , Σ ) 2 + t ( δ , Σ ) 2 , (4.15)
with

Z ( δ , Σ ) = 1 + δE + -δE - E + -E - (4.16)
and:

t ( δ , Σ ) = δt b (4.17)
It is important to notice here the major modification induced by the superconductor:

the transition energy of the circuit does now also depend on the sum of the two dot energies Σ , rather than only on their difference δ .

Before concluding this section, we would like to point out that the calculation is essentially the same if there is an additional quantum number (e.g. a valley quantum Chapter 4. The superconducting double quantum dot 105 number) ruling the states of the double quantum dot. This naturally leads to a second transition similar to the one considered above but with different parameters. This can account for the two transition structure which is used to understand quantitatively our experimental findings.

Electron-photon coupling of the hybrid DQD

We now evaluate the effect of the superconductor on the coupling to the cavity. The conventional coupling mechanism of a double quantum dot to the cavity in the absence of the superconductor takes the form (cf: formula 2.6):

+ (g L nL + g R nR ) â + â † -= uv (g L -g R ) â + â † (4.18)
where â is the annihilation operator of the photonic cavity mode.

This leads to the usual coupling mechanism of a double quantum dot to a cavity mode which vanishes in case of a symmetric coupling (g L = g R ), ie L and R orbitals couple equally to the cavity electric field. However, the superconductor gives rise to a new coupling mechanism which works also in case of symmetric coupling. In order to evaluate it, one can still rely on the Schrieffer-Wolf transformation by including formally the operator â + â † into the derivation and expanding in powers of g L + g R (since one

assumes that g L -g R g L + g R ).
To first order, this adds a term of the form:

2 ∂δt b ∂ Σ (g L + g R ) â + â † (4.19)
It is important to note that although the above expression is in principle perturbative, it can in fact be much larger than the usual coupling term since in case of symmetric or nearly symmetric coupling (g L -g R g L + g R ). One can note that this "common mode " coupling mechanism is in fact not restricted to our situation but would hold for any double quantum dot with a tunable barrier [START_REF] Loss | Quantum computation with quantum dots[END_REF]. In our case, since the energy scale ruling the barrier tunability is the superconducting gap which is smaller than the semiconducting gaps of usual semiconducting materials, our tunability is very efficient.

Finally, the effective Hamiltonian of our device in cavity, projected on the |+ , |-states Chapter 4. The superconducting double quantum dot 106 reads:

Ĥeff,c = Ĥeff + ω cav â † â + ĤBaths + g(|+ -| + |-+|)(â + â † ) (4.20)
With the following coupling strength:

g ≈(g L -g R ) 2t b 2 δ + 4t 2 b (4.21) + (g L + g R ) 4t b 2 δ + 4t 2 b Γ SR -Γ SL ∆ Σ ∆ -(g L + g R )π t 0 eh ∆ 2 2 δ + 4t 2 b
The above equation contains three different terms: the first is the usual coupling term between a double quantum dot bonding/anti-bonding transition and a microwave cavity which needs to have asymmetric g L and g R . The second term corresponds to the fact that the superconductor renormalizes the energy levels of each dot with a strength proportional to each tunnel coupling Γ SL(R) . It corresponds to an indirect Cooper pair assisted tunneling modulation between the two dots. The last term arises from the direct modulation of the Cooper pair assisted tunneling between the two dots by the cavity photons. The two last terms only exist in the presence of a superconductor. In the next section, we will illustrate with experimental values that the "common mode " coupling mechanism can yield a sizable electron-photon coupling strength.

Resonant interaction between the hybrid DQD and the cavity

The interaction between our hybrid double quantum dot and the cavity photons is conveniently probed by measuring the microwave signal transmitted through the cavity. and the right(R) dot, but is distorted in the perpendicular direction. This means that the transition frequency of our circuit does not only depend on δ , but also on the average energy of the two dots Σ = L + R + U m . Qualitatively, this agrees with the theoretical expression for the transition frequency which we derived in section 4.3 from a microscopic theory of our hybrid superconductor-double quantum dot:

ω S-DQD = ( 2 δ + 4t 2 b )Z ( δ , Σ ) 2 + t ( δ , Σ ) 2 (4.22)
The crescent shape of the transition line can be recast from the dependence of the functions Z and t on δ and Σ (see formulas 4.16 and 4.17). As shown in figure 4.4a, the transition frequency map expected from the theory as a function of δ and Σ , displayed in light brown is cut by the blue plane at the cavity frequency. This results naturally in a crescent shaped transition frequency contour line.

All these experimental signatures can be captured more quantitatively by an inputoutput formalism of the coupled equations of the cavity field in the semiclassical limit and the electronic degrees of freedom [START_REF] Cottet | Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena[END_REF][START_REF] Bruhat | Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua[END_REF]. Specifically, the transmission t through the cavity reads:

t = √ κ L κ R ω -ω cav -iκ/2 + g 2 χ dot-dot (ω) + g c 2 χ dot-lead (ω) (4.23)
where κ L (resp. κ R ) is the photon loss rates through the left (resp. right) coupling ports of the resonator and κ is the total cavity photon loss rate. The bare cavity transition is modified by χ dot-dot (ω) (the charge susceptibility associated to internal transitions of the circuit), and χ dot-lead (ω) (the charge susceptibility associated to electronic transitions between the dot and the leads). In our case, these susceptibilities read:

χ dot-dot (ω) = 1 ω -ω S-DQD -iΓ/2 (4.24) χ dot-lead (ω) = 1 1 -iω/γ 4k B T cosh E N -E N +1 2k B T
where Γ is the decoherence rate of the internal transition of the DQD and E N is energy of the DQD for N charges in total. The χ dot-dot (ω) susceptibility can give the strong electron-photon coupling if g > Γ, κ. This susceptibility yields a resonance in the trans-

U L /h = 32 GHz U R /h = 42 GHz U m /h = 24 GHz
The quantitative agreement with the ω cav = ω S-DQD (internal) transition lines validates the low energy spectrum of our device and is the first direct observation of Cooper pair assisted tunneling between two quantum dots.

Noticeably, the renormalization of the hopping between the two dots is related to the Cooper pair splitting amplitude t 0 eh (see section 4.4). Therefore, we can extract a value for t 0 eh /h = 400 MHz even without direct observation of Cooper pair splitting. Our measurement demonstrates that cavity photons can be used to probe very small energy scales, inaccessible to transport, related to superconducting proximity effect in quantum dots. Such a scheme could be generalized to superconducting hybrid structures with topological properties.

Additionally, it is important to notice that the dependence of t ( δ , Σ ) upon Σ yields a new light-matter coupling term for our device as shown in the previous section. This follows the original Loss and DiVincenzo proposal [START_REF] Loss | Quantum computation with quantum dots[END_REF][START_REF] Bertrand | Quantum Manipulation of Two-Electron Spin States in Isolated Double Quantum Dots[END_REF][START_REF] Reed | Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation[END_REF][START_REF] Martins | Noise Suppression Using Symmetric Exchange Gates in Spin Qubits[END_REF] and recent cavity-double quantum dot coupling proposals [START_REF] Friesen | A decoherence-free subspace in a charge quadrupole qubit[END_REF][START_REF] Srinivasa | Entangling distant resonant exchange qubits via circuit quantum electrodynamics[END_REF][START_REF] Russ | Long distance coupling of resonant exchange qubits[END_REF]. Our work provides the first example of this common mode coupling to a microwave cavity. Indeed, in the Bloch sphere representation of figure 4.4-b, the north and south poles are more along the detuning axis δ and the light-matter coupling indicated by a red arrow is mainly along the tunnel coupling axis, in stark contrast with the usual case for double quantum dots [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF][START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF][START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF],

where it is along δ . The electron-photon coupling strength is controlled by the sum g L + g R which can easily be of the order of 2π× 100 MHz, as shown for example in reference [START_REF] Bruhat | Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua[END_REF], which is a large magnitude. Using expression 4.21 of section 4.4 with the circuit parameters of sample Aa given above, we get along the crescent contour a g between 2π× 1 MHz ( δ ≈ 0, Σ ≈ ∆/2) and 2π× 4.7 MHz ( δ ≈ 2t b , Σ ≈ 0). This shows that the "common mode " coupling mechanisms can yield a sizable electronphoton coupling strength, even if it originates from second order tunneling through the superconductor.

Finally, we briefly comment on an interesting feature of our light-matter interface which appears on figure 4.3-d. The amplitude map displays a "bright" spot, in blue, corresponding to photon gain (of about 1.3). Microwave photon emission from quantum dot circuits was recently investigated in double quantum dots [START_REF] Stockklauser | Microwave Emission from Hybridized States in a Semiconductor Charge Qubit[END_REF][START_REF] Stehlik | Double Quantum Dot Floquet Gain Medium[END_REF][START_REF] Liu | Semiconductor double quantum dot micromaser[END_REF][START_REF] Liu | Injection locking of a semiconductor double-quantum-dot micromaser[END_REF] and hybrid superconductor single quantum dot [START_REF] Bruhat | Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua[END_REF]. It is interesting to see that it also appears in our hybrid double quantum dot circuit, although we did not study this effect quantitatively.

Qualitatively, it is consistent with having a coherent interface. In the next session, we show an even more striking consequence of the high cooperativity of our device, namely the vacuum Rabi splitting of the cavity when it is brought into resonance with the DQD transition properly tuned. In the previous section, we have focused on characterizing our hybrid double dot circuit, using the microwave resonator as a spectroscopic probe. However, looking at the large The fact that we observe this splitting implies that the coupling strength g between the circuit transition and the cavity is larger than half the linewidth of the cavity κ/2

and half the linewidth of the double quantum dot transition involved Γ/2. The intrinsic linewidth of the cavity can be directly measured from the transmission spectrum when the double dot is detuned and is κ/2π = 0.5 MHz. Therefore, the linewidth of the double dot transition can be inferred from the linewidth of each peak observed in figure 4.5-a, of about 3 MHz. For the simplest case of a single transition [START_REF] Chiorescu | Coherent dynamics of a flux qubit coupled to a harmonic oscillator[END_REF], the linewidth is equal to (Γ/2π + κ/2π)/2. This would lead to Γ/2π ∼ 5.5 MHz. In order to account quantitatively for the observed transmission spectrum (and in particular for the very low transmission maximum), we can rely on a two transition scheme (one very coherent, one less coherent). Using a modeling based on two independent transitions depicted in figure 4.5-b, we are able to fit the data using a fully quantum numerical code (QuTip, see Appendix B.3 for details). The use of two transitions anticipates on the existence of a K/K' valley degree of freedom commonly observed in nanotubes. We use the following parameters g K =2π×4. As expected for a few level system, we are able to saturate the transitions and to recover the bare transmission of the cavity by injecting a large number of photons inside the cavity. In the present case, this saturation occurs for n ≈ 100. As shown in figure 4.5-c, there is a continuous evolution from the vacuum Rabi splitting to a single off-centered lorentzian peak from n ≈ 0.1 to n ≈ 300. Such a peculiar saturation is well reproduced by the QuTip numerical simulation and arises from the two transition structure mentioned above. Note that the splitting at low power is slightly smaller (about 6 MHz) in figure 4.5-c than in figure 4.5-a because this measurement was done for a different working point of our device.

Reaching the strong coupling regime with an excitation, which is primarily charge-like is non-trivial and has been the main challenge of the mesoscopic cQED community for years until recently. The main limitation of all the charge qubit like setups in cavity is the linewidth of the double dot transition which is typically reported to be at least in the few 100 MHz range [START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF]. One important decoherence source explaining such a large linewidth is the background charge noise. One can think of several strategies to overcome this difficulty. One possible path is to reduce the charge noise in the material. Recently, this idea was successfully implemented by Mi et al in a SiGe based two-dimensional electron gas double quantum dot [START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF], with a linewidth Γ/2π =2.6 MHz lower than the coupling g/2π=13. [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF] MHz. An alternative strategy to reach the strong coupling regime despite the presence of large charge noise was demonstrated by Stockklauser et al. [START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF]: by using a squid array resonator, the electron-photon coupling was increased to g/2π = 238 MHz, which exceeds the DQD charge qubit linewidth Γ/2π = 93 MHz. Our ability to reach the strong coupling regime relies on a third approach, namely reducing our qubit linewidth by reducing the device sensitivity to charge noise. In this scheme, our specific common mode coupling mechanism plays a crucial role to keep a sufficient coupling strength. Now we successively detail each of these two key ingredients.

In presence of the noise spectral density S(f ) = δn 2 /f , where f is the frequency, the dephasing rate Γ ϕ of the double dot can be expressed as:

Γ ϕ ≈ ∂ω ∂ E C δn 2 + 1 2 ∂ 2 ω ∂ 2 E C 2 δn 2 + . . . (4.25)
which is strongly influenced by the charging energy E C ∼ e 2 /C Σ , where C Σ is the total capacitance of the device [START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF][START_REF] Koch | Chargeinsensitive qubit design derived from the Cooper pair box[END_REF][START_REF] Cottet | Implementation of a combined charge-phase quantum bit in a superconducting circuit[END_REF]. The expression of Γ ϕ takes the above simple form only if the transition frequency ω depends on a single parameter . This is the case for the standard double quantum dot charge qubit transition ω DQD , which dispersion relation is governed by δ . In our case, the expression of Γ ϕ is more complex since it involves all the derivatives of the transition ω S-DQD , with respect to its control parameters δ and Σ . Nevertheless, its dependence as a function of the total charging energy and the transition frequency derivatives remains qualitatively the same. The points where all the first order derivatives vanish are called sweet spots [START_REF] Friesen | A decoherence-free subspace in a charge quadrupole qubit[END_REF][START_REF] Srinivasa | Entangling distant resonant exchange qubits via circuit quantum electrodynamics[END_REF][START_REF] Russ | Long distance coupling of resonant exchange qubits[END_REF] because the double dot is insensitive at first order to charge noise. The usual method to reduce Γ ϕ is therefore to operate the system close to a sweet spot which implies that only the second order terms are present in the expression of Γ ϕ . To reduce the second order term at constant noise density and without engineering the dispersion relation [START_REF] Russ | Coupling of three-spin qubits to their electric environment[END_REF], it is a priori very efficient to go towards small charging energy, in analogy with the transmon qubit Importantly, this reduction of E C also implies a decrease of the lever arm between the orbital energies L , R of the dots and the cavity potentials. The coupling of photons through the variable δ used in former experiments [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF] thus becomes too small to be exploited. However, our hopping t ( δ , Σ ) is tunable with the parameter Σ , which is naturally more strongly coupled to the cavity potential than δ . This compensates the decrease of E C and gives us a large charge-photon coupling strength of about 2π×10

MHz which allows us to reach the strong coupling regime.

Conclusion

We have presented the first experimental study of a hybrid superconductor-double quantum dot in a microwave cavity. The resonant interaction between cavity photons and a charge qubit-like transition of our circuit reveals a peculiar dispersion map as a function of the dot gate voltages. We are able to interpret our data by theoretically deriving the device energy level structure, which is dressed by cotunnelling processes between the left and the right dot induced by the superconductor. This is the first direct observation of Cooper pair assisted cotunneling in a double quantum dot. Due to its relation to the Cooper pair splitting, we are able to extract a value for t 0 eh /h ∼ 400MHz, which is inaccessible to transport measurements. A natural perspective of our work is to use the theoretical and experimental tools developed here to study the same type of device with a more transparent superconducting contact, i.e. with larger t 0 eh . In principle, such a regime would allow to study the physics of Cooper pair splitting more directly [START_REF] Cottet | Probing coherent Cooper pair splitting with cavity photons[END_REF][START_REF] Cottet | Subradiant Split Cooper Pairs[END_REF].

The same methods could also be instrumental to the study of Majorana bound states through microwave cavities [START_REF] Cottet | Squeezing light with Majorana fermions[END_REF][START_REF] Dartiailh | Direct Cavity Detection of Majorana Pairs[END_REF]. Importantly, our novel qubit design demonstrates a new way of reaching the strong electron-photon coupling based on a tunable hopping barrier and a low charging energy. These ingredients are very generic and could be used in many other setups [START_REF] Friesen | A decoherence-free subspace in a charge quadrupole qubit[END_REF][START_REF] Srinivasa | Entangling distant resonant exchange qubits via circuit quantum electrodynamics[END_REF][START_REF] Russ | Long distance coupling of resonant exchange qubits[END_REF].

In our case, the tunable hopping is due to the use of a hybrid superconductor double quantum dot setup, thanks to superconductor induced cotunneling processes. However, using local gates, one could also engineer a direct electrostatic control over the hopping strength. Our findings open the path for entanglement of individual electron states [START_REF] Burkard | Ultra-long-distance interaction between spin qubits[END_REF] and teleportation of electronic entanglement over macroscopic distances.

Chapter 5

Carbon nanotube platform for spin qubit

We use a circuit QED spin-photon interface to drive a single electronic spin in a carbon nanotube based double quantum dot using cavity photons. The microwave spectroscopy allows us to identify an electrically controlled spin transition with a decay rate which can be tuned to be as low as 250kHz. By performing time domain manipulations via pulses in the cavity field, we demonstrate a Rabi decay time of about 2µs. We also extract both the spectral linewidth and spin-photon coupling attributed to this transition, allowing us to establish a decoherence model and identify an optimal operating point. These coherence properties, which are attributed to the use of pristine carbon nanotubes stapled inside the cavity, should enable coherent spin-spin interaction via cavity photons and compare favorably to the ones recently demonstrated in Si-based circuit QED experiments.

Highly coherent spin transition

Introduction

Spins confined in quantum dots are considered as a promising platform for quantum information processing [START_REF] Loss | Quantum computation with quantum dots[END_REF]. While many advanced quantum operations have been demonstrated, experimental as well as theoretical efforts are now focusing on the development of scalable spin quantum bit architectures. One particularly promising method relies on the coupling of single spins to microwave cavity photons [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. This would enable the coupling of distant spins via the exchange of virtual photons [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF][START_REF] Trif | Spin dynamics in InAs nanowire quantum dots coupled to a transmission line[END_REF][START_REF] Hu | Strong coupling of a spin qubit to a superconducting stripline cavity[END_REF][START_REF] Harvey | Coupling two spin qubits with a high-impedance resonator[END_REF] for two qubit gate applications, which still remains to be demonstrated with spin qubits.

Very recently, the observation of strong coupling between the charge or the spin confined in a quantum dot circuit and cavity photons has been reported [12-14, 53, 55, 151], bringing closer the demonstration of distant spin-spin interaction. One critical parameter of a spin-photon platform is the linewidth of the spin transition which sets the maximum number of coherent swap operations between a spin and a photon. Whereas they are well documented in Si or GaAs, the coherence properties of single electronic spins in carbon nanotubes are still debated [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF][START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF].

Here, we use a spin qubit scheme based on a carbon nanotube embedded in a microwave cavity. Our device is made using a stapling technique developed for cQED architectures, it aligns along the right spin quantization axis. Since the two dots are separated by few hundreds of nm, there is a large (mesoscopic) electric dipole between the left and the right dots which is given to the spin thanks to the non-collinear magnetizations.

The photons of the cavity convey an electrical field which couples to this electric dipole and therefore to the spin. An alternative wording is to state that the ferromagnetic electrodes give rise to a two-site artificial spin orbit coupling, which makes the spin sensitive to the cavity electric field [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. It is interesting to note that such an "orbitally" mediated spin-photon coupling allows one to increase the natural spin-photon coupling by about 5 orders of magnitude [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Mi | A coherent spinphoton interface in silicon[END_REF] without degrading substantially the inherent good coherence properties of a single spin if the device is used in the limit where the electron is trapped almost completely in one of the two dots (left or right). This can be achieved by detuning the left gate with voltage V L from the right gate with voltage V R , as sketched in figure 5.1-b. At this working point, the charge noise sensitivity of the spin-qubit is low because of the energy dispersion but keeps a large coupling to the cavity field [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF][START_REF] Benito | Input-output theory for spin-photon coupling in Si double quantum dots[END_REF].

Nano-fabrication

Our devices are made with a complete dry transfer nanotube technique adapted from previous works [START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF][START_REF] Viennot | Stamping single wall nanotubes for circuit quantum electrodynamics[END_REF][START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF][START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF][START_REF] Ranjan | Clean carbon nanotubes coupled to superconducting impedance-matching circuits[END_REF] which allows us to integrate as-grown carbon nanotubes in a microwave cavity. The full chip, comprising the cavity, the bottom gates and the non-collinear ferromagnetic contacts shown in figures 5.2-a, 5.2-b, and 5.2-c respectively, is placed in a vacuum chamber with a base pressure of 5 × 10 -7 mbar. The zig-zag contacts visible in figure 5.2-b and partially in figure 5.2-c are NiPd ferromagnetic contacts with transverse magnetization [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. Carbon nanotubes are grown on a Si comb with a standard Chemical Vapor Deposition (CVD) recipe with CH 4 as feedstock gas.

The comb is mounted inside the chamber on a stage with micro-and nano-manipulators which allow us to place the nanotube on the chip with controlled approach steps of about 100nm. The assembly of the carbon nanotube and the ferromagnetic contacts is done under vacuum in order to ensure a clean interface. This results in the device shown in figure 5.2-c where a nanotube bridges the two ferromagnetic contacts and is a priori suspended over bottom gates. The wider gate visible in the SEM picture of figure

Stability diagram

The devices obtained with our fabrication technique are more tunable than previous nanotube based spin quantum bits and much less disordered [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. The measurement setup is similar to the setup of ref. [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. We measure simultaneously the DC current I flowing through the device and the microwave signal transmitted through the cavity in

Two-tone spectroscopy

In the double quantum dot regime, the tunnel coupling between the two dots and the non-collinear ferromagnetic contacts generate a spin dependent hybridization between the left and the right dot which allows to implement a spin photon coupling [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF].

This implies that the phase can a priori read-out the spin state of a trapped electron.

The measured phase is determined by an average of the dispersive shifts induced by each transition, weighted by the steady-state occupation of each state (see figure 5.3-b).

Applying a second tone allows to individually address the different transitions, and to recover their respective coupling strength to cavity photons, as explained in the next paragraph.

The microwave spectroscopy of our ferromagnetic spin qubit is conveniently done by reading out in the dispersive regime the phase φ of the cavity signal when a second tone is applied through the cavity and its frequency is swept, which is mainly sensitive to the expectation value of the spin projection on the Z axis of the left (right) dot, σz , at large detuning δ . In the dispersive regime, at large detuning δ , the expression of the The saturation value is given by the effective Zeeman field felt by the (pure) spin state at large detuning. The fact that we observe several spin transitions is attributed to the K/K' valley degeneracy of the nanotube as well as from the fact that we are not in the single electron regime. As expected for a spin transition, we can tune the value of this saturation with the external magnetic field. The resulting linear dependence is shown in figure 5.4-b. The low slope is consistent previous measurements in a similar architecture with non-stapled nanotube material [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF].

A cut along the lowest resonance at large detuning is shown in figure 5.4-d. This measurement fitted by a lorentzian has a full width at half maximum of 2γ S = 2π×(498±80)kHz which corresponds to a decay rate about 2π × 249kHz. Such a narrow linewidth is two orders of magnitude lower than that found in the valley-spin qubit in previous work with carbon nanotubes [START_REF] Pei | A valleyspin qubit in a carbon nanotube[END_REF] and compares favorably to the very recent figures of merit reported for Si based platforms [START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Mi | A coherent spinphoton interface in silicon[END_REF]. From the phase contrast of about 4 o , and assuming that the transition is fully saturated (∆ σz = 1 )we can estimate a lower bound of the spin photon coupling strength g S ≈ 2π × 2.0M Hz, which exceeds the decoherence rate of the spin states and of the cavity. This implies that the spin is strongly coupled to the cavity photons although they are not resonant. As a comparison, we have g S /γ S ≥ 8

for the spin transition whereas g C /γ C ≈ 0.015 for the charge like transition. There is therefore a very large gain in the coherence properties of our device when we switch from the charge like transitions to the spin transitions.

Time-domain measurement

We further substantiate our microwave spectroscopy measurements by time domain measurements. For that purpose, we use a similar pulse sequence than the one shown in ). The Rabi frequency Ω R = 2π × 140kHz found is low due to the limited coupling strength of the spin to the photons and to the fact that we drive the spin through cavity photons 1 .

Cavity-Bloch equations

To account for the measured cavity field quadratures I and Q shown in figure 5.5, we used the cavity-Bloch equations (given in ref. [START_REF] Bianchetti | Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics[END_REF]). The dynamics of the coupled qubit-cavity system is described by the master equation:

d dt P = M P + E source (5.2)
where M is the matrix containing the prefactors of the coupled equations (those prefactor can be found in ref. [START_REF] Bianchetti | Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics[END_REF]), E source is the vector containing source terms such as the externally applied microwave field, and P is the population vector and reads:

P = â σz σx σy âσ z âσ x âσ y â † â (5.3)
To account for the nonlinear terms in the numerical solving, the matrix M contains populations such as â † â obtained at the previous iteration. We then simulate the full pulse sequence used in the Rabi measurement including the drive of the qubit (with the envelop Ω(t)) during a time τ , the drive of the cavity (with the envelop m (t)) during 1.5µs, and the time window over which the cavity output signal is acquired, which last 1µs. The fit procedure is done on the two quantities I = Re â and Q = Im â , and the result is superimposed in red in figure 5.5.

1 In such a situation, the Rabi frequency reads: ΩR = g S ∆ d , where d is the qubit pulse amplitude (see supplemental material of ref. [START_REF] Murch | Cavity-Assisted Quantum Bath Engineering[END_REF]). Therefore, the prefactor g S ∆ limits the Rabi frequency ΩR. In the measurement presented in figure 5.5, we have g S ∆ ≈ 0.025. In figure 5.6 we show the σ z evolution, for the pulse sequence with the longest qubitpulse: τ pulse = 8µs (corresponding to the last point of the Rabi measurement). Importantly one can see that σ z saturates at -0.5, instead of 0 as one would expect for a strong qubit-drive with respect to the relaxation rate of the qubit. Such a saturation can also be explained by a finite detuning between the qubit frequency and the qubit drive frequency. From the saturation value of ∆ σ z = 0.5, one can deduce the spin-photon coupling strength :

g S = ∆ qc κ∆φ 2∆ σ z 3M Hz (5.4)

Decoherence model

In order to specify the decoherence mechanism explaining the linewidth found for our spin transition, we have measured the dependence of the decoherence rate as a function of the detuning between the left and right dots. Such a measurement is displayed in which yields γ S ≈ 2π × 200kHz if A = 0.1µeV . Our measurements are therefore fully in agreement with the tabulated values for the hyperfine coupling expected in CNTs of A ≈ 0.1 -0.5µeV contrarily to previous work [START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF][START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]. In addition to the decoherence rate, we also present the spin-photon coupling strength and the cooperativity of the spin-photon interface C = 2g 2 S κγ S as a function of the detuning (figure 5.7-b and 5.7-c respectively). Interestingly this last quantity allows to identify an optimal detuning working point around δ = -18GHz. For this detuning point, the hybridization with the charge creates a sizeable spin-photon coupling while maintaining a low decoherence rate. Our results suggests that carbon, like silicon, can be a promising host for electronic spins encoding quantum information. This is enabled by our clean and controlled nanoassembly technique of carbon nanotubes in cavity.

In summary, we have demonstrated that carbon nanotube based double quantum dots can provide a tunable and coherent spin-photon interface. The figures of merit of coupling strength of g S = 2π × 2.0M Hz and low decoherence rate γ S = 2π × 250kHz are suitable for future swap experiments.

Design and characterization of the spin-qubit device

Design of Ferromagnetic contacts

The design of the ferromagnetic contacts is determinant to the spin-charge hybridization. It has been shown that the geometry of the magnetic domains of PdNi nanostrip is fixed by the anisotropy of the nanostructure and results from the so-called inverse magnetostriction effect [START_REF] Chauleau | Magnetization textures in NiPd nanostructures[END_REF]. In the case of PdNi electrode with thickness about 30nm, and width below 500nm the magnetization is perpendicular to the electrode direction (see figure 5.8). Therefore, one can control the magnetization axis by changing the electrode orientation. In our architecture, suspended carbon nanotubes requires high enough contacts electrodes (with a typical height of 200nm, see section 3.2.7 of the chapter 3), which is incompatible with the appearance of perpendicular magnetic domain in the ferromagnetic contact. Thus, a first thick metallic layer is used to elevate the PdNi magnetic texture without affecting to much the inverse magnetostriction effect. The emergence of constraints in the PdNi is due to a mismatch between thermal expansion coeficient of PdNi and the thick layer metal. Indeed, two main cool-down occurs during the fabrication: first, when the sample is cooled down at room temperature after the metal evaporation, and then at the cryogenic cool-down (expected to reinforce the effect [START_REF] Chauleau | Magnetization textures in NiPd nanostructures[END_REF]). In the device presented in this chapter, we use Titanium as the thick layer metal, which is the metal maximizing the thermal expansion coefficient mismatch among the metals directly accessible in the clean room facility (thermal expansion coefficient for Ti and PdNi: α T i = 8.6 × 10 -6 and α P dN i 12.5 × 10 -6 ). The magnetic domains obtained with such a metallic stack are shown in figure 5.8-a.

The angle between the two ferromagnetic contacts is a key parameter in the spin-photon interface. Indeed, the hamiltonian of a double quantum dot with only one orbital in each dots, and two Zeeman fields of amplitude E z , with an axis misalignment defined by an angle θ reads [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF]:

Ĥ = - δ 2 τ3 σ0 + tτ 1 σ0 -E z σ3 τ0 + τ3 2 + cos(θ)σ 3 + sin(θ)σ 1 τ0 -τ3 2 (5.5)
Where σi with i ∈ [0, 3] are the Pauli matrices acting on the spin degree of freedom ( σ0 is the identity), and τi with i ∈ [0, 3] are the Pauli matrices acting on the orbital degree of freedom. The spin-charge hybridization arises from the term sin(θ)σ 1 τ0 -τ 3 2 , which is the strongest for θ = π/2. Although a large spin-charge hybridization provides a large coupling to the cavity photons, it also strongly affects the sensitivity of the 0-1 transition (see figure 5 Concerning the spin-qubit experiment (chapter 5), it would be very instructive to probe the full tunability of the ferromagnetic spin-qubit by changing in-situ the two local effective Zeeman fields using gate voltages on the side of the double quantum dot. This features is specific to this spin-qubit design, hence, it would be very instructive to demonstrate them. In particular, one could identify the mechanism responsible for the polarization of the spectrum: transmission or reflection in the spin valves. Another direction of research would be to couple two ferromagnetic spin-qubits through exchange of virtual photons [START_REF] Majer | Coupling superconducting qubits via a cavity bus[END_REF] or direct energy exchange [START_REF] Sillanpää | Coherent quantum state storage and transfer between two phase qubits via a resonant cavity[END_REF].

Further characterization of the circuit

Finally, both of these experiments could benefit from a better cavity-carbon nanotube interface by increasing the charge-photon coupling strength. One could use an higher impedance resonator, either by changing the cavity material (for instance: NbTi) or the cavity design (lumped-element resonator, Josephson junction array, ...), or combine the two approaches. Simulating the cavity field inside the quantum dot circuit geometry using softwares such as HFSS (High Frequency Simulation Software) could also be very instructive to estimate the charge-photon coupling strength in a double quantum dot, and therefore to tune the cavity field gradient over the circuit in order to increase the charge-photon coupling strength.

• open the bottle of Argon, Methane, and Dihydrogene, and note the pressures before and after the pressure regulator in the notebook. After the regulator, all pressure should be set at 1 bar.

• Launch the monitoring program (control f our2), and note the three measured flows (which correspond to the offset value of the flowmeters). • When the temperature displayed by the oven is below 80 • C (this is the temperature inside the oven close to the resistors and does not correspond to the real temperature of the chip.), close all the valves, and open the oven.

• Do not forget to close all the bottles.

where t i ≡ √ Γ Si and:

A j kσ = kσ,j∈{L,R} e ik.r j u k γ kσ + σe -ik.r j v -k γ † σ (B.4) with u k and v k as conventionally defined in the BCS theory.

The main step of the derivation is to write an effective hamiltonian to second order in the tunnel couplings t i of the superconductor to the two dots using a Schrieffer-Wolf transformation, also called adiabatic elimination in atomic physics [START_REF] Recher | Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons[END_REF][START_REF] Simon E Nigg | Detecting nonlocal Cooper pair entanglement by optical Bell inequality violation[END_REF]. The idea is to find a unitary transformation which cancels the tunneling term up to second order in tunneling. The effective hamiltonian reads:

Ĥeff = e -S Ĥe S ≈ Ĥ -S, Ĥ + 1 2 S, S, Ĥ + . . . The constraints for the operator S can be fulfilled if X j kσ has the following matrix elements: where ω D is the Debye angular frequency, used as a cut-off and ∆ is the superconducting gap. It is interesting to note that √ Γ SL Γ SR cosk F δr e -δr/ξ 0 is the Cooper pair splitting amplitude which appears both here in the renormalization of the bonding/anti-bonding states as well as in the hybridization between the |S and the |0, 0 . We note:

k |u k | 2 ( 1 E k -E + + 1 E k -E - ){-uv(|t L | 2 -|t R | 2 ) + t * L t R e ik.
t 0 eh = Γ SL Γ SR cosk F δr e -δr/ξ 0 . (B.9)

Criteria to choose a CNT The SEM is definitely not the best tool to characterize carbon nanotubes (single or multi-walled, bundle or not, diameter, chirality), but it allows at least to know their position and whether or not they are huge bundles. Here are a few criteria to identify CNTs good for stapling:

• The CNT has to be in the first 10µm starting from the tips of the cantilevers.

• It has to be well separated from other suspended CNTs (more than 5µm, see figure

C.2.c).

• It should looks like a single-walled carbon nanotube, or at least not a huge bundle of CNTs. This means that it should be thin, and there should be no visible braiding Tuning of the cantilever chip angle: The idea is to have a small angle between the cantilevers and the circuit chip in order to know which side of the cantilever chip will touch the circuit chip first. Thus, you have to choose if you want the cantilever 48 to touch first or the cantilever 1, depending on where are the good looking CNTs.

• Rotate back the piezo-motor block so that the cantilevers are pointing down (in the stapling position). Normally, the cantilever 1 should be on the side of the turbomolecular pump, and the cantilever 48 should on the side of the rotary feedthrough (to which the sample holder is attached).

• Screw the piezo-motor block, but not too tight so that it is still possible to rotate it.

• With the micromanipulator place the cantilever chip over a reflecting surface, for instance you can use the cover of the sample holder.

• To estimate the angle α (see figure C.3.b) of the cantilever chip, first one has to do the focus on the tips of the cantilever 1 for instance, and then check if the cantilever 48 is above or below the focus plan of cantilever 1. Then adjust the angle α manually, so that the desired cantilever (1 or 48) touch first. A difference of 50 graduations between the focus on the cantilever 1 and the cantilever 48 is sufficient.

• Set the detection setup (see figure C.5), with a bias voltage V app = 500mV . • Lower the cantilever chip first using the "continuous" option (see figure C.6) as long as the cantilever chip is far enough from the circuit (more than 500 graduations on the focus between the circuit and the cantilevers), and then for the fine approach use the "step" option. You know when a cantilever is touching the bottom of the trench, when it start to elongate from our top view (due to its deflection). • Depending on the contacts width, the detection signal can be a current spike as low as 100pA simultaneous to a step of the piezo motors. Thus it is important to look constantly at the current measurement during the approach stage.

• Once the contact with a CNT has been detected, the next step is to cut sequentially it on the two external sections sequentially. Apply the bias voltage on one of the external cutting contact, put the neighboring contact to the ground, and all the

The risk here is to accidentally cut the CNT. The rule of thumb is to not exceed the cutting current of the two external sections. Measure again the resistance of the central portion at V SD = 10mV to quantify the improvement of the contact resistance using the setup shown in figure C.9.

• One can measure the gate dependence of the Source-Drain current (sweepVg ... C. [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF] Tranfering the circuit to the cryostat

• Put 10M Ω resistors on the DC box of the cryostat on the lines you will use as the source and drain contacts. This is a safety to limit possible transient currents during the connection of the stapled CNT to the cryostat DC lines.

• Put the switches on the DC box in the connected position (acting on the switches while the CNT is in ambient air may burn the CNT due to the believed high transient currents at the switching moment).

• Check the DC setup of the cryostat by testing the DC lines you will use with a resistor box.

• Check that the voltage source you will use is set to 0 Volt, and OUTPUT ON.

• Take the ground of the cryostat using the cable with 10M Ω inside (see figure C.10).

You will connect the DC lines of the sample holder to it.

To limit the exposition of the stapled CNT to ambient air, it is important to do the transfer to cryostat as fast as possible. Thus, prior to vent the stapler vacuum chamber, one should rehears the different following steps:

• On the cryostat computer, launch a monitoring ... measurement on Exopy to be able to follow the resistance of the CNT once it has been connected to the cyostat lines.

• Close the sample holder with the cover using the wobble stick.

• Close V3, and V2.

• Put yourself to the ground of the cryostat using the cable with 10M Ω inside (see figure C.10) and the dedicated wristband.

• Vent the vacuum chamber of the stapler with argon, while maintaining the cover on the sample holder.

• Connect the DC lines of the sample holder to the grounding cable (which is connected to the ground of the cryostat), then disconnect them from the lines of the stapler.

The signal at the output of the cavity reads:

a

  In this thesis, we study carbon nanotubes based quantum dot circuits embedded in a microwave cavity. This general architecture allows one to simultaneously probe the circuit via quantum transport measurements and using circuit quantum electrodynamics techniques. The two experiments realized in this thesis use metallic contacts of the circuit as a resource to engineer a spin sensi�ve spectrum in the quantum dots. The first one is a Cooper pair spli�er which was originally proposed as a source of non-local entangled electrons. By using cavity photons as a probe of the circuit internal dynamics, we observed a charge transi�on dressed by coherent Cooper pair spli�ng. Strong charge-photon coupling in a quantum dot circuit was demonstrated for the first �me in such a circuit. A new fabrica�on technique has also been developed to integrate pris�ne carbon nanotubes inside quantum dot circuits. The purity and tunability of this new genera�on of devices has made possible the realiza�on of the second experiment. In the la�er, we use two noncollinear spin-valves to create a coherent interface between an electronic spin in a double quantum dot and a photon in a cavity. Highly coherent spin transi�ons have been observed. We provide a model for the decoherence based on charge noise and nuclear spin fluctua�ons. Shaping the spectrum of carbon nanotube quantum dots with superconduc�vity and ferromagne�sm for mesoscopic quantum electrodynamics iii Résumé Dans cette thèse, nous étudions des circuits de boîtes quantiques à base de nanotubes de carbone intégrés dans une cavité micro-onde. Cette architecture générale permet de sonder le circuit en utilisant simultanément des mesures de transport et des techniques propre au domaine de l'Electrodynamique quantique sur circuit. Les deux expériences réalisées durant cette thèse exploitent la capacité des métaux de contact à induire des corrélations de spins dans les boîtes quantiques. La première expérience est l'étude d'une lame séparatrice à paires de Cooper, initialement imaginée comme une source d'électrons intriqués. Le couplage du circuit aux photons dans la cavité permet de sonder la dynamique interne du circuit, et a permis d'observer des transitions de charge habillées par le processus de séparation des paires de Cooper. Le couplage fort entre une transition de charge dans un circuit de boîtes quantiques et des photons en cavité, a été observé pour la première fois dans ce circuit. Une nouvelle technique de fabrication a aussi été développé pour intégrer un nanotube de carbone cristallin au sein du circuit de boîtes quantiques. La pureté et l'accordabilité de cette nouvelle génération de circuit a rendu possible la seconde expérience. Cette dernière utilise deux vannes de spins non colinéaire afin de produire une interface cohérente entre le spin d'un électron dans une double boîte quantique, et un photon dans une cavité. Des transitions de spins très cohérentes ont été observé, et nous donnons un modèle sur l'origine de la décohérence du spin comprenant le bruit en charge et les fluctuations des spins nucléaires.

Figure 2 :

 2 Figure 2: Cavity and circuit QED: a. Schematic of a cavity-QED experiment such as the one performed in the group of Serge Haroche. Rubidium Rydberg atoms are produce in a oven labeled B and are then flying through a high finesse microwave cavity C. Two adjacent low finesse cavities, R 1 and R 2 are used to manipulate the state of the flying Rydberg atoms. Source: [7]. b. False-color scanning electron micrograph of a typical circuit-QED chip. The long conductor forming meanders is a coplanar waveguide microwave cavity. c. At one of the antinode of the cavity electric field aCooper pair box is embedded playing the role of the atom. Source:[START_REF] Frunzio | Fabrication and Characterization of Superconducting Circuit QED Devices for Quantum Computation[END_REF]

  Finally Chapters four and five present the two experiments realized in this thesis, namely the indirect observation of coherent Cooper pair splitting and the strong charge-photon

Figure 1 . 1 :

 11 Figure 1.1: The zone folding approximation: The rolling-up of a graphene sheet impose new boundary conditions to the transverse part of the wave vector. The Brillouin zone of a carbon nanotube is form of parallel lines. Whether or not one of the line is passing right through the middle of a Dirac cone will determine if the carbon nanotube is semiconducting or metallic. Source: [3].

Figure 1 . 2 :

 12 Figure 1.2: Double quantum dot circuit: a. Circuit representation of a double quantum dot, in which the three tunnel barriers can be tuned with the gate voltages: V Γ L , V t , and V Γ R . The two quantum dots are capacitively coupled to gate electrodes with voltages V L and V R , allowing to tune independently the chemical potential in each dot. b. Landscape of the confinement potential in the carbon nanotube felt by conduction electrons. The two curvy lines delimit the semiconducting gap inside the carbon nanotube. The two fermionic reservoirs associated to the source and drain electrodes are shown in blue. c. Side view of a suspended carbon nanotube quantum dot circuit.

Figure 1 . 3 :

 13 Figure 1.3: Charge Stability diagram of a double quantum dot: a. Current through a double quantum dot as a function of the gates voltage V L V Rshowing the characteristic honeycomb pattern of double quantum dots (measurement taken on device SPN15R). The points with high current are called "triple points" and correspond to the situation where levels in each of the dots are in the bias window. The white lines describing the hexagons are cotunneling lines. In such second order process, one of the dot is detunned from the bias window, and electrons tunnel through this dots via a state only virtually accessible. b. Schematic of the region in between two triple points. The grey lines represent delimitation between lower-energy electronic configuration considering only capacitances between dots and conductors. The avoiding crossing in the current line (in blue) is due to the interdot tunneling. This region is of paramount importance since it is where (0,1) and (1,0) charge configurations are degenerate, thus an electron is allowed to jump between the two dots. the new eigenstates are the molecular bonding and anti-bonding state in which a charge qubit can be encoded (see section 1.2.3).

Figure 1 . 4 :

 14 Figure 1.4: Stability diagram in a narrow-gap carbon nanotube: a. The four regimes (electron,electron), (electron, hole), (hole, hole), and (hole, electron) are visible on this measurement. When the circuit is deeply in the (e, e) or the (h, h) regime, it does not behaves as a double quantum dot anymore, but as a large single quantum dot. In the bottom left region, one can observe a four-fold degeneracy. b. Coulomb peaks extracted from a. along the black line. Every four peaks the spacing is larger, which is reminiscent of the four-fold degeneracy in carbon nanotubes due to spin and valley degeneracy. c. Coulomb peak spacing ∆V L+R (as defined in section 1.1.2) as a function of the peak index. This representation highlights the four fold degeneracy.Measurement taken on device SPN8R

Figure 1 . 5 :

 15 Figure 1.5: Charge qubit in a double quantum dot: a. Zoom on the stability diagram in the region between two triple points. The red arrow define the detuning axis which control the energy difference between the states |0, 1 and |1, 0 . In this area |B and |AB are the eigenstates states of the double quantum hamiltonian. b.The subspace of the charge qubit can be represented on a Bloch sphere. The two poles are the two logical states and can be reach only for δ = 0. Close to δ = 0 the energy transition is mainly set by the tunnel coupling t. Like in NMR experiment[START_REF] Vandersypen | NMR techniques for quantum control and computation[END_REF], small amplitude modulations of δ at the frequency of the transition between |B and |AB (originating from a cavity electric field for instance) will make the state vector rotate about the δ axis. Such a coupling scenario is called transverse coupling since the rotation axis is perpendicular to the quantization axis. c. Energy dispersion of the double quantum dot states. For δ = 0, the first derivative of Ω( δ , t) is zero, which means that the energy transition is insensitive at first order to δ -noise. This region is called a sweet spot. Placing the qubit in such region allows to limit the decoherence of the qubit due to charge noise.

Figure 1 . 7 :

 17 Figure 1.7: Andreev reflection and Andreev Bound States: a. Schematic of the Andreev reflection process. An electron incoming from a normal metal with a wave-vector k has a finite probability to be reflected on a normalsuperconductor interface as a hole of opposite wave-vector and spin. b. Mechanism responsible for the emergence of Andreev bound states in a superconductor -coherent conductor -superconductor junction. c. First observation of individually resolved Andreev bound states in a tunnelling spectroscopy experiment. In this experiment, acarbon nanotube plays the role of the coherent conductor. source:[START_REF] Pillet | Andreev bound states in supercurrent-carrying carbon nanotubes revealed[END_REF]

  splitting of Cooper pairs by measuring a supercurrent in a superconductor -parallel DQD -superconductor junction[START_REF] R S Deacon | Cooper pair splitting in parallel quantum dot Josephson junctions[END_REF]. They measure the supercurrent in their circuit while turning independently the two dots ON and OFF. Nevertheless, in such experiment, it is tricky to discriminates the respective contribution of each processes, local and non-local. In addition, this experiment requires to recombine the Cooper pairs, thus it doest not constitute a splitter.

Figure 1 . 8 :

 18 Figure 1.8: Double quantum dot dressed by Cooper pairs: a. Schematic of a closed Cooper pair. The two normal contacts are not represented since not considered in the hamiltonian. b. The coherent Cooper pair splitting of amplitude t eh give rise to hybridization of |0 = (0, 0) and |S = 1/ √ 2(| ↑, ↓ -| ↓, ↑ ). c. The superconductor can also induce a second order process where an electron from one of the dot can virtually excite a Bogoliubon in the superconductor and then tunnel in the next dot.

figure 1 .

 1 figure 1.10, J L ( δ ) and J R ( δ ) drive rotations about two axis that are 120 o apart from each other. When moving in the detuning space , M (where = ( 1 -3 )/2 and M = 2 -( 1 + 3 )/2), one change the strength of one exchange coupling relatively to the other (see formula 1.26) hence changing the global effective rotation axis via -pulses. An otherstrategy is to control the J's through their dependence of the tunnel couplings t's (see formula 1.26). It allows to stay at fixed position in the detuning space, and maintaining the qubit at a charge double sweet spot (where first derivatives of (E 1 -E 0 )( , M ) with respect to and M are zero), hence reducing decoherence due to charge noise during manipulation steps. Such a procedure is often called: symmetric operation point[START_REF] Bertrand | Quantum Manipulation of Two-Electron Spin States in Isolated Double Quantum Dots[END_REF][START_REF] Reed | Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation[END_REF][START_REF] Martins | Noise Suppression Using Symmetric Exchange Gates in Spin Qubits[END_REF].

Figure 1 . 10 :

 110 Figure 1.10: Principle of a three electron spin qubit: a. Energy dispersion of the two relevant states of the spin qubit according to the detuning = ( 1 -3 )/2. The detuning is spanning three different charge stability regions (2,0,1), (1,1,1), (1,0,2). b. Charge stability diagram in the -M plan. The position of the 1D cut shown in a. is approximatively indicated. c. Bloch sphere showing the two axis σ L and σ R .

Figure 1 . 12 :

 112 Figure 1.12: Principle of the reflection-induced Zeeman splitting:The energy of an electron confined in a coherent conductor is set by the constructive interference condition. If the coherent conductor is connected to a ferromagnet via a tunnel barrier, when the electron reflect on this interface, it will acquire a spindependent phase φ σ that adds to the phase acquired during the propagation into the conductor η. This results in a Zeeman splitting of the spectrum in the conductor.

Figure 2 . 1 :

 21 Figure 2.1: Light-matter coupling in mesoscopic-QED: a. Schematic of a mesoscopic-QED device.The cavity electric field in yellow is strongly modified due to the presence of cavity conductor protrusion, fermionic reservoirs, and also electrostatic gates. b. By designing protrusion in the cavity conductor (as shown in a.), one can make the amplitude of the AC electric field vary over the size of the circuit, thus the amplitude of the modulation of the orbital levels, the tunnel rates, and the chemical potentials can be very different (represented by the varying size of the zigzag arrows). One can choose which dipole in the circuit to address with light by designing the shape of these protrusion.

  the two eigenstates of the double quantum dot hamiltonian (see formula 4.1) are the bonding state |-= -v|L + u|R and the anti-bonding state |+ = u|L + v|R (expressions of u and v are given in section 4.3). The electron-photon coupling to the bonding/antibonding transition then reads:

Figure 2 . 2 :

 22 Figure 2.2: Input-Output representation of the relevant modes: The microwave cavities used in this thesis work are probed via transmission measurement. In a simple representation only two outside modes are considered bin,1 and bout,2 . A third mode is commonly used to describe internal losses of the cavity: bout,L .

Figure 2 . 3 :

 23 Figure 2.3: Energy spectrum of the qubit-cavity system

  Driving tone +H diss (2.20) Chapter 2. Mesoscopic -QED 53 With σ-= |g e| and σ+ = |e g|. in is the amplitude and ω RF the frequency of the drive applied on one of the two ports of the cavity. Dissipation in the cavity is taken into account by considering its coupling to an external bosonic bath b : H diss = d η(ω b † b + β b † â + β * b â † ). When going into the rotating frame of the drive tone, the Heisenberg equation of â and σ-read:

Figure 2 . 4 :

 24 Figure 2.4: Parameter space diagram for cavity, circuit and meso-QED:The Γ normalizing the two axis contains all sources of decoherence, arising both from the cavity and the qubit. Adapted from:[START_REF] Schuster | Resolving photon number states in a superconducting circuit[END_REF]

2 .

 2 30) ∆ω c is the cavity resonance frequency shift and ∆(κ/2) is the broadening of the cavity resonance. Those two expressions are valid in the limit of ∆ω c ω c and ∆(κ/2) κ/Notice that one can recover the frequency shift ∆ω c by measuring only the phase shift ∆φ at ω RF = ω c . Hence, ∆φ informs us about the dispersive effect of the circuit on the cavity. Similarly, the broadening of the resonance is proportional to ∆A and is related to the circuit induced dissipation. This free ourself from measuring the full cavity resonance, and allows a much faster characterization of the circuit state. This informs us also on the central role of the susceptibility χ(ω RF which encompass these two phenomenon through its real and imaginary part. Typical phase and amplitude signals are shown in figure 2.5.

Figure 2 . 5 :

 25 Figure 2.5: Cavity signal for Double quantum dot: a. Schematic of the coupling scenario to a double a quantum dot. In order to couple photons to left-right electronic transition, a protrusion of the cavity central conductor is created close to one of the quantum dot, to maximize the gradient of V ⊥ (r) (schematic taken from ref. [113]). b. (top panel) Energy dispersion of the left/right hopping transition inside a double quantum dot as a function of the detuning δ = L -R . The minimum of the transition energy is set by the tunnel rate t. (middle and bottom panels) Phase and amplitude signals in a resonant situation (ω DQD = ω cavity ), along the detuning axis δ . c., d. Measured phase and amplitude in the region of the stability diagram where (0, 1) and (1, 0) are the stable electronic configuration in the double quantum dot. The visible diagonal is the degeneracy line between the two configuration (0, 1) and (1, 0).Measurement take on device SP N 15R

Figure 3 . 1 :

 31 Figure 3.1: The development of this stapling technique started in the group of Z.Zhong[START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF], and inspired a few other groups. Each of them have different technical approach but the principle remains the same. Such a method allows to envision arbitrarily complex carbon nanotube circuit. Up to now the most successful one has been developed in the group of S. Ilani[START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF], where circuits with up to 16 gates, and multiple carbon nanotubes have been realized as shown in the top left box. Sources:[START_REF] Chiang Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF],[START_REF] Pei | Valleyspin blockade and spin resonance in carbon nanotubes[END_REF],[START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF],[START_REF] Gramich | Fork stamping of pristine carbon nanotubes onto ferromagnetic contacts for spin-valve devices[END_REF],[START_REF] Blien | Quartz tuning-fork based carbon nanotube transfer into quantum device geometries[END_REF]

Figure 3 . 2 :

 32 Figure 3.2: The impact of environment on charge noise of single CNT device: a. pristine suspended CNT, b. pristine CNT laying on Si substrate, c. Polymer coated suspended CNT, d. photoresist processed CNT laying on Si substrate. Source:[START_REF] Sharf | Origins of Charge Noise in Carbon Nanotube Field-Effect Transistor Biosensors[END_REF]

Figure 3 . 3 :

 33 Figure 3.3: Quantum dot circuit coupled to a microwave resonator a. Optical image of the whole device. RF pads used for wire-bonding to the PCB board are visible on both sides of the microwave cavity. Two trenches for each QD-circuit region ensure the smooth running of the integration of the carbon nanotube (colored in blue). DC bonding pads are also visible in the lower part of the picture. b. SEM micrograph of the QD-circuit. In the left part of the image we can see the central conductor of the resonator to which the QD circuit is capacitively coupled, and on the right part the different electrodes are visible. c. Zoom-in on the QD-circuit where the carbon nanotube is positioned. The gate array below the carbon nanotube is dedicated to the shaping of the confinement potential for electron inside the CNT.

Figure 3 . 4 :

 34 Figure 3.4: Lithography steps:The lithography technique is used whenever one want to pattern a structure on the device. It can be applied for metal deposition as well as for etching a defined area. The fabrication of a our samples typically contains around five different lithography.

Chapter 3 .techniques 69 3 . 4 . 5 . 1 . 3 . 4 .

 3345134 Experimental Carbon nanotube growth and stamping carbon nanotube are grown with a Chemical Vapor Deposition technique (CVD) on a quartz substrate with a predefined mesa structures. Then they are stamped on the circuit chip in a MJB4 mask aligner. The mesa structure allows for a very local transfer of carbon nanotube, hence maintaining a good quality factor of the microwave cavity. Carbon nanotube localization Prior to the fabrication of contacts and gates that realize the quantum dot circuit, one need to localize the transferred carbon nanotubes on the circuit chip and choose the most suitable one. Contact and gates of the nano-circuit This step include the smallest structure of the device, namely the QD-circuit. It typically consists in two or three electronic lithography steps for which the design and the evaporated metals vary greatly with the realized circuit. Stapling process Alignment crosses and pre-contact 2. Niobium cavity Contact and gates of the nano-circuit Etching side trenches Two trenches on both sides of the quantum dot circuit are needed for the stapling steps. This step combines optical lithography and dry etching of the silicon oxide and silicon preformed in a Reactive Ion Etching (RIE) equipment.

6 .

 6 Stapling Last step before mounting in the cryostat. It is performed in a dedicated vacuum chamber where a nano-motor stage allows us to accurately position of the Chapter 3. Experimental techniques 70 cantilever comb with respect to the circuit chip. The alignment is made possible thanks to an optical access in the chamber.

2 Figure 3 . 5 :

 235 Figure 3.5: PMMA/MMA stack Thanks to this resist structure, the deposited metal is not in contact with resist, which facilitate the following lift-off.

2 O 3

 23 nanoparticules diluted in 30 mL of IPA. The Al 2 O 3 nanoparticules serve as a support for iron and molybdenum and tend to form clusters. Hence, before the deposition of the catalyst solution on the growth chip, one need to sonicate it for one hour in order to break apart alumina clusters which are believed to favor the formation of bundles of CNT. Then the catalyst solution is left for decantation during 45 minutes and a small volume is taken off at the surface of the solution. A few drops are deposited on the growth chip, then it is partially rinsed in IPA before drying it with nitrogen. Rinsing in IPA allows to obtain a more uniform catalyst deposition.

Figure 3 . 6 :

 36 Figure 3.6: CVD growth on a comb of cantilevers: a. The CVD setup consists in a 1.5m long quartz tube whose central portion is caught in a furnace reaching 900 • C. Growth gases are injected at one extremity of the quartz tube. b. The growth of carbon nanotubes is performed on commercial combs with about 50 cantilevers. We obtain typically between 5 and 10 suspended carbon nanotube suitable for the stapling. Prior to use commercial cantilevers we have been using homemade one whose process development have been carried out by J.Palomo. The process is based on a wet etching process of the silicon (see appendix A). Scale bar: 5 µm.

Figure 3 . 7 :

 37 Figure 3.7: Directional growth based low-flow recipe:a. The catalyst deposition is critical for getting long CNTs. It has been found that depositing the catalyst on the edge is a efficient way to obtain a descent number of long CNTs (longer than 500 µm). In this low flow growth recipe, the kite-mechanism[START_REF] Huang | Growth Mechanism of Oriented Long Single Walled Carbon Nanotubes Using Fast-Heating Chemical Vapor Deposition Process[END_REF] is taking place, where the catalyst nanoparticles are floating above the substrate and dragging the growing CNTs. The edge is believed to lift the nanoparticles because of local turbulent flow, then they are maintained thanks to the buoyant force. In such situation, CNTs can easily reach few millimeters. b. Similar growth with pillars prepatterned on the substrate resulting in suspended CNTS. The fact that the trajectories of the long CNTs are not modified by the pillars is consistent with the kite-mechanism.Scale bar: 5 µm.

  Examples of I cut (V bias ) line traces are shown in figure 3.10. The current increases until it reaches a threshold value I threshold , then it abruptly fall to zero indicating the cutting of the CNT. The value of I threshold is instructive about the

  absence of local disorder, one expect similar cutting curves at different position of the CNT, which are only defined by intrinsic properties of the CNT. This symmetric behavior has been observed on several devices (see figure 3.11).

Figure 3 . 11 :Figure 3 . 12 :

 311312 Figure 3.11: Cutting I-V curves on two distant sections of a CNT

O 2 .

 2 The vacuum chamber is vented with nitrogen, and right after the opening of the chamber, we close the chip carrier, then the circuit lines are connected to the ground of the cryostat via a 10 MΩ resistor to avoid high transient currents. All DC lines of the circuit are connected to two output lines, allowing for a smooth ground transfer and preventing the circuit lines to be at a floating potential during the transfer into the cryostat. During the transfer the inside of the chip carrier is kept under nitrogen atmosphere by flushing it via holes in the chip carrier. Electrical discharges are more likely to happen during the connection of the DC lines to the cryostat lines. To limit such transient current during the connection of the DC lines, additional 10 MΩ resistors are added on the lines of the cryostat. Ongoing work focus on improving the yield of transferred CNTs by designing a vacuum tight sample holder (see figure 3.13). Although maintaining the sample holder under nitrogen atmosphere has revealed to be very efficient, and brought the success yield close to 100%, strong variations of the circuit resistance still remain. Keeping the circuit chip under vacuum during all electrical connections should limit those uncontrolled variations of resistance.Circuit chip design Dimensions of the circuit play a critical role for the smooth running of the stapling. The ratio between the QD-circuit length and the height of the contacts with respect to the gates is decisive to obtain a suspended carbon nanotube (see figure3.14). Another important parameter which affect the suspended nature of the CNT once it is stapled, is the slack of grown CNTs in between cantilevers. Although, our growth recipe does not provide any control over the slack of CNTs, most of them are taut. The rule of thumb is to keep this aspect ratio below 15.

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: Vacuum tight sample holder Such a sample holder is expected to diminish the variations of resistance during the connection to the cryostat. It also reduces the overall atmosphere exposition time of the CNT. Cu�ng sec�on Contact width QD Circuit length Cu�ng contact width

  Chip carrier and PCB Once the chip has been cut in pieces of 5 × 10 mm, it is glued with PMMA resist to the sample holder and can be micro-bonded to the Printed Circuit Board (PCB). A multi-layer PCB is used where DC and RF lines are on two different layers with a ground plane in between to suppress cross-talk between RF and DC signals. Two additional ground planes serve to encapsulate the whole stack. These three ground planes are connected to each other with via holes. One of the objective of the PCB design is to eliminate RF modes in the vicinity of the cavity resonance (around 6.5 GHz). Because of the dimensions of the inside sample holder (15 mm × 45 mm × 5 mm) 3D box modes can act as parasitic resonance. In order to push at higher frequency those modes, a copper cover has been design to reduce the box volume over the chip to 10 mm × 5 mm × 5 mm and push 3D modes above 9 GHz. To ensure a good electrical contact an Indium seal is used between the top layer ground plane and the cover. Another possible parasitic mode is the slot line mode where the two ground planes oscillate in phase opposition. Micro-bonds are used to link the two ground planes, hence supressing the slot line mode.Cryostat and wiringAll the experiments presented in this thesis work have been carried out in a dry dilution refrigerator (HEXADRY 200 Cryoconcept) at a base temperature of 20 mK. The RF and DC wiring of this cryostat has been done by a previous PhD student, Laure Bruhat. The temperature needed for an experiment is determined by the smaller energy scale one want to resolve. In the case of the Cooper pair splitter experiment (Chapter 4) this energy is the proximitized superconducting gap of Pd/Al which is of the order of 100 µeV. This set the condition on the temperature: 3.5 k B T 100 µeV which correspond to T 335 mK. For the spin-qubit experiment (Chapter 5) the smaller energy scale is the Zeeman splitting (3.5 k B T g µ B B giving rise to T 385 mK at 1 Tesla). One thing worth notice is that with such base temperature, the microwave cavity is not thermally populated (photon number = 1 e ωc/k B T -1 10 -2 ),

  . The room-temperature CNT resistance typically ranges from 50 kΩ to 2 MΩ. Differential conductance provides direct measurement of the density of states in the quantum dots which is highly desirable in such experiments. The measurement of the differential conductance is based on a lock-in detection scheme where an AC bias voltage (at 77.77 Hz) is applied to the CNT circuit. The output current is then demodulated by this input frequency, thus moving the relevant signal to DC, which justify the use of a subsequent low pass filter inside the lock-in amplifier (SIGNAL RECOVERY 7265 ).This technique is very efficient to extract a specific signal from a noisy environment.

Figure 3 . 15 :Figure 3 . 16 :Figure 3 . 17 :

 315316317 Figure 3.15: Schematic of the cryostat wiring:In addition to the cavity input and output lines, there are two other RF lines designed to address the quantum dot circuit in the microwave range (they are less attenuated).Further details about the wiring are given in the PhD thesis of Laure Bruhat[START_REF] Laure | Microwave as a probe of quantum dot circuits: From Kondo dynamics to mesoscopic quantum electrodynamics[END_REF].

(Figure 3 . 18 :Chapter 3 .techniques 90 3. 3 . 3 . 2

 318390332 Figure 3.18: Schematic of the RF measurement setup

Figure 3 . 19 :

 319 Figure 3.19: Cavity transmission measurement: a. RF transmission measurement betwen 6 GHz and 7 GHz of a coplanar waveguide cavity as shown in sectin 3.2.1. b. Zoom-in on the cavity fundamental mode resonance.The FWHM (Full Width Half Maximum) of the resonance is ∆f -3dB = 530 kHz corresponding to a quality factor Q = 12500. The measurement is performed at a input power such that the average photon number at the resonance is about n ph 10. The estimation of the photon number requires to know the input and output powers at the cavity ports, which are known with a 3 dB precision.

Figure 3 . 20 :

 320 Figure 3.20: Pulse sequence for qubit spectroscopy

Figure 3 .

 3 Figure 3.21: Pulse sequences for T 1 -measurement and Rabi oscillationsThe delay between the filling and the read-out of the cavity is set to 1/κ. In all the presented pulse sequences the re-initialization of the system is done by waiting long enough for the cavity and the qubit to relax in their ground states.

  Our findings could be adapted to many other circuit designs and shed new light on the coupling of superconducting nanoscale devices to microwave fields. This chapter reproduces an article: Circuit-QED with a quantum dot charge qubit dressed by Cooper pairs, published in Phys. Rev. B 98, 155313. Besides a preliminary version of this work was communicated at the conference ICPS (August 2016) in Beijing and on condmat as arXiv:1612.05214.

4. 3 Figure 4 . 1 :

 341 Figure 4.1: a. Optical photograph of the layout of our cavity QED architecture on a large scale. b. and c. SEM micrographs of our devices on two different scales in false colours. The 'fork' coupling gate is coloured in red. The superconducting electrode is coloured in orange. The normal (non-superconducting) electrodes are coloured blue. The gates are coloured in green. d. Circuit diagram of our hybrid double quantum dot highlighting the symmetric coupling scheme between the two dots and the resonator in red.

Chapter 4 .

 4 The superconducting double quantum dot 100 waveguide cavity with resonance frequency of about 6.636 GHz and a quality factor of about 16000. Throughout the paper, we describe results obtained with two different devices (sample A and sample B) which had exactly the same layout.

Figure 4 .

 4 1-a, b and c show optical as well as scanning electron microscope pictures of one of our devices. A single wall carbon nanotube is tunnel-coupled to a central superconducting finger (in orange) and two outer non-superconducting electrodes (in blue). Two side gates (in green) are used to tune the double dot energy levels. A finger galvanically coupled to the central conductor of our cavity (in red) is attached to two top gates in a fork geometry. This coupling scheme is markedly different from the double-dot/cavity coupling schemes used so far in that context[START_REF] Mi | Strong coupling of a single electron in silicon to a microwave photon[END_REF][START_REF] Stockklauser | Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator[END_REF][START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Frey | Dipole Coupling of a Double Quantum Dot to a Microwave Resonator[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] K D Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF]. Instead of favoring a microwave modulation of the difference of the energy between the left and the right dot, the fork geometry shown in figure4.1c favors the modulation of the sum of the left and right dot energies by microwave photons.All the measurements have been carried out at about 18 mK. We simultaneously measure DC transport through the quantum dot device and microwave transmission through the coplanar waveguide resonator. Our control parameters of the quantum dot circuit are the bias voltage V S applied to the superconducting electrode and the gate voltages V L and V R . For convenience, measurements are often taken in the rotated frame V Σ -V δ , as defined in Appendix B.2. Concerning the cavity, the tunable parameters are the frequency and power of the probe tone. Details about sample fabrication and measurement setup are given in Appendix B.2.

Figure 4 . 2 :

 42 Figure 4.2: a. Colorscale map of the current I L flowing through the left (L) normal metal contact as a function of bias voltage V S and the gate voltage V Σ for sample A. From this map, we read-off a superconducting gap ∆ ∼ 150µeV. b. Cotunneling scheme accounting for the renormalization of the hopping constant between the left and the right quantum dot.

4. 3

 3 Low energy spectrum of the hybrid superconductor DQD in the absence of the cavity Contrarily to normal contacts, a superconducting contact is expected to modify coherently the spectrum of a double quantum dot. Let us first recall the double dot spectrum in absence of superconductor. The double dot hamiltonian reads:

Figure 4 . 2 -

 42 Figure 4.2-b gives a qualitative picture of the main process responsible for dressing the bonding and antibonding states of a double dot charge qubit. In addition to the bare tunneling between the two dots (grey solid arrow), the superconductor induces cotunneling processes: an electron from one dot can virtually excite a quasiparticle in the superconductor and tunnel to the other dot (blue dotted arrows).

Figure 4 . 3 -Figure 4 . 3 :Chapter 4 .

 43434 Figure 4.3-a and 4.3-b (resp. 4.3-c and 4.3-d) display the phase (resp. amplitude) contrasts for sample A for two different gate voltage tunings called Aa and Ab. The avoided crossing lines are characteristic of a double dot stability diagram and correspond to tunneling between the dots and the leads. We focus now on the most striking features, which lie within the area delimited by the avoided crossing. In figure 4.3-a, one observes

  mission t when ω cav = ω S-DQD and allows to map the dispersion relation of our hybrid DQD. The χ dot-lead (ω) susceptibility is resonant when E N = E N +1 and allows to map the stability diagram of the DQD. We display in figure 4.3-a and 4.3-b the theory for the lines ω cav = ω S-DQD and E N = E N +1 in black dashed lines, using the following parameters: besides the measured value ∆/h= 37.5 GHz (see figure 4.2-a), we have for sample Aa: t b /h = 6.3 GHz Γ SR /h = 400 MHz Γ SL /h = 900 MHz t 0 eh /h = 400 MHz U L /h = 29 GHz U R /h = 71 GHz U m /h = 16 GHz and for sample Ab, we have: t b /h = 5.5 GHz Γ SR /h = 330 MHz Γ SL /h = 900 MHz t 0 eh /h = 350 MHz

Figure 4 . 4 :

 44 Figure 4.4: a. Diagram of the transition map of the hybrid double quantum dot intersecting with the cavity resonance frequency. This results in the phase contrast maps of figure 4.3-a and 4.3-b. The axis are the orbital detuning δ and the average orbital energy Σ of the double dot. b. Bloch sphere diagram depicting the active states of our hybrid double quantum dot and the tunable hoping strength. This symmetric coupling scheme is crucial for the strong electron-photon coupling.

phase and amplitude contrasts measured in figure 4 . 3 , 1 .

 431 one can wonder what happens to the cavity spectrum when the cavity is resonant with the DQD circuit transition. We tune the double dot gate voltages to the point of maximum phase contrast and there, we measure the resonator transmission as a function of frequency (supplementary data in appendix B.2). The top panel of figure4.5-a shows the result of the measurement for sample B, for which we measured the strongest effect. We observe a splitting of the order of 10 MHz in the cavity resonance for an average number of photons n of about 1. This observation persists down to the lowest input power which corresponds to n This is the hallmark of a vacuum Rabi splitting which indicates the strong coupling between our hybrid double quantum dot and the microwave cavity photons.

Figure 4 . 5 :

 45 Photon number

[ 119 ]Chapter 4 .

 1194 . The charging energy of sample A and B can simply be read-off from the transport stability diagram which is shown in figure4.2-a for sample A. Due to the fork-shaped top gates that increase the capacitance to the ground, our charging energy is 2 meV, about 10 times smaller than what we find typically for similar devices with a conventional top gate setting[START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF][START_REF] Viennot | Out-of-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF]. Since Γ ϕ /2π ≈ 400 MHz in those conventional settings, a reduction of 10 of E C is expected to reduce Γ ϕ by a factor of 100, i.e. Γ ϕ /2π ≈ 4 MHz, which is consistent with the order of magnitude of Γ/2π ∼ few MHz inferred from the cavity spectroscopy of figure4.5-a. The superconducting double quantum dot 115

Figure 5 . 2 :

 52 Figure 5.2: circuit-QED device with a carbon nanotube spin-qubit: a. Large scale view of our circuit QED setup. Two carbon nanotubes spin qubits can be integrate in such a geometry, in between the two pairs of trenches (shaded blue). b. Scanning electron micrograph of one of the two spin-qubit region. The circuit is realized in between two trenches 15µm deep, enabling the insertion of the carbon nanotube. c. Zoom on the embedded carbon nanotube showing the two non-colinear ferromagnetic contacts, and the array of gates below the suspended carbon nanotube.

Figure 5 . 3 :

 53 Figure 5.3: Charge stability diagram of the double quantum dot: a. Current measured throught the spin-qubit circuit as a function of the left gate V L and the right gate V R . The Semiconducting gap of the carbon nanotube is visible in the bottom left corner. This indicates that the two dots are negatively charged. b. Phase of the transmitted cavity signal measured in the gate-gate region indicated by the white square in a.. In this region left/right tunneling transition occurs an manifests in the strong phase shift.

5 .

 5 amplitude A and phase φ. The control which we have on the spectrum of the device is visible from the stability diagram shown in figure 5.3-a which displays the current under a bias of V SD = 100µV . Several features indicate very weak disorder and electrostatic control of the potential landscape of confined electrons via the bottom gates: a clear "electron-electron quadrant" delimited by the semiconducting gap controlled by two of the bottom gates (V L and V R ), continuous transition from double-dot spectrum (triple points and avoided crossings) to single dot spectrum (parallel transverse lines) and rather regularly spaced Coulomb blockade peaks. At the edges of the electron-electron quadrant of the stability diagram, we can form a double quantum dot in a controlled way in the few electron regime. We focus here on a degeneracy line between two charge states highlighted by a white square in figure 5.3-a. The phase contrast of the microwave Chapter Carbon nanotube platform for spin qubit 123 signal in this region is displayed on figure 5.3-b. This allows to define the detuning δ = α L V L -α R V R + 0 along the white arrow, α L(R) being given by the slope of the degeneracy line. The phase of the microwave signal displays the characteristic sign change of a resonant interaction between the cavity and the double quantum dot. A transition to the dispersive (off-resonant) regime is also visible by a gradual change of phase contrast along this degeneracy line. In the resonant regime, the dependence of the phase contrast ∆φ as a function of δ which has maxima/minima of about ±15% provides an estimate of the charge coupling strength g C ≈ 2π × (21 ± 1)M Hz and of the charge decay rate γ C ≈ 2π × (1.35 ± 0.16)GHz.

Figure 5 . 4 : 5 .

 545 Figure 5.4: Two-tone spectroscopy of the spin transition: a. Phase contrast measured during a two-tone spectroscopy of the ferromagnetic spin-qubit as a function of the detuning δ . The overall behavior of the transition: saturation at large detuning δ and a minimum at δ = 0 is reminiscent of a spin transition such as the 01-transition shown in fig 5.1-a. The presence of three different spectroscopic lines can be explained by the lifting of the K/K' degeneracy in the carbon nanotube. b. Magnetic field dependence of the lower transition in a.. c. Pulse sequence used for the two tone spectroscopy. The qubit drive pulse last t = 3µs, then the cavity is filled for 2µs, during when the cavity output signal is read-out using a fast data acquisition card for t = 700ns. d. Phase contrast as a function of the qubit drive frequency f pump taken for the lower transition shown in a.. The spectroscopic linewidth of this transition correspond to a decoherence rate of γ S = 2π × 249kHz.

figure 5 . 5 . 2 S

 552 figure 5.4-c, except that we have a varying qubit drive pulse length. Such a pulse sequence is the one traditionally used to observe Rabi oscillations. The resulting variations of the in-phase quadrature I and the out-of-phase quadrature Q of the cavity field as a function

Figure 5 . 5 :

 55 Figure 5.5: Rabi measurement: a. Pulse sequence used for the Rabi measurement with qubit drive pulse duration varying τ ∈ [0µs, 8µs] . b. and c. In-phase quadrature I and out-of-phase quadrature Q of the cavity field respectively, as a function of the qubit pulse duration τ . d. Extracted Rabi frequency for a varying qubit drive amplitude. A roughly linear dependence is obtained. e. Reconstructed σz evolution while the qubit is driven, using cavity-Bloch equations [140] (see section 5.1.7).

Chapter 5 . 128 Figure 5 . 6 :

 512856 Figure 5.6: σ z evolution for the pulse sequence with τ pulse = 8µs.

figure 5 .Chapter 5 .

 55 figure 5.7-a. Two main decoherence sources are expected for the electronic spin in double quantum dots: charge noise and nuclear spin. Our 12 C platform is grown from a natural CH 4 feedstock gas and therefore is expected to have a low concentration of nuclear spins (1.1% of13 C with a nuclear spin I = 1/2). The charge noise is related to the fact that the transition frequency may fluctuate if offset charges nearby the device change the detuning. Therefore, it should induce a decoherence rate γ S proportional to the derivative of the transition with respect to the detuning[START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF]. For a large detuning δ , the nuclear spin bath is on the contrary expected to give a nearly independent contribution as a function of the detuning. The decoherence rate γ S and the derivative ∂ω/∂ δ as a function of detuning δ are shown to overlap well provided we add a constant of about 500kHz to the derivative in figure5.7-a. The linear behavior of the decoherence rate γ S

Figure 5 . 7 :

 57 Figure 5.7: Charge stability diagram of the double quantum dot:

Chapter 5 .Figure 5 . 8 :

 558 Figure 5.8: MFM image of the ferromagnetic contacts: a. Magnetic force micrograph (MFM) of the ferromagnetic contact geometry used for the spin-qubit experiment. The two larger contacts are used only for the integration of the carbon nanotube, hence they do not belong to the spin-qubit circuit. b., c. Magnetic force micrograph (MFM) for single ferromagnetic electrode lying directly on a SiO 2 substrate.

  .

  1-a) to charge noise. The sensitivity to charge noise is captured (at first order) by the derivative ∂ω 01 ∂ δ and is shown in figure5.9 along with the coupling strength between cavity photons and excitations in the 01 transition.

Figure 5 . 9 :

 59 Figure 5.9: Influence of the Zeeman angle: a. Coupling strength between cavity photons and excitation in the 01 transition, as a function of the detuning (here labeled D = L -R ) and the Zeeman angle θ. b. the derivative ∂ω01 ∂ δ as a function of same two parameters. The red point labeled ON, corresponds to the working point for which D = 2.8δ and θ = π/6.

Charge transition propertiesFigure 5 . 10 : 2 δarg 1 iκ/ 2 - g 2 c( 5 . 6 )

 510212256 figure 5.3-b (and reproduce in figure 5.10-a), the phase signal display the characteristic profil of a resonant situation between a transition in the quantum dot circuit and the cavity. A cut along the V L -axis in this region in shown in figure 5.10.b. The strong

Figure 5 . 11 : 2 δ

 5112 Figure 5.11: Amplitude signal for the two-tone spectroscopy: The black arrow is pointing toward the region where the spin transition caused a 4 o phase shift in the cavity transmission signal.

Figure 5 . 12 :

 512 Figure 5.12: Tunnel barrier tunability: The height of the tunnel barrier is proportional to -eV tunnel , which corresponds to the middle gate in a five gate geometry.In a. the charge transition is fully resonant with the cavity frequency. In b., c., d., e. the dispersive region progress from the top right corner, and in f. the charge transition frequency is almost completely dispersive with respect to the cavity frequency.

•• H 2 •

 2 Purge of lines (5 min): Open all the three valves on the valve panel plus the main valve at the exit of the panel. Note the three pressures. The target values (subtracting the offset values) are: -Ar: 700 sccm -H 2 : 100 sccm -CH 4 : 50 sccm • 900 • C heating ( ≈ 20 min): Heating of the oven up to 900 • C under Ar and H 2 flow. Flash (20 min): Inject only H 2 during 20 minutes. This step is supposed to chemically reduce the possibly oxidized catalyst. Heating from 900 • C to 950 • C (5 min): Heating of the oven up to 950 • C under H 2 flow only. • Growth (15 min): Inject only CH 4 and H 2 . After 15 minutes, close the CH 4 valve, open the Ar valve, and turn-off the heating of the oven. • Cool-down (2 hours): When the temperature goes below 400 • C, it is possible to open the oven to speed-up the cool-down. When the temperature is below 200 • C (10 minutes later), one can close the H 2 valve, to save some H 2 .

If S is constructed

  such that S, Ĥ0 = ĤS-DQD , the effective hamiltonian becomes:Ĥeff ≈ Ĥ0 -1 2 S, ĤS-DQDOne seeks for the operator S in the following form:S = k,σ,j∈{L,R}γ kσ X j kσ -h.c

  l|X j kσ |m = -t j σe ik.r j v * -k l|d jσ |m + t * j e ik.r j u k l|d † jσ |m E k + E m -E l (B.5) This allows us to calculate explicit expressions of the double quantum dot effective hamiltonian by using the ground state of the superconductor in the initial and final states and states in which one quasiparticle is excited in the superconductor as intermediate states. In the bonding/antibonding basis, we get, setting δr = r L -r R the distance between tunneling to left and right dots: +|[S, ĤS-DQD ]|-= (B.6)

  Figure C.2: a. This CNT fulfills all the criteria, and it is quite well taut. b. This CNT seems to be a bundle, based on the V-shape at the right cantilever, its high contrast, and its width. c. There are multiple CNTs, but the first one is sufficiently separated from the other ones.

Figure C. 3 :

 3 Figure C.3: The stapling chamber: a. The circuit and the cantilevers in the stapling configuration. The piezo-motor block is quite delicate and should be manipulated only via the screws block. b. View of the cantilever holder in the axis of the accordion. The angle α determines which cantilever will touch first the circuit chip.

Figure C. 5 :

 5 Figure C.5: Detection setup

Figure C. 6 :

 6 Figure C.6: Piezo-motor controller

Figure C. 8 :

 8 Figure C.8: I-V curves during the cutting of the CNT: 1. Improvement of the contact resistance. 2.Before the cutting of the CNT, the current saturates, and in some case it can even decrease with increasing bias voltage. This last behavior can be explained by electron scattering by optical phonons. The cutting current typically lay in between 10µA and 20µA. The cutting current value is connected to the ability of the CNT to dissipate heat. Thus it will depends on the intrinsic properties of the CNT, but also on the length over which the CNT is suspended. A large cutting current value (> 20-30µA) can have several reasons: a short distance between the two contacts involved in the cutting (make the heat dissipation easier), the presence of multiple CNT (a bundle), a CNT with a large diameter. 3. The presence of multiple drop is a strong indicator for bundles. 4. A remaining current after the current fall can be either due to a second CNT still standing, or due to a contact between the CNT and the silicon at the bottom of the trench. To check this last possibility, one can put all the contacts to a floating potential (except the one which is biased) and see if the current remains. The CNT cutting can be seen as a first electrical characterization of the stapled CNT.

Figure C. 9 :

 9 Figure C.9: Setup for measurement of the resistance of the circuit and its gate dependence.

  , andV SD ∆E conf,L , ∆E conf,R . An example of

such 2D plot is shown in figure

1

.3. This plot is the charge stability diagram of a double quantum dot measured on the device nicknamed SP N 15R. Inside each hexagon of this honeycomb pattern the number of electrons in each dot is fixed.

  The excitation and semiconductor pictures: a. Representation of the BCS ground state in the excitation picture. The ground state is by definition the vacuum of Bogoliubon excitations. b. Schematic of a tunneling event where an electron is tunneling from a superconductor into a normal metal. The decomposition of Bogoliubon operators into electron-like and hole-like parts allows to conveniently represent tunneling event. The lower band correspond to the hole-like part of Bogoliubons. This representation is widely used to describe transport experiments. Notice that as it is represented the BCS density of state is twice smaller than in the excitation picture.phase, hence it is not adapted to describe coherent phase effects such as the Josephson effect, or to predict the appearance of Andreev bound states (see next section).

	Chapter 1. Hybrid double quantum dots	27
	Energy	Energy
	Figure 1.6: 1.3.1.3 Andreev reflections
	At the interface between a superconductor and a normal metal, an incident electron
	from the normal metal with an energy E << ∆ has no directly accessible states in the
	superconductor. It can be either reflected back into the normal metal (specular reflec-
	tion), or as a hole of equal energy and opposite spin. This last process is called Andreev
	reflection. It is the basic mechanism responsible for superconducting proximity effect,
	namely the fact that normal metal contacted by a superconductor becomes supercon-
	ducting over a length characterized by ξ 0 . During Andreev reflection, particle number
	conservation implies the creation of two electrons inside the superconductor which then
	decay into the Cooper pair condensate.
	Crossed Andreev reflection When two spatially separated normal metal conduc-
	tors are connected to the same superconductor, the reflected hole can propagate in a
	different conductor from which the incident electron is coming, provided that the two

  [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] MHz, Γ K =2π×2 MHz, g K =2π×16.8 MHz, Γ K =2π×100 MHz (see TableB.1 in Appendix B.3 for all the parameters). It is important to note here that the K/K' valleys are in general coupled by weak disorder in carbon nanotubes[START_REF] Laird | Quantum transport in carbon nanotubes[END_REF]. Therefore, the K/K' eigenstates correspond to linear combinations of the original (degenerate) valley states. Their coupling to the field and decoherence rates are therefore different in general. As one can see in TableB.1, a low coupling strength is accompanied by a low decoherence rate (K mode) whereas a large coupling strength is accompanied by a large decoherence rate (K' mode).

  δr u 2 -t * R t L e -ik.δr v 2 } -|v -k | 2 ( 1 E k -E + + Σ + 1 E k -E -+ Σ ){(uv(|t L | 2 -|t R | 2 ) + t * L t R e -ik.δr u 2 -t * R t L e ik.δr v 2 }We get similar expressions for +|[S, ĤS-DQD ]|+ and -|[S, ĤS-DQD ]|-. In order to obtain useful analytical expressions, we use the following identities, taking the 1D limit similarly to ref.[START_REF] Leijnse | Coupling Spin Qubits via Superconductors[END_REF], assuming the tunnel coupling t i is real and setting t i = √ Γ Si , to second order in ∆ where is the energy of the state: Si Γ Sj cos(k F δr)e -δr /ξ 0 1 -Si Γ Sj cos(k F δr)e -δr/ξ 0

	t i t j	k	u k v -k	e ik.δr E k +	=	(B.7)
		Γ Si Γ Sj π -	2 ∆	+	π 2
								2 |δr| πξ 0 ∆	+	|δr| 2ξ 0	2 ∆ 2 + . . . for δr	ξ 0
	t i t j	k	|u k | 2 e ik.δr E k +	=	(B.8)
	Γ Si Γ Sj Ln	2ω D ∆	-	π 2 ∆	+
								πξ 0 2 |δr|	-	π 2 ∆	+	π |δr| 2ξ 0	2 ∆ 2 + . . . for δr	ξ 0

2 ∆ 2 + . . . for δr = 0 π Γ 2 ∆ 2 + . . . for δr = 0 π Γ

  RF (t) = I(t)cos(ω C t) + Q(t)sin(ω C t) + n(t) (D.[START_REF] Van Wees | Quantized conductance of point contacts in a two-dimensional electron gas[END_REF] with n(t) the noise in the cavity output signal:n(t) = n 1 (t)cos(ω C t) + n 1 (t)sin(ω C t) (D.2) + n 2 (t)cos(ω C t + 2∆t) + n 2 (t)sin(ω C t + 2∆t) Here we only focus on the noise at the pulsations ω C , and ω C +∆, where ∆ = ω LO -ω C 20M Hz, because these are the two frequencies that will not be attenuated by the bandpass filter (minicircuit SBP 21.4+). After the demodulation by the local oscillator signal (a LO (t) = a LO cos(ω LO t)), the incoming signal on port A (connected to the I-port of the mixer !), S A (t) reads: Hence, one can define I tot (t) and Q tot (t) such that: one can notice that in the terms I A (t), I B (t), Q A (t), and Q B (t) there are always two terms of noise, while in I tot (t), and Q tot (t) there is only one. A sanity check consists in measuring I A (t) -Q B (t), and I B (t) + Q A (t) which should give only noise.

	Similarly, we have:
			I B (t)		Q B (t)
	S B (t) =	a LO (t) 2	Q(t) + n 1 (t) + n 2 (t) cos(∆t) +	a LO (t) 2	I(t) -n 2 (t) + n 1 (t) sin(∆t)
					(D.5)

S A (t) = a LO (t) 2 I(t)cos((ω C -ω LO )t) + Q(t)sin((ω C -ω LO )t) (D.3) + n 1 (t)cos((ω C -ω LO )t) + n 1 (t)sin((ω C -ω LO )t) + n 2 (t)cos((ω C -ω LO + 2∆)t) + n 2 (t)sin((ω C -ω LO + 2∆)t) S A (t) = I A (t) a LO (t) 2 I(t) + n 1 (t) + n 2 (t) cos(∆t) + Q A (t) a LO (t) 2 -Q(t) -n 1 (t) + n 2 (t) sin(∆t) (D.4)

where the terms with a frequency out of the filter band have already been eliminated.

I tot (t) = I A (t) + Q B (t) = a LO (t)(I(t) + n 1 (t)) (D.6) Q tot (t) = I B (t) -Q A (t) = a LO (t)(Q(t) + n 1 (t)) (D.7)

v F /3D ≈ 700 meV for D = 1 nm) or metallic. Nevertheless, when one starts to consider the overlap between p z orbital, the Dirac cones can be slightly shifted in the

Baker-Hausdorff-Campbell formula: HSW = H0 + Ht + i[S, H0] + i[S, Ht] -1 2 [S, [S, H0]] + O(t 3 )

This reformulation is valid only if tunneling events are perturbative, so that the discrete orbitals still form an orthogonal basis on which one can expand the field operator.

When reaching the ultra-strong regime coupling (g ωc) the counter rotating terms σ+â † , and σ-â are not negligible anymore.

Figure 3.8:The stapler: a. The system is placed on top of a air-cushion table to mechanically decoupled it from its environment. b. Zoom-in on the inside of the main chamber. Here the cantilever chip and the circuit chip are in the stapling configuration.

• Do not forget to close all the bottles.

Acknowledgements

Schematic of the double quantum dot electronic states, showing the concept of the spin-photon coupling. When the energy difference between the two dots is such that one electron is mostly localized on one dot but slightly spread over the other one, the DQD-dipole modulation due to cavity photons allows to manipulate this electron spin.

which produces ultraclean double quantum dot devices with near-ideal spectra. We use the circuit QED platform to perform a microwave spectroscopy of the spin transition as well as time domain experiments. We observe the characteristic dispersion of the spin transition of our spin qubit. When the qubit state are tuned to be almost pure spin states (at large detuning, see figure 5.1-a), the measured decoherence rate is found to be as low as 250kHz. Strikingly, such a figure of merit is more than 100 times better than in previous work on carbon nanotubes [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF][START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF] and compares favorably to the very recent values reported for Si based devices in a circuit QED environment [START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Mi | A coherent spinphoton interface in silicon[END_REF]. From the gate dependence of the decoherence rate, we show that the charge noise is the main source of decoherence for the spin at low detuning, but that it can be made negligible compare to nuclear spin induced noise at large detuning.

Working principle

The principle of our spin photon coupling relies on two non-collinear Zeeman fields on each quantum dots (see figure 5.1-b and 5.2-c) in a double quantum dot, originating from zig-zag shaped ferromagnetic contacts. These non-collinear Zeeman fields can be achieved by interface exchange fields [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF] or by stray magnetic fields [START_REF] Hu | Strong coupling of a spin qubit to a superconducting stripline cavity[END_REF][START_REF] Benito | Input-output theory for spin-photon coupling in Si double quantum dots[END_REF] which both give the same hamiltonian. In our case, the interface exchange fields a priori dominate [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. In the adiabatic regime, if an electronic spin is located on the left dot, it aligns along the left spin quantization axis whereas if it is located on the right dot,

Conclusion and perspectives

Summary The experiments presented in this thesis are performed in carbon nanotube based quantum dot circuits coupled to a microwave cavity. Such circuit-QED platform has proven to be very powerful to probe the internal dynamics of quantum dot circuits.

While the contact electrodes in quantum dot circuits are necessary to perform quantum transport, the choice of a specific contact metal is also instrumental to engineer specific electronic states in the circuit. In chapter 4, we presented results on a Cooper pair splitter experiment, in which a superconducting contact is used as a resource to generate non-local entangled electronic states in a double quantum dot. We show that the bonding and anti-bonding states in the double quantum dot are dressed by Cooper pair injection, giving rise to a renormalized tunneling rate between the two dots. This effect has been exploited to engineer a new coupling scheme between cavity photons and electronic transitions, and enabled us to reach the strong charge-photon coupling in this quantum-dot circuit [START_REF] Bruhat | Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs[END_REF]. A nanofabrication technique has also been developed during this thesis, allowing to integrate pristine carbon nanotubes in a circuit-QED device architecture. This technique produces quantum dot circuit with a much higher electrical tunability. In the second experiment we make use of this integration technique to create a ferromagnetic spin-qubit displaying unprecedented long coherence time of a spin transition in carbon nanotubes.

Perspectives

First cleaning

The wafer used is a undoped high resistivity silicon (10 kΩ.cm) covered with 500 nm of silicon oxide, and it is dinced into 10mm×10mm chips. The first step is to remove the PMMA resist layer used to protect the wafer during the dicing.

• 20 minutes: Acetone at 50 • C (all the chips inside the same becher).

• 30 secondes: Shaking in acetone.

• 1 minutes: (Chip one by one) Acetone at 50 • C + sonic-bath (force 9). Use a second acetone becher for this step.

• 30 secondes: (Chip one by one) Move the acetone becher from the sonic-bath and vigorously shake the chip in order to remove the fragment of silicon that sonic-bath may have created.

• 30 secondes: Vigorous shaking in IPA.

140

A.1.2 Gold crosses and device name

A.1.2.1 PMMA 500 spin coating

• Launch the pumping of the chip and remove the possible dirt from your chip using the nitrogen blower.

• Depose 3 drops of PMMA 550 resist, then launch the 4000-4000-30 program (4000 turns/s 2 , 4000 turn/s, 30 s).

• Bake the chip at 185 • C for 3 minutes.

• check at the microscope if there are bubbles, or dirt on the chip. If so, you should start over again the first cleaning step and the PMMA spin coating.

A.1.2.2 Two-current e-beam lithography SEM paramters:

• Aperture: 7.5 µm and 120 µm

• Acceleration voltage: 20 kV Procedure:

• Measure the currents for the 7.5 µm-aperture and the 120 µm-aperture.

• Localize the origin and the angle of the chip.

• Make a contamination dot at the center of the chip (with the 7.5 µm-aperture with a zoom = x200k): Focus on the dot + WF alignment at 5 µm then at 1 µm.

• Save the column parameter for the 7.5 µm-aperture.

• Call the 120 µm-aperture column parameter, and the WF parameter for the 120 µm-aperture.

• Zoom-out up to x10k-20k, Focus on the dot + center the contamination by adjusting the beam shift (DO NOT TOUCH THE JOYSTICK).

• Launch the WF alignment procedure at 5 µm.

• Save the column parameter for the 120 µm-aperture.

• Scan the following position list.

Position list:

• Loading of the 7.5 µm-aperture parameters.

• Loading of the writefield parameters for the 7.5 µm-aperture (zoom, shift, rotation).

• Exposition of the layer 10 (pre-contact + small crosses for the layer 63).

• Loading of the 120 µm-aperture parameters.

• Loading of the writefield parameters for the 120 µm-aperture (zoom, shift, rotation)

• Exposition of the layer 11 (name of the chip + large crosses for the 3-point alignment)

Exposure properties:

• Structure: Wafer

• Exposed layer: 10 or 11

• Working Area: "Boundaries"

• Position: U = 0.05 mm, V = 0.05 mm Exposure parameters:

• Area step sizes: U = V = 20 nm

Reveal and evaporation

• 1 min 10 sec: Reveal in MIBK/IPA solution (1:MIBK, 3:IPA).

• 30 secondes: Rinse in IPA then dry. Check if the lithography went well at the optical microscope.

• 10 secondes Stripping.

• Evaporate 5 nm of Ti, and 45 nm of Au.

A.1.2.4 Lift-off

• 20 minutes: Acetone at 50 • C.

• 1 minutes: Acetone at 50 • C + pipette, until there is no visible remaining of metal on the chip.

• 1 minutes: (Chips one by one) Acetone at 50 • C + sonic-bath (force 9).

• 30 secondes: Vigorous shaking in acetone.

• 30 secondes: Vigorous shaking in IPA.

A.1.3 Cavity

A.1.3.1 Niobium evaporation

• 3 minutes: Stripping.

• Load the chip in the load-lock of the Plassys evaporator immediately after the stripping.

• Launch the cool-down of the evaporator cryo-panels by filling them with liquid nitrogen ( ≈ 30 minutes).

• Evaporate 100 nm of Nb at 10 Å/sec.

A.1.3.2 Laser lithography

• Launch the pumping of the chip and remove the possible dirt from your chip using the nitrogen blower.

• Depose 3 drops of S1805 resist, then launch the program 10.

• Bake the chip at 115 • C for 1 minutes.

• check at the microscope if there are bubbles, or dirt on the chip. If so, you should start over again the first cleaning step and the PMMA spin coating.

• Laser lithography parameters: alignment procedure A+B, lens: 5, filter: 3 %, gain: 11 (203 mJ.cm -2 ), D-step: 1, pos: 1 mm.s -1 .

• Reveal: 1 min in MF319 + 1 min in DI H 2 O + blow dry.

A.1.3.3 Niobium etching

• Load the chips, then when the penning pressure is below 15 nbars, start the "test"

program and purge the SF 6 line.

• Launch the "Nb" program, and monitor the progress using the reflectometry measurement. First the reflectivity slowly increases, then it decreases abruptly and reach a minimum signaling that the Nb has been etched away.

• Do 20 secondes of overetch after that reflectivity minimum.

A.1.3.4 Cleaning

• 20 minutes: Acetone at 50 • C.

• 30 secondes: Vigorous shaking in acetone.

• 1 minutes: (Chips one by one) Acetone at 50 • C + sonic-bath (force 9).

• 30 secondes: Vigorous shaking in acetone.

• 30 secondes: Vigorous shaking in IPA.

• 2 minutes: Stripping. • Launch the pumping of the chip, and remove the possible dirt from your chip using the nitrogen blower.

• Depose 3 drops of diluted MMA resist (from the HQC bottle with "110 nm" written on it), then launch the program number 10 (called "jpc").

• Bake the chip at 185 • C for 3 minutes.

• Cool-down the chip for 20 secondes on the metal of the table.

• Depose 3 drops of diluted PMMA resist (from the HQC bottle with "40 nm" written on it), then launch the 4000-4000-30 program.

• Bake the chip at 185 • C for 3 minutes.

• check at the microscope if there are bubbles, or dirt on the chip. If so, you should clean again the chip and spin-coat the bi-layer MMA/PMMA.

A.1.4.2 Lithography

The dose used for the gate lithography can be quite critical, so it is better to perform a dose test prior to the real lithography (if there has not been one in the past few weeks).

With the current design (gate width = 60 nm, gate spacing = 140 nm, cavity gate width = 150 nm), the good dose has been found to fluctuate between 320 and 360.

• Measure current with the 7.5 µm aperture.

• Origin/Angle alignment.

• Pre-focus on a defect close to the edge of the chip.

• 3-point alignment on the large crosses (position with respect to the center of the chip: U = ± 450 µm, V = ± 550 µm).

• As the focus can vary from positions that are few millimeters away, it is preferable to adjust the focus on a contamination dot each time you go to an other region (basically this means to do 4 contamination dots for one chip). The dot can be done 100-200 µm away from the quantum dot circuit region, in a niobium region.

• After the focus on the dot, you can launch the lithography (Structure: Wafer, Exposed layer: 63 and 4, Working Area: one of the six predefined 100µm×100µm, Area step sizes: U = V = 20 nm).

A.1.4.3 Reveal, evaporation, and lift-off

• 18 sec: Reveal in MIBK/IPA solution (1:MIBK, 3:IPA).

• 30 secondes: Rinse in IPA then dry. Check if the lithography went well at the optical microscope.

• 10 secondes Stripping, then load in the load-lock right after, and let it pump overnight.

• Evaporate 5 nm of Ti, and 50 nm of Al, then 10 minutes of oxidation at 1 mbar of O 2 in the SAS.

• (×2) Evaporate 1.5 nm of Al, then 10 minutes of oxidation at 1 mbar of O 2 in the load-lock.

• Use the same lift-off procedure than the one for the gold crosses.

A.1.5 Contact electrodes

This step is very similar to the one of the gate electrodes, so I will not go into the details.

• Spin-coating of PMMA 500 resist.

• E-beam lithography: 7.5 µm-aperture, dose: 360, layer: 6.

• Reveal: 1 min MIBK/IPA (1/3) + 30 sec IPA + 10 sec stripping.

• Evaporation: 211 nm of Ti + WAIT 14 min + 20 nm of PdNi (use the film parameters of Pd and not the one of PdNi) + 4 nm of Pd.

• Use the same lift-off procedure than the one for the gold crosses.

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A.1.6 Trenches

• Launch the pumping of the chip, and remove dirt using the nitrogen blower.

• Depose 3 drops of AZ5214, then launch the 4000-4000-30 program.

• Bake the chip at 115 • C for 1 min 30 sec.

• Check at the microscope if there are bubbles, or dirt on the chip.

• Laser lithography parameters: alignment procedure A+B, lens: 5, filter: 3 %, gain: 13.6 (250 mJ.cm -2 ), D-step: 1, pos: 1 mm.s -1 .

• Reveal: 40 sec in AZ726MIF + 1 min in DI H 2 O + blow dry.

• SiO 2 etching: program "DavidSiO 2 " (CHF 3 only), at the end of the SiO 2 etching the reflectivity starts to be constant at the summit of an oscillation (last approximately 25 min) + 2 minutes of overetch.

• Si etching: program "Si100W O 2 80mT (SF 6 + O 2 ) for 7 minutes.

• Cleaving of the chip using the scriber, with the chip still covered with AZ5214 resist.

• Cleaning: Same procedure than the first cleaning + 2 min stripping + 1 min of vigorous shaking in IPA.

A.2 Carbon nanotube growth A.2.1 Low flow recipe

This recipe is largely inspired from ref. [START_REF] Zhong | Ultralow Feeding Gas Flow Guiding Growth of Large-Scale Horizontally Aligned Single-Walled Carbon Nanotube Arrays[END_REF], and is supposed to provide longer and oriented carbon nanotubes. In the oven of the ENS, millimeter-long CNT has been obtained (see chapter 3) only when the catalyst was deposited on the edge of the chip.

• Be careful to use the "CNT std" labeled quartz tube.

• Load the sample into the quartz tube by the right extremity, and connect it to the "CNT" output circuit.

A.2.2 high flow (standard) recipe

This growth recipe has been developed in Basel by J. Furer [START_REF] Furer | Growth of Single-Wall Carbon Nanotubes by Chemical Vapor Deposition for Electrical Devices[END_REF]. The growth should be done as close as possible to the stapling step (ideally the day before).

• Be careful to use the "CNT std" labeled quartz tube.

• Load the sample into the quartz tube by the right extremity, and connect it to the

• open the bottle of Argon, Methane, and Dihydrogene, and note the pressures before and after the pressure regulator in the notebook. After the regulator, all pressure should be set at 1 bar.

• Launch the monitoring program (control f our2), and note the three measured flows (which correspond to the offset value of the flowmeters).

• Purge of lines ( 5 The hamiltonian of a double dot with a central superconducting lead can be written as

, where H 0 = H DQD + H S is the sum of the individual hamiltonians and H S-DQD describes the coupling between them. Let us recall the explicit expressions for each term:

2)

Projecting the effective hamiltonian on the {|+ , |-}, we get:

where the expressions for the perturbative elements close to the degeneracy line between (0,1)/(1,0) charge states are given in the main text.

B.2 Experimental details B.2.1 Sample fabrication and measurement setup

The sample fabrication process is the following. A 150nm thick Nb film is first evaporated on a thermal silicon oxide (500nm)/high resistivity (10kΩ.cm) silicon substrate at rate of 1nm/s and a pressure of 10 -9 mbar. The cavity is made subsequently using photolithography combined with reactive ion etching (SF6 process). Carbon nanotubes are grown with Chemical Vapor Deposition technique (CVD) at about 900 • C using a methane process on a separate quartz substrate and stamped inside the cavity. The nanotubes are then localized. The fork top gate oxide is made using 3 evaporation steps of Al (2nm) followed each by an oxidation of 10 min under an O 2 pressure of 1 mbar.

The Alox is covered by a Al(40nm)/Pd(20nm) layer. The nanotube is contacted with a central Pd(4nm)/Al(80nm) finger and two Pd(70nm) outer electrodes.

The DC measurements are carried out using standard lock-in detection techniques with a modulation frequency of 137 Hz and an amplitude of 10 µV . The base temperature of the experiment is 18 mK. The microwave measurements are carried out using room temperature microwave amplifiers and a cryogenic amplifier (noise temperature about 5K) with a total gain of about 90 dB. We measure both quadratures of the transmitted microwave signal using an I-Q mixer and low frequency modulation at 2.7 kHz.

B.2.2 Supplementary data: double dot stability diagram

B.2.3 Supplementary data: hybridization of cavity-dot system

B.3 Vacuum Rabi splitting power dependence modelling

In this section we present the modelling of the vacuum Rabi splittings shown in figure 4.5a and 4.5c of the main text. In both figures, a misalignement is visible between the center of the Rabi splitting peaks and the recovered coherent state at high power, whereas the spectral weight of the two peaks at low power remains the same. Such a misalignement can be accounted for by a two transition structure depicted in figure 4.5b.

This contrasts with the case of a single electronic transition involved, where this shift would automatically be accompanied by asymmetric spectral weights on the two peaks. Table B.1: Summary of parameters used in the two transitions modelling (in MHz). κ, f c are respectively the decay rate and the resonance frequency of the cavity. g i , f i , Γ ϕ,i , and γ i are respectively the coupling strength, the frequency, the dephasing rate, and the decay rate of transition i ∈ {K, K }.

We can account for this asymmetric splitting with respect to the coherent state peak at the cavity frequency recovered at high power, while keeping similar spectral weight on each peak, with the two transition scheme depicted in figure 4 • Rinse the catalyst bottle with IPA in the sonic-bath for 10 minutes.

• With a buret, take 60mL of IPA.

• Weigh 78mg of F e(N O 3 ) 3 -H 2 O in a handmade aluminum cup using the weighing scale of Vincent Croquette. Put the F e(N O 3 ) 3 -H 2 O in the catalyst bottle, and rinse the aluminum cup with a small portion of the 60mL of IPA into the catalyst bottle.

• Do the same with 15.8mg of M oO 2 , and 64mg of Al 2 O 3 .

• Put the remaining of IPA in the catalyst bottle.

C.1.2 Catalyst deposition

• Put the catalyst solution in the sonic-bath for 1 hour.

• Let the catalyst solution decant for 45 minutes. unscrew the cover before this step in order to shake the solution as less as possible once the decantation is over.

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• Fill a becher with IPA. This becher will be used for rinsing the cantilever chip after the catalyst deposition, so it is better to use a dedicated becher that is polluted with catalyst nanoparticles.

• With a pipette, take a bit of the catalyst solution from the surface. Maintain the cantilever chip on the clean room paper using the tweezers (with the cantilever surface on the top), and put 2 or 3 drops on the chip.

• Quickly after the deposition, rinse the cantilever chip with IPA for 10 seconds in the dedicated becher. This step allows to obtain a more homogeneous catalyst deposition, in particular, catalyst particles reach more easily the tips of the cantilever.

• Dry the cantilever chip using the pressurized nitrogen gun, while maintaining it on the clean room paper with tweezers.

• Observe with the optical microscope in the "Dark field" mode the catalyst density at the tips of the cantilevers. If the catalyst density is too low, the deposition of few drops of catalyst can be repeated using the same technique.

C.1.3 CNT growth

The growth recipe used is the so-called "standard" recipe:

• In order to limit the exposition of the CNT to the air, it is important to schedule a SEM time slot for the localization of the CNT the same day of the growth, and then store the CNT gelpak in the stapler chamber.

C.1.4 CNT localization

To localize the CNTs suspended between cantilevers, an easy way is to use SEM (even if the common belief is that it creates amorphous carbon around the CNT). An undamaged chip possesses 48 cantilevers. In case of broken cantilevers, one should take a picture of all the cantilever to avoid mistakes in the numbering of the CNTs. One can do the focus/stigmatism tuning on the catalyst nanoparticles at the tips of the cantilevers, which is the area of interest. The column parameters are:

• Magnification: ×2000

• EHT: 2kV

• aperture size: 30µm

• High brightness: the cantilevers have to be completely white to hope seeing suspended CNTs in between.

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C.2 Preparation of the stapling C.2.1 Fixing of the sample holder to the rotating arm

The positioning of the sample holder can be tricky. First, it has to be low enough such that the cantilever holder (see figure C.3) fit in between the sample holder and the window of the cover. Secondly, it has to be high enough to be able to do the focus with the microscope on the circuit chip surface through the window of the stapler's cover.

It is possible that those two conditions are not fulfilled (depending on the geometry of the sample holder), in this case, one can envision to rotate the sample holder when the cantilever chip is brought over the circuit.

One also has to check the possible shunts in the circuit chip by testing individually each contacts and gates while leaving the others grounded (this can be done using the cutting electrical setup (see figure C.7).

C.2.2 Fixing of the cantilever chip and tuning its angle

Choose the cantilever chip with most of the CNTs fulfilling the previous criteria.

Fixing the cantilever chip:

• Unscrew the piezo-motor block (see figure C.3), and rotate it so that the cantilever chip holder (see figure C.3) is pointing up.

• Glue the chosen cantilever on the cantilever chip holder with PMMA (you can use a toothpick to drop the PMMA off), and wait 5 minutes for the PMMA to dry. During this waiting time, one can close the vacuum chamber with the cover (without pumping) to protect the CNTs.

Pumping

• Before to launch the pumping of the stapling chamber, be sure that the cantilever chip is far from any object in the chamber (more than 1 centimetre), since at the beginning of the pumping the accordion is contracting and this make the cantilever chip to translate by few millimetres.

• Open the valve to the primary pump, V1 (see figure C.4).

• Wait for the pressure to be below 5 × 10

C.3 Stapling

• Wait for the pressure to be in the range of 10 -7 mbar.

• Align the pair of cantilevers with the desired CNT with respect to the circuit lines using the micromanipulator for the coarse alignment, and the piezo-motors for the fine one (the range of each piezo-motor is ±1mm). At this stage, one has to be very careful not to touch the stapler table. It is quite easy to break all the cantilevers in a single shot... • once the CNT has been cut on the two external sections, one can check if the CNT is still connected to the source and drain contacts (it happened that the cutting procedure also cut the central portion...).

• Now one can safely raise the cantilever using the piezo-motor, while measuring the current in the central portion (but in principle it should not be affected by the raising of the cantilever chip). Retract the cantilever chip using the micromanipulator and put it back in the accordion.

• Measure the resistance of the central portion at V SD = 10mV (including the voltage divider). To characterize the quality of the contacts, it is important to always measure the resistance at the same bias since the current does not have a linear behavior with respect to the bias voltage.

• If the resistance of the central portion is too high (few M Ω), one can try to decrease it by driving a large current through the central portion by means of the cutting setup. One can launch a monitoring ... measurement on Exopy to follow the evolution of the current with time, while increasing step by step the bias voltage.

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• Unscrew the sample holder from the rotary arm.

• Screw it to the cold arm of the cryostat.

• Connect the DC lines of the sample holder to the lines of the stapler, then disconnect them from the grounding cable.

MΩ

To cryostat ground