Initially spin-orbit coupling was thought to be a purely relativistic quantum effect described in the context of Dirac equation. However L. Thomas found an explanation without relying on the Dirac equation which "pushed Pauli to admit that relativistic quantum theory is not the only way to handle the spin

Introduction

This manuscript presents the research activity I have conducted during the last seven years. Since my PhD my research activity focused either directly or indirectly on the manipulation the electron spin. The spin of the electron (and other particles) is a purely quantum mechanical property arising in the Dirac equation for relativistic particles which Pauli extended to non-relativistic quantum theory. By isolating a spin 1/2, it is possible to realize close to a true and ideal two-level system which is the simplest quantum system to manipulate and investigate. It is probably the reason why historically many experimental observations of the spin where made before the theory was drawn, including the seminal Stern-Gerlach experiment. Addressed by the magnetic field, ensembles of electron and atom spins were exploited as a powerful resource to characterize and explore condensed matter, with nuclear magnetic resonance (NMR) and electron spin resonance (ESR). The former lead to one of the key technologies of the now called first quantum revolution, namely magnetic resonance imaging (MRI).

With the advent of sophisticated micro-fabrication techniques in research laboratories in the 80's, coming from the semiconductor industry, the field of mesoscopic physics could emerge. It made it possible to isolate and manipulate quantum phenomena and single quantum excitations in macroscopic size circuits (to be understood as circuits containing a huge number of atoms, comparable to condensed matter systems and to be compared to atomic physics). It was quickly realized that the quantum two-level system -which by essence is the definition of a qubit-formed by a single electron spin isolated in a quantum dot could be an ideal system to manipulate quantum information. Any quantum system encodes quantum information but spins are particularly simple systems to address and manipulate. Beyond being individually used as qubits, the spin texturing of electrons in conductors can lead, in particular in low dimensional systems, to exotic states of matter which can also encode quantum information, topologically protected from decoherence. To isolate individual spins or to shape the spin texture of electrons in low dimensional conductors, one needs to exploit the combination of various materials presenting different electronic degrees of freedom and different dimensions. This is my definition of hybrid -quantum-circuits.

During my postdoctoral stay in RIKEN, Japan, I have worked on the development of spin qubits devices for quantum information manipulation. These are made in quantum dots, formed in a semiconducting heterostructure hosting a two-dimensional electron gas (2DEG), by electrostatic gating with normal metal electrodes. A cobalt micromagnet generates a local slanted magnetic field which induces a spin-electric-coupling (SEC), allowing for electrically controlling individual spins. While very attractive qubit candidates for the long coherence and lifetime of the electron spin as well as for the scalability potential offered by the semiconductor industry, these kind of spin qubits have long suffered from the presence of fluctuating nuclear spins in the host material, which induce strong dephasing. Fighting against this nuclear spin noise was thus a major task to undertake. With my colleagues in RIKEN, we exploited the slow dynamics of the nuclear spin bath, which I previously investigated in detail, to perform real time Hamiltonian estimation with a spin qubit in GaAs, which host 100% nuclear spins. We also developed spin qubit devices in host materials containing less (natural Si) to almost no (purified 28 Si) nuclear spins, ultimately demonstrating the most promising spin qubit so far. In all three platforms, we reached the fault tolerance threshold of quantum error correction. Carbon is another promising host material for spin qubits because it contains even fewer nuclear spins than silicon and can also be isotopically purified. In particular carbon nanotubes are promising host materials, however the promises have not been fulfilled yet, mainly because of the lack of electronically clean carbon nanotubes compared to GaAs or Si wafers, the difference coming from decades of industrial research and development on semiconductors. After developing an ultra-clean carbon nanotube platform at the ENS, we demonstrated highly coherent spin states in a microwave cavity competing with silicon devices in similar architectures.

Besides continuing an activity on spin qubits in carbon nanotubes at the ENS, I am also investigating the realization of exotic topological states of matter in low dimensional conductors. The most sought for of these excitations are Majorana zero modes, which are expected to be their own anti-particle and to possess non-abelian exchange statistics that could be used to encode information through topologically protected operations known as braiding. The intense worldwide research activity in this field is mainly focusing on using semiconducting nanowires having a strong intrinsic spin-orbit interaction with superconducting correlations induced by an epitaxial superconducting layer, the whole being subject to a large external magnetic field. These are the necessary ingredients to realize the topological phase in which the Majorana zero modes can emerge. Rather, I am adopting a different strategy in which a strong spin-orbit interaction is synthetically induced in the low dimensional conductor, concomitantly to an autonomous local magnetic field, with the use of a patterned magnetic texture. With this Hamiltonian engineering, we demonstrated a large spin-orbit interaction in a carbon nanotube, stronger than in semiconducting nanowires and we observed a possible signature of Majorana zero modes at zero external magnetic field.

The manuscript is organized as follows. In chapter 1, I briefly discuss the theoretical aspects of spin-electric-coupling using local magnets as well as spin texturing using a magnetic texture. We will see that all approaches essentially reduce to implementing a synthetic or artificial spin-orbit interaction. In chapter 2, I discuss spin qubits in nuclear spins environment. The first section will theoretically describe the hyperfine interaction leading to an equation for nuclear spin dephasing that is usually missing in the literature. Then the chapter will summarize my work on spin qubits in GaAs, SiGe, 28 Si and carbon nanotube as published in Refs. [START_REF] Delbecq | Quantum Dephasing in a Gated GaAs Triple Quantum Dot due to Nonergodic Noise[END_REF][START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF][START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF][START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF][START_REF] Cubaynes | Highly coherent spin states in carbon nanotubes coupled to cavity photons[END_REF]. Finally in chapter 3 I will discuss the experimental results on the synthetic spin-orbit interaction in carbon nanotubes published in Ref. [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF] as well as some preliminary unpublished results on the sensing of the induced spin texture with microwave photons. Both chapters 2 and 3 conclude with a perspective section in which I present my research projects for the coming years in these two activities.

Chapter 1

Controlled spin-orbit interaction

Spin or intrinsic angular momentum is a purely quantum property that is particularly attractive to encode quantum information. It generally induces a "spin" magnetic dipole moment which makes it addressable by a magnetic field. An external constant magnetic field is generally used to control the spectrum of electronic systems, lifting the spin degeneracy, on time scales much longer than the individual spins dynamics. Radio frequency external magnetic fields work well for manipulating ensemble of spins as in NMR or ESR [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. Local magnetic fields from ferromagnets in combination with external magnetic fields or local electric fields can be used to generate and control spin currents [START_REF] Žutić | Spintronics: Fundamentals and applications[END_REF]. When it comes to dynamically controlling individual spins, it becomes more challenging as the direct coupling to the magnetic field of a single spin is very weak (and typically grows as the square root of the number of spins). Sticking to a magnetic field drive, it is thus necessary to apply locally, close to the isolated electron spin, large currents of the order of several mA at the gigahertz frequency, which in turn heat up the device that is at a typical sub kelvin temperature. Despite these difficulties, the first demonstration of the manipulation of a single electron spin in a quantum dot was realized using this technique in GaAs [START_REF] Koppens | Driven coherent oscillations of a single electron spin in a quantum dot[END_REF]. The major drawback of this technique (heating due to large currents) comes from the fact that it is needed to reach high Rabi frequencies in GaAs spin qubits to observe the estimated qubit state evolution as oscillations, because of the large dephasing due to nuclear spins (this will be extensively detailed in chapter 2). In isotopically purified 28 Si where the dephasing time of spin qubits is orders of magnitude larger than in GaAs, much smaller Rabi frequencies are enough, which explains why this technique was still successfully employed until recently by some groups [START_REF] Veldhorst | An addressable quantum dot qubit with fault-tolerant control-fidelity[END_REF][START_REF] Veldhorst | A two-qubit logic gate in silicon[END_REF][START_REF] Petit | Universal quantum logic in hot silicon qubits[END_REF] although it might not be good enough any more for recent architecture development at the Kelvin range [START_REF] Yang | Operation of a silicon quantum processor unit cell above one kelvin[END_REF]. Standard ESR drive was not suitable at the time for developing spin qubits in GaAs (which was then the main working horse) and it does not provide an easy scheme to develop scalability.

The idea then arose that a specifically designed magnetic field, local to the electron spin, could be used to control it electrically through a so called spin-electric-coupling (SEC) [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF], lifting the hindrances of the pure magnetic control. As we will discuss in this chapter, it turns out that such a spin-electric-coupling is equivalent to inducing an artificial spin-orbit coupling or interaction (SOC or SOI). Spin-orbit coupling is a relativistic effect initially described in the context of atomic physics with electrons moving in the electric field of the nucleus 1 . It can also arise in crystals with broken space inversion symmetry, thus becoming 1.1. ELECTRICAL CONTROL OF INDIVIDUAL SPINS a material property. It shapes the electronic band structure (see Figure 1.1), lifting spin degeneracy at zero external magnetic field, except at zero momentum. Adding an external magnetic field (external or intrinsic, the latter usually being the case with topological insulators) can lead (in the case of Rashba spin-orbit) to the opening of an helical gap which results in spin-momentum locking. Such helical gaps are highly sought for because they are a key signature and ingredient for the realization of non trivial topological phases of matter. The number of materials with the right and strong enough intrinsic spin-orbit interaction is limited and it is therefore attractive to synthetically shape it in other materials where it is weak or absent. Spin-orbit coupling being equivalent to an oscillating magnetic field [START_REF] Braunecker | Spin-selective Peierls transition in interacting one-dimensional conductors with spin-orbit interaction[END_REF], it can be synthetically realized using a strong and inhomogeneous, on short distances, electric field (which is very challenging) or by using a spatially oscillating local magnetic field which is the approach that we chose and that we will discuss in this chapter. 1.1 Electrical control of individual spins 1.1.1 Electron spin in a slanted magnetic field

In this subsection, we will discuss single spin (spin-1/2) qubits defined in a single quantum dot. The discussion is based on Ref. [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF] where technical details can be found.

An efficient spin-electric-coupling scheme was first proposed in 2006 based on a local slanting, or slanted, magnetic field [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF]. It was experimentally realized in 2008 in the group of S. Tarucha with the electrical control of a single electron spin-1/2 in a GaAs quantum dot [START_REF] Pioro-Ladrière | Electrically driven single-electron spin resonance in a slanting Zeeman field[END_REF]. The slanted magnetic field is created by a ferromagnet, usually in cobalt, deposited on top of the spin qubit quantum dot device. Its working principle is presented in Figure 1.2(a) and a design for optimal field gradients [START_REF] Yoneda | Robust micromagnet design for fast electrical manipulations of single spins in quantum dots[END_REF] is shown in Figure 1.2(b). An external magnetic field B ext defines the spin quantization axis and the longitudinal direction z. The transverse component of the micromagnet field, along the x direction, is designed to have a constant gradient along z which is referred to as the transverse slanted properly". The interested reader can find a detailed and clear demonstration of Thomas' reasoning in appendix A of Ref. [START_REF] Combescot | Spin-orbit coupling: Atom versus semiconductor crystal[END_REF] CHAPTER 1. CONTROLLED SPIN-ORBIT INTERACTION term b t = (∇ • B MM ) x with the subscript MM referring to micromagnet. Such a constant gradient can be realized by the micromagnet design presented in Figure 1.2(b), whose simulated magnetic field is shown in Figure 1. 3(a). Around the center of the micromagnet with respect to the z axis, B x MM noted b t in further is linear. Now, considering this term only, the magnetic field at the quantum dot position is B = b t zu x + B ext u z with u x and u z unit vectors along x and z respectively. The Hamiltonian of the system writes

H = p2 z 2m + V (ẑ) - 1 2 gµ B (B ext σz + b t ẑσ x ) , (1.1)
with V the confinement potential assumed to be symmetric for simplicity, g the electron g-factor, µ B the Bohr magneton and σ the Pauli matrices for the spin. We see that the slanted term of the Hamiltonian couples position and spin and is therefore suggestive of a spin-orbit coupling. It is the general term that we will encounter in this chapter for artificial spin-orbit interaction. However, it is not similar to any spin-orbit coupling Hamiltonian like Rashba or Dresselhaus spin-orbit terms which explicitly couple spin and momentum. It is needed to do perturbation theory or (not exclusive) to restrict the Hilbert space to reveal the spin-orbit character of this term. Here second order perturbation with respect to the slanting field energy E SL = -gµ B b t L, with L the potential confinement characteristics length, yields a restricted two-state Hilbert space defining the spin qubit. By periodically displacing the quantum dot in space along the z direction using an electric field, the confined electron spin will experience an oscillating transverse magnetic field. The time dependent perturbation Hamiltonian H ac (t) = eV 0 f (t)ẑ/L is an odd function of space and is spin independent, with f (t) usually a cosine function. It therefore couples off-diagonal terms, which are in this case the two spin states of the qubit, leading to the effective Hamiltonian

H ef f = 1 2 z σz + 1 2 x f (t)σ x , (1.2)
which is the Hamiltonian of ESR that is now called within this scheme electron dipole spin resonance (EDSR) [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF]. A key aspect here is that due to second order perturbation theory, the two states defining the qubit are combinations of the quantum dot orbitals. Therefore the reduced Hamiltonian (1.2) effectively hybridizes spin and orbital degrees of freedom and EDSR can be interpreted as an artificial spin-orbit coupling which adds to any intrinsic spin-orbit coupling.

To complete the picture of the micromagnet inhomogeneous magnetic field attractive features, we also note that there is a finite gradient of the z component of the field along the y direction. This gradient is provided by the asymmetry of the magnet with respect to the z axis through the "bridge" that connects its two large parts. This results in a y dependent B z MM (y) which allows for selectively addressing several spin qubits located at various positions along the y axis. Indeed, to first order, z = 1 2 gµ B [B ext + B z MM (y)] and EDSR is performed by applying f (t) = cos(ω EDSR t) with ω EDSR = z / . It is therefore possible to control selectively several spin qubits with the same micromagnet which has already been successfully done with 3 and 4 spin qubits [START_REF] Noiri | Coherent electron-spinresonance manipulation of three individual spins in a triple quantum dot[END_REF][START_REF] Ito | Four single-spin Rabi oscillations in a quadruple quantum dot[END_REF]. This makes the micromagnet a versatile and instrumental element of spin qubit devices in semiconducting materials which is at the heart of the discussion of chapter 2.

Two-site spin-orbit interaction

In this subsection, we will discuss spin-1/2 qubits defined in a double quantum dot. The discussion is mostly based on Refs. [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF][START_REF] Hu | Strong coupling of a spin qubit to a superconducting stripline cavity[END_REF] where technical details can be found. In the previous subsection, we have seen how a periodic displacement in continuous coordinates of an electron spin in a slanted magnetic field induces EDSR through an artificial spin-orbit interaction. It is also possible to reduce the continuous coordinates to discrete sites, down to two, using a double quantum dot. This scheme was first proposed in 2010 for a double quantum dot system with non-collinear ferromagnetic electrodes, using carbon nanotube as an example [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF], and then in 2012 for a double quantum dot in a semiconducting heterostructure 2DEG with a micromagnet similar to the one discussed previously [START_REF] Hu | Strong coupling of a spin qubit to a superconducting stripline cavity[END_REF]. Both schemes are identical and result in the same Hamiltonian and artificial two-site spin-orbit interaction. These schemes have been proposed in the context of hybrid (or mesoscopic) circuit quantum electrodynamics (cQED) to achieve spin-photon coupling and were successfully implemented first in a carbon nanotube device with the demonstration of coherent spin-photon coupling [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF] and then in silicon devices with the demonstration of the strong spin-photon coupling [START_REF] Mi | A coherent spin-photon interface in silicon[END_REF][START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF].

The reasons behind using a double quantum dot are twofold : 1) Since the two dots are separated by few hundreds nanometers (this is much larger than the displacement of the single electron in the previous section, which is of the order of the quantum dot confinement length ∼ 20 nm), there is a large (mesoscopic) electric dipole between the two 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. 2) A double quantum dot is a closed system and it is possible to finely tune the internal transition between the levels of the two dots to match the cavity frequency. This results in the cavity photons driving the transition that controls the localization of the electron on one dot or the other, effectively activating the large dipole. This working principle is depicted in Figure 1.4(a). In the proposal with non-collinear ferromagnetic electrodes, the quantization axis in each quantum dot is set by the local non-collinear magnetic fields (induced by exchange interaction, see Figure 1.4(b), rather than through the stray field), B L and B R with L, R referring to the left and right dots and B L • B R = cos(θ) with θ the angle between the electrodes. In the proposal with a micromagnet (see Figure 1.4(c,d)), the two specific locations of the quantum dot in space also set two discrete quantization axis for the two quantum dots with

B α = B z MM (x α )u z +B x MM (x α )u x with α in { L, R
} and x α the dot position along the x axis. In the ideal situation of a double quantum dot system exactly in the center of the micromagnet, we have

B z MM (x L ) = B z MM (x R ) and B x MM (x L ) = -B x MM (x R
) giving a similar non-collinear configuration with θ = arctan(2B x MM /B z MM ). With both approaches, the term coupling spin and position discussed in the previous section transforms from the continuous to discrete coordinates by projecting the system Hamiltonian onto the two orbitals, on the left and right dots, with spin subspace as

b t ẑσ x -→ B x τz σx , (1.3)
with τ the Pauli operators in position (L,R), σ the Pauli operators in spin and B x ≡ B x MM . The total Hamiltonian in this subspace is

H = 1 2 ( τz σ0 + 2tτ x σ0 + gµ B B z τ0 σz + gµ B B x τz σx ) , (1.4)
with the subscript 0 used for the identity operator in the corresponding subspace, the energy detuning, t the tunnel coupling and

B z = B ext + B z MM .
It is interesting to include the intrinsic, if present, spin-orbit term (Rashba here) H R = α R (σ x py -σy px ) with α R the Rashba spin-orbit strength and p the momentum operator. A similar transformation to (1.3) assuming a Gaussian wave function gives [START_REF] Hu | Strong coupling of a spin qubit to a superconducting stripline cavity[END_REF] α R (σ x py -σy px ) -→ λ x τz σx , (1.5)

with λ x = α R lS/(a 2 √ 1 -S 2 ), S = L|R = exp[-(l/a) 2 ]
, a the radius of the single dot ground state Gaussian wave function and l the half distance between the two dots. The two transformed terms of Eqs. (1.3) and (1.5) have the same structure, pinning down the artificial spin-orbit equivalence of the two sites with non-collinear magnetic fields in the qubit reduced subspace. To finalize the discussion on the spin-photon coupling, we can look at the problem in the basis of bonding and antibonding orbitals n = -, + with spin

Synthetic spin-orbit interaction with magnetic textures

In this subsection, we will discuss how to induce a synthetic spin-orbit interaction with a spatially periodic magnetic field. We will first consider the case of a quantum dot subject to such a field, similarly to previous sections, and then the ideal case of a infinite conductor. An in-depth overview with all the technical details can be found in the thesis of L. Contamin [START_REF] Contamin | Mise en évidence de textures de spin synthétiques par des mesures de transport et de champ microonde[END_REF].

As outlined in the introduction of this chapter, spin-orbit coupling is equivalent to a spatially oscillating magnetic field [START_REF] Braunecker | Spin-selective Peierls transition in interacting one-dimensional conductors with spin-orbit interaction[END_REF]. This can be generated either by localized magnetic moments that order through the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism, thanks to their coupling to conduction electrons. This was studied both in atomic chains [START_REF] Nadj-Perge | Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor[END_REF][START_REF] Klinovaja | Topological Superconductivity and Majorana Fermions in RKKY Systems[END_REF][START_REF] Pientka | Topological superconducting phase in helical Shiba chains[END_REF] and in 13 C carbon nanotubes [START_REF] Braunecker | Nuclear magnetism and electronic order in C13 nanotubes[END_REF][START_REF] Hsu | Antiferromagnetic nuclear spin helix and topological superconductivity in ¡math¿ ¡mmultiscripts¿ ¡mtext¿C¡/mtext¿ ¡mprescripts/¿ ¡none/¿ ¡mn¿13¡/mn¿ ¡/mmultiscripts¿ ¡/math¿ nanotubes[END_REF]. Or it can be the stray field of a nearby magnetic material [START_REF] Klinovaja | Transition from Fractional to Majorana Fermions in Rashba Nanowires[END_REF][START_REF] Kjaergaard | Majorana fermions in superconducting nanowires without spin-orbit coupling[END_REF][START_REF] Egger | Emerging Dirac and Majorana fermions for carbon nanotubes with proximity-induced pairing and spiral magnetic field[END_REF][START_REF] Matos-Abiague | Tunable magnetic textures: From Majorana bound states to braiding[END_REF] (either a ferromagnet with domains, an array of magnets or an array of magnetic tunnel junctions). We chose the latter option, an interesting aspect of this choice being that one can in theory control the magnetic domains and thus the effective spin-orbit interaction, either by applying a magnetic field [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF], as will be discussed in chapter 3, or by flowing a current through the ferromagnet.

Figure 1.5: Schematics of a magnetic texture's stray field. Oscillating magnetic field considered in the calculation. For a succession of domains (or nanomagnet, drawn in blue), the field lines follow the black curve. At a given altitude x (close to the surface), one can in first approximation consider that the amplitude of the oscillating field B osc is constant along z, with an oscillation of its direction. The period of this oscillation is λ, corresponding to two domains/magnets with opposite magnetization. Figure taken from Ref. [START_REF] Contamin | Mise en évidence de textures de spin synthétiques par des mesures de transport et de champ microonde[END_REF].

The periodically oscillating magnetic field that we will consider is generated by mag-1.2. SYNTHETIC SPIN-ORBIT INTERACTION WITH MAGNETIC TEXTURES netic domains whose magnetization alternately point upward and downward out of the plane, as depicted in Figure 1.5. Before looking at the 1D limit, we will investigate the quantum dot case, in line with the discussion of previous sections. The magnetic texture stray field can be written as

B osc = B osc   sin(k λ z) 0 cos(k λ z)   , (1.6)
with k λ = 2π/λ, λ being the spatial period of the texture. We consider a quantum dot confined in a 1D conductor, as we will discuss the case of a carbon nanotube in the last chapter. In this case, in the absence of magnetic texture, the wave function corresponding to the n-th orbital of the dot with spin σ is, not taking into account the CNT band structure for a simplified discussion

ψ n,σ = 2 L sin(k n,σ z), (1.7)
with L the length of the dot and k n,σ = nπ/L. The eigenenergies are E n,σ = v F k n,σ with v F the Fermi velocity. We will describe the effect of the oscillating field, using again perturbation theory, considering 1 2 gµ B B osc ∆E with ∆E = hv F /2L the level spacing of the dot. This time, we have to do perturbation theory of degenerated states as the spin degeneracy is not lifted in the unperturbed Hamiltonian. We thus have

E (2) n,σ =E (0) n,σ + σ ={ σ,σ } 1 2 gµ B n, σ|B osc • σ|n, σ + (1.8) n =n σ ,σ ( 1 2 gµ B ) 2 n, σ|B osc • σ|n , σ n , σ | 1 2 gµ B B osc • σ|n, σ E n -E n .
The bracket terms write generically

n, σ|B osc • σ|n, σ = B z eff (n, σ, n , σ ) σ|σ z |σ + B x eff (n, σ, n , σ ) σ|σ x |σ , (1.9)
with

B z eff (n, σ, n , σ ) = B osc 2 sin[(k λ -k n + k n )L] (k λ -k n + k n )L + sin[(k λ + k n -k n )L] (k λ + k n -k n )L (1.10) B x eff (n, σ, n , σ ) = B osc 2 1 -cos[(k λ -k n + k n )L] (k λ -k n + k n )L + 1 -cos[(k λ + k n -k n )L] (k λ + k n -k n )L (1.11)
We see that the first order correction already introduces a mixing of the two spins within the same orbital (n = n ) through the σx operator, given L = pλ with p ∈ Z, meaning that the size of the dot should not be commensurable with the magnetic texture period to see this correction. Then of course, the second order correction term mixes spins and orbitals through the σx operator again, leading to the synthetic spin-orbit interaction. We note that these terms quickly decrease with the orbital separation (as ∼ 1/(n-n )∆E) and can show resonant condition for k λ ≈ ±(k n -k n ) + 2πp/L. The overlap between the electron wave function and the oscillating magnetic field sets the amplitude of the synthetic spin-orbit coupling and is a critical aspect of the phenomenon. High in the electron band (at high chemical potential), wave functions oscillates fast in space, therefore giving effectively an average of the magnetic texture field with a small evolution from one orbital to the other. On the contrary, low in the electron band, close to the helical gap opened by the magnetic texture (we will shortly come back to this point), it will be possible to have k n close to k λ , thus giving an overlap of the electron wave function on one dot with the magnetic field strongly dependent on the orbital, which in turn can yield a potentially strong spin-orbit coupling.

The synthetic spin-orbit nature induced by the magnetic texture is more clearly seen in the limit of an infinite 1D conductor. In this ideal case, a unitary transformation to the rotating frame of the cycloidal magnetic field directly gives a term in the Hamiltonian which has the same structure as Rashba spin-orbit interaction. Indeed, if we consider the 1D Hamiltonian for a free electron [START_REF] Kjaergaard | Majorana fermions in superconducting nanowires without spin-orbit coupling[END_REF]

H = p2 z 2m -µ γz + 1 2 gµ B B osc • σ, (1.12)
with γ Pauli matrices operating in sublattice space and apply the unitary transformation H = U HU † with

U = e iΦσxy/2 , cos(Φ) = B osc • u z B osc , σxy = (B osc × u z ) • u z B osc × u z , we find H = p2 z 2m -µ + E so 2 γz + B osc σz + α pz σy γz . (1.13)
We thus see that the oscillating magnetic field is equivalent to both a constant Zeeman effect of amplitude B osc and a synthetic spin-orbit interaction of Rashba form with perpendicular quantization axis. The combination of both terms opens a helical gap in the band structure as recalled in Figure 1.1(c). The spin-orbit coupling strength is α = πh 2 /mλ and the corresponding spin-orbit energy is E so = h 2 /4mλ 2 . The synthetic spin-orbit interaction also shifts the chemical potential to μ = µ -E so /2, which in turns shifts the position of the helical gap in energy. It is interesting to note that a similar helical band gap can be found in a finite length wire with a cycloidal magnetic field using scattering formalism (see the supplementary information of Ref. [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF] or the thesis of L. Contamin for in-depth analysis [START_REF] Contamin | Mise en évidence de textures de spin synthétiques par des mesures de transport et de champ microonde[END_REF]), without the Rashba term explicitly showing-up.

Generically, we have seen that an electron moving in an inhomogeneous magnetic field (either within quantum dots or as a free electron) leads to a spin-orbit interaction with more or less direct evidence. It turns out that the two situations of the previous slanted magnetic field section for spin qubits and the periodic magnetic field for helical band gap shaping of this section are very much similar. Indeed, the periodic displacement of the electron in the slanted magnetic field is equivalent to having a free electron in a cycloidal magnetic field with an offset on the z component of the field, as shown in Figure 1. 3(b,c). In this situation, the magnetic field vector does not fully rotate by 2π during a period as compared to the ideal cycloidal case (see Figure 1.6), yet still yields an artificial spinorbit interaction. We note that adding a constant external magnetic field to the cycloidal magnetic field of this section (Figure 1.6(a)) actually gives the same field structure as the slanted field configuration (Figure 1.6(b)) which is the experimental realization that we 1.2. SYNTHETIC SPIN-ORBIT INTERACTION WITH MAGNETIC TEXTURES will discuss in the last chapter. However it does not seem obvious from the ideal infinite 1D conductor unitary transformation that adding a constant external magnetic field would also provide a synthetic spin-orbit interaction. Indeed, in the rotating frame of the magnetic texture field, the constant external magnetic field becomes rotating. As we experimentally apply an external magnetic field to investigate the effect of the magnetic texture on the electronic excitations, we would need to look deeper into this theoretical aspect. Finally, to close the loop of this chapter back to spin qubits, we should mention that spin qubits can be defined with states lying in a helical gap, which could be engineered, and be coupled to microwave photons [START_REF] Kloeffel | Circuit QED with hole-spin qubits in Ge/Si nanowire quantum dots[END_REF].

Chapter 2

Spin qubits in a nuclear spin bath environment

Electron spin qubits are attractive candidates for manipulating quantum information. One of the primary reason is their supposedly long coherence time due to the weak coupling that spins have to their environment. As discussed in section 1.1, the weak coupling of spins to environment impedes their control and electrical control through spin-electric-coupling schemes have been accordingly devised. These in turn obviously open a breach in the long coherence time origin of spin qubits, by adding a noise source, namely charge noise. Additionally, these spin qubits are not hanging in vacuum but are hosted in a material. The nuclear spins of the host material atoms interact with the electron spin qubit through hyperfine coupling. The spin qubit environment is therefore a bath of fluctuating nuclear spins which plays the role of a magnetic noise source. It turns out that the fluctuating magnetic field generated by nuclear spins, also called the Overhauser field, is the largest source of dephasing for spin qubits. As such, most efforts have been put for years on understanding and fighting it. We should stress here that nuclear spins in semiconductor materials were well investigated and manipulated for decades before the arrival of spin qubits, in particular in the context of nuclear magnetic resonance (NMR) [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. However, it is only after the seminal theoretical proposal to manipulate quantum information with electron spins [START_REF] Loss | Quantum computation with quantum dots[END_REF], and the first experimental hints of their long relaxation times [START_REF] Kikkawa | Resonant spin amplification in n-type GaAs[END_REF] that nuclear spin dephasing of electron spins was theoretically investigated in-depth [START_REF] Merkulov | Electron spin relaxation by nuclei in semiconductor quantum dots[END_REF][START_REF] Khaetskii | Electron spin decoherence in quantum dots due to interaction with nuclei[END_REF][START_REF] Schliemann | Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei[END_REF].

Nuclear spins dephasing 2.1.1 Hyperfine coupling

In this section, we will briefly derive the dephasing time T * 2 of an electron spin due to the hyperfine coupling to nuclear spins of the environment, finding an expression that is absent (to our knowledge) from the literature, explicitly showing the nuclear spins concentrations.

In classical electrodynamics, in the limit of zero distance between two magnetic dipoles1 , the dipole-dipole interaction or energy is given by the Fermi contact interaction [START_REF] Jackson | Classical electrodynamics[END_REF] The proportionality factor between the electron and the orbital and magnetic moment is the gyromagnetic ratio γ e , alternatively parametrized by the g-factor g e , quantifying how the magnetic moment differs from the classical value of e/2m e with m e the electron mass. For nuclei, analogous quantities are defined, with the electron mass replaced by the proton mass m p and I the vector of nuclear spin operators.

E F c = - 2µ 0 3 m 1 • m 2 δ(r 1 -r 2 ), ( 2
The resulting Hamiltonian follows

H hf = n βn δ(r -r n )I n • σ. (2.4)
The sum goes over all nuclear spins, labelled by a discrete index n, which are located at positions r n . The electron position is at r, its spin is σ. Sometimes s = σ/2 is used instead, giving twice larger interaction constants. From Eqs. (2.2) and (2.3) we have βn = 2µ 0 3

g e 2 µ B µ N g n . (2.5)
Two points have to be commented here. First, the g-factor of the electron is the vacuum one, not the band structure one, meaning that it is the same for GaAs, Silicon or Carbon [START_REF] Coish | Nuclear spins in nanostructures[END_REF]. Second, the position operator in Eq. (2.4) acts on the full wave function of the electron Φ(r) = Ψ(r)ψ(r) including its Bloch part ψ(r) and its envelope Ψ(r). The former is periodic in the crystal lattice, dimensionless and normalized such that the integral over the unit cell of its modulus squared is equal to the volume of the unit cell. The volume integral of Φ(r) is then equal to the volume integral of Ψ(r). With this, the matrix elements of any operator smoothly varying on the inter-atomic distances can be calculated ignoring the Bloch part. However here, the delta function in Eq. (2.4) is not smooth and has to be dealt with explicitly. To keep the envelope part, which is convenient, the prefactors change to (removing the tilde)

β n = 2µ 0 3 µ B µ N g n |ψ(r n )| 2 , (2.6)
using g e = 2. Thus we have

H hf = n β n δ(r n -r)I n • σ, (2.7)
in the microscopic form of the hyperfine interaction, useful for quantum dots. The interaction strengths β n have dimensions of energy times volume and their values for different isotopes in GaAs, Silicon and Carbon are given in Table 2 [START_REF] Schliemann | Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei[END_REF][START_REF] Coish | Nuclear spins in nanostructures[END_REF] for GaAs and Si and from Refs. [START_REF] Yazyev | Hyperfine Interactions in Graphene and Related Carbon Nanostructures[END_REF][START_REF] Bulaev | Spin-orbit interaction and anomalous spin relaxation in carbon nanotube quantum dots[END_REF] for carbon which are calculated for carbon nanotubes. The values of A n from Ref. [START_REF] Coish | Nuclear spins in nanostructures[END_REF] are twice bigger as they use s and not σ.

Refs. [START_REF] Erlingsson | Hyperfine-mediated transitions between a Zeeman split doublet in GaAs quantum dots: The role of the internal field[END_REF][START_REF] Camenzind | Hyperfine-phonon spin relaxation in a single-electron GaAs quantum dot[END_REF] for calculation of off-diagonal elements both in orbital and spin indexes).

The expectation value of Eq. (2.7) in the orbital state Ψ(r) is

H hf Ψ ≡ Ψ| n β n δ(r n -r)I n • σ|Ψ = n β n |Ψ(r n )| 2 I n • σ (2.8)
From here it is convenient to multiply and divide by the volume corresponding to one spin because we will later go from the discrete sum to a volume integral using

n f (r n ) = 1 v 0 n v 0 f (r n ) ≈ 1 v 0 drf (r), (2.9)
valid as long as the function is smooth. The volume v 0 corresponding to one spin is the volume corresponding to one atom, for zinc-blende crystals like GaAs with 8 atoms per cell units, v 0 = a 3 0 /8 with a 0 the lattice constant. Using A n = β n /v 0 (given in Table 2.1), we get

H hf Ψ = v 0 n A n |Ψ(r n )| 2 I n • σ.
(2.10)

This form is appealing as A n have the dimension of energy and the rest together is dimensionless. Now comes the tricky part to properly deal with elements and isotopes. GaAs is easy as all nuclei and their isotopes have a nuclear spin. We can simplify things by averaging over the isotopes to get a single value of A ≈ 45 µeV. In silicon or carbon, not all nuclei have a spin, only a proportion p of the isotopes. We can do two things: 1) analogously to GaAs, using the volume v 0 and averaging A n including the zero-spin isotope, giving A ≡ pA n , which does not corresponding to any atom, or 2) using the "true" A n of 29 Si or 13 C and multiplying and dividing by v 0 /p instead of v 0 . The first choice hides the prefactor p inside A, defining it as pA n , the second leaves it with v 0 , but both give the same result

H hf Ψ = v 0 p n A n |Ψ(r n )| 2 I n • σ, (2.11)
where the 100% nuclear spin isotopes expression is naturally recovered for p = 1.

The dephasing time

We now calculate the average value of the modulus of the Overhauser field, defined as the vector multiplying the sigma matrices in Eq. (2.11). The average here is a statistical 2.1. NUCLEAR SPINS DEPHASING average over unpolarized and uncorrelated nuclear spins defined by

I α n = 0, I α n I β m = δ αβ δ nm I(I + 1) 3 , (2.12)
with n, m atom indexes and α, β Cartesian coordinate indexes. The square of the Overhauser field along an arbitrary axis (z for example) is

∆ 2 = v 0 p n A n |Ψ(r n )| 2 I z n v 0 p m A m |Ψ(r m )| 2 I z m (2.13) = v 0 p 2 I(I + 1) 3 v 0 n A 2 n |Ψ(r n )| 4 . (2.14)
Using an averaged A (actually one should calculate an averaged A 2 n , which is very close for GaAs A 2 n ≈ A n 2 and the same for silicon or carbon) and Eq. (2.9), we get

∆ 2 = v 0 p 2 A 2 I(I + 1) 3 dr|Ψ(r)| 4 = v 0 V D p 2 A 2 I(I + 1) 3 , (2.15)
with V D the quantum dot volume, equal to 2πl x l y l z for a harmonic confinement. Finally, we reach our initial goal, the dephasing rate is obtained as

T * 2 = 2∆ = 2pA 3 I(I + 1) V D v 0 . (2.16)
This equation is the central result of this section which is surprisingly difficult to find, if present, in the literature. It calls for several important comments. For the following, N S = pN = pV D /v 0 the total number of nuclear spins within the quantum dot is also a useful quantity to manipulate. The question which drives everyone involved in spin qubits is how to improve the dephasing time T * 2 . We have the following options:

1. The dephasing rate is inversely proportional to A and I so it is wise to choose a material where both are small. Looking at Table 2.1 we see that carbon is better than silicium which is much better than GaAs for both quantities.

2. Fixing the material, hence A, and not changing the isotopic proportions (p is constant), we have

T * 2 ∝ √ V D or equivalently T * 2 ∝ √ N S .
Larger quantum dots with more nuclear spins improves the dephasing time, due to the statistical narrowing of the variance of the Overhauser field, which can be observed through motional motional narrowing for example [START_REF] Bloembergen | Relaxation Effects in Nuclear Magnetic Resonance Absorption[END_REF][START_REF] Flentje | Coherent long-distance displacement of individual electron spins[END_REF].

3. Now fixing the quantum dot volume V D and changing p through isotopic purification to change the nuclear spins concentration, we have

T * 2 ∝ 1/p or equivalently T * 2 ∝ 1/N S .
Here we find the intuitive result that less nuclear spins is better for the dephasing time. Current isotopic purification technologies that remove 29 Si or 13 C typically reach p ∼ 8 × 10 -4 (800 ppm).

Dephasing times T *

2 for GaAs, silicon and carbon nanotubes with various isotopic concentrations are presented in Table 2.2. For carbon nanotubes, the number of atoms is calculated using

N = 4π 3 √ 3a 2 CC LR, (2.17)
CHAPTER 2. SPIN QUBITS IN A NUCLEAR SPIN BATH ENVIRONMENT with a CC = 1.42 Å the lattice constant, L ≈ 200 nm the length and R ≈ 2 nm the radius of the CNT. This expression is obtained by combining Eqs. (3.2), (3.7) and (3.9) of Ref. [START_REF] Saito | Physical properties of carbon nanotubes[END_REF]. For isotopically purified materials, the estimated hyperfine induced dephasing times are so large that most often charge noise becomes the dominant dephasing source [START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF] as will be discussed in subsection 2.3.2. We note also the very large predicted values for CNT (calculated using A = 0.5 µeV) which are far from what has been measured experimentally so far. We will come back to this point in more details in subsection 2.3.3.

Before discussing actual experimental realizations of spin qubits in silicon, purified silicon and carbon nanotubes where the dephasing time should be significantly enhanced compared to GaAs spin qubits, we will first have a look at another important aspect of hyperfine induced dephasing, the dynamics of the nuclear spin bath.

GaAs

Dynamics of the nuclear spins bath

This section mostly discusses results published in Ref. [START_REF] Delbecq | Quantum Dephasing in a Gated GaAs Triple Quantum Dot due to Nonergodic Noise[END_REF] as well as some results published in Refs. [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF][START_REF] Noiri | A fast quantum interface between different spin qubit encodings[END_REF].

The dephasing time described in the previous section was originally defined in the context of NMR as the source of the observed finite resonance linewidth [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. As such experiments are usually performed in the steady state and on large ensembles of spins, this dephasing time reflects the system inhomogeneity over a large range both in space and time. The hallmark of the nuclear spin environment is its very slow internal dynamics [START_REF] Urbaszek | Nuclear spin physics in quantum dots: An optical investigation[END_REF], due to the weakness of nuclear spin-spin interactions [START_REF] Paget | Low field electron-nuclear spin coupling in gallium arsenide under optical pumping conditions[END_REF][START_REF] Maletinsky | Dynamics of Quantum Dot Nuclear Spin Polarization Controlled by a Single Electron[END_REF]. Due to this slowness, the typical total operation time of spin qubits is usually much shorter than the decorrelation time of the noise. This is a very different regime than in NMR where the "infinite" averaging over space or time renders the noise uncorrelated or white. Therefore it is expected that in the correlated noise, or non-ergodic, regime, the dephasing time might be strongly affected.

To briefly give a picture of the decorrelation time of the noise, we can model the dipoledipole hyperfine noise as being diffusive, and equivalently following a random walk [START_REF] Wang | On the Theory of the Brownian Motion II[END_REF].

A warning about this model which is widely used and accepted has to be made however.

The diffusiveness of the nuclear spin bath is an assumption with which calculations can be performed but, quoting A. G. Redfield [START_REF] Redfield | Spatial Diffusion of Spin Energy[END_REF] "In many cases it is difficult, if not impossible, to prove theoretically whether or not such spin diffusion actually occurs, but if we assume that it does, then it is possible in principle to find a unique value for the diffusion coefficient D"; we see that we should keep in mind that diffusion is only an assumption. As the Overhauser field is bounded in amplitude to ∆ = σ B (∞), the diffusion cannot spread infinitely, and must be bounded as well. A simple model to account for such boundary effect is the Ornstein-Uhlenbeck process [START_REF] Uhlenbeck | On the Theory of the Brownian Motion[END_REF]. Assuming that at the beginning of operations,

DYNAMICS OF THE NUCLEAR SPINS BATH

the Overhauser field is B(t = 0), the mean value of B decays exponentially to zero as B(t) = B(0)e -t/T 0 , the memory of the initial value also decays exponentially to zero as B(t)B(0) = B(0)2 e -t/T 0 and the variance grows linearly until it saturates exponentially as

[B(t) -B(t) ] 2 = σ 2 B (∞)(1 -e -2t/T 0 ). Within this model, the decorrelation time is T 0 = 2σ 2
B (∞)/D with D the diffusion coefficient of the diffusive process. For GaAs, reported decorrelation times ranges from seconds to hours due to variations in doping, strain and nanostructure confinement. As the measurement of the qubit is of projective nature and one has therefore to average many observations to infer the estimated quantum state probability, sticking to the uncorrelated white noise regime of the Overhauser field would impose to have a repetition rate of several T 0 , which is obviously too slow for quantum information manipulation, if not simply impractical. This is the motivation for investigating the nuclear spin bath dynamics and the dephasing time in the non ergodic regime using the spin qubit as a fast and sensitive probe.

The use of a singlet-triplet (ST) qubit [START_REF] Petta | Coherent manipulation of coupled electron spins in semiconductor quantum dots[END_REF][START_REF] Hanson | Spins in few-electron quantum dots[END_REF] is well suited for this purpose as its oscillation frequency is directly subject to the nuclear field fluctuations. ST qubits require two quantum dots, which in this work are the two rightmost dots of a triple quantum dot device shown in The qubit dephasing time T * 2 is extracted from the free induction decay of the ST qubit. The measurement scheme is organized such that we can access the evolution of the dephasing as a function of the acquistion time. The basic unit is a "cycle" (index c) during which the qubit is initialized in the state |↑, S(0, 2) , then quickly moved to the |↑, S(1, 1) state where it precesses with |↑, T 0 (1, 1) for the qubit evolution time τ c before undergoing a Pauli spin blockade measurement deep in the (1,0,2) region [START_REF] Barthel | Relaxation and readout visibility of a singlet-triplet qubit in an Overhauser field gradient[END_REF]. The cycle duration is set to 15.192 µs independent of τ c by adjusting the initialization time. The next level is a "record", which comprises 250 consecutive cycles with qubit evolution times increased by 4 ns steps, restarting each record from zero. A single record takes time t rec = 3.8 ms = 250 × 15.192 µs to acquire, covering the qubit evolution for τ c ∈ [0, 996] ns. Finally, we form a set R by selecting N R records from all measured ones. We extract the projection of the qubit state on the S-T 0 axis of the Bloch sphere, s(τ c ), by averaging over data in R, using s(τ c ) = 2P S (τ c ) -1 R with P S ∈ {0, 1} the results of projective measurements of the singlet state as shown in Figure 2.2(a) for a particular record. The simplest, and most standard, choice is to take R as a block of N consecutive records. The time to acquire such data is ∆t = N t rec , referred to as the acquisition time, setting it as the natural parameter for dephasing rates. Indeed, even though we are usually interested in the qubit evolution on times of the order of τ c T * 2 , the acquisition time needed to sample the continuous function s(τ c ) from binary data of projective measurements is typically orders of magnitude larger, as is clear from the above measurement description.

The time evolution of ∆B z is first extracted over 40000 consecutive records spanning more than two minutes as shown in Figure 2.2(c)]. It fluctuates around a finite value of 30 MHz, set by the micromagnet, by ±20 MHz due to nuclei. With our measurement sequence we can follow the nuclei dynamics down to the time t rec = 3.8 µs. Namely, using a Bayesian estimation algorithm [START_REF] Sergeevich | Characterization of a qubit Hamiltonian using adaptive measurements in a fixed basis[END_REF][START_REF] Shulman | Suppressing qubit dephasing using real-time Hamiltonian estimation[END_REF] on the data of a single record, we estimate the mean and variance of the qubit frequency as it evolved during that record. As an example, the Bayesian inference algorithm applied to the record of Interestingly, the exponent α < 1 indicates a sub-diffusive behaviour. This differs from the normal diffusion (corresponding to α = 1) that is assumed for dipole-dipole interactions that should dominate at times equal or larger than our t rec , as discussed previously, and super-diffusion expected for electron-mediated interactions which should dominate at much shorter times [START_REF] Klauser | Nuclear spin dynamics and Zeno effect in quantum dots and defect centers[END_REF]. Non-Markovian nuclear dynamics could result in such subdiffusion [START_REF] Bouchaud | Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications[END_REF], it would however also imply a non-Gaussian noise correlator [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF], at odds with the observations. Since it is difficult to infer the correlator functional form in the time domain from its noise power spectrum [START_REF] Li | Intrinsic Spin Fluctuations Reveal the Dynamical Response Function of Holes Coupled to Nuclear Spin Baths in (In,Ga)As Quantum Dots[END_REF] if the latter is known only within a limited frequency range, other investigations [START_REF] Reilly | Measurement of Temporal Correlations of the Overhauser Field in a Double Quantum Dot[END_REF][START_REF] Medford | Scaling of dynamical decoupling for spin qubits[END_REF][START_REF] Malinowski | Spectrum of the nuclear environment for gaas spin qubits[END_REF] do not necessarily contradict our observation (in these works, the lorentzian shaped power spectral density of a normal diffusion process is fitted). It is to be noted that a behaviour closer to standard diffusion cannot be completely excluded at the shortest times that have been reached where the correlator variance appears to bend slightly towards the α = 1 slope. A similar approach to extract the correlator variance was used also with a ST qubit in GaAs where α = 1 is reported for times between ≈ 1 ms and 50 ms [START_REF] Shulman | Suppressing qubit dephasing using real-time Hamiltonian estimation[END_REF]. This could indicate that normal diffusion occurs at time scales shorter than anticipated. The measurement of the Overhauser field variance correlator as a function of acquisition time has been reproduced in two other experiments. In the first one, a ST qubit was also used and manipulated in the same manner (initialization, operation and readout) [START_REF] Noiri | A fast quantum interface between different spin qubit encodings[END_REF] and in the second one, a different type of qubit was manipulated, a spin-1/2 spin qubit [START_REF] Nakajima | Quantum non-demolition measurement of an electron spin qubit[END_REF]. In both of these two subsequent works, the GaAs wafer is identical but it is different from the one used in the previously described work [START_REF] Delbecq | Quantum Dephasing in a Gated GaAs Triple Quantum Dot due to Nonergodic Noise[END_REF]. The corresponding variance correlators, shown in Figure 2.3 (c) and (d), both exhibit a subdiffuse behaviour with α = 0.8 [START_REF] Noiri | A fast quantum interface between different spin qubit encodings[END_REF] and α = 0.84 [START_REF] Nakajima | Quantum non-demolition measurement of an electron spin qubit[END_REF]. The reproducibility of the subdiffusive behaviour with different wafer and type of qubit is encouraging to exclude a material origin or a difference of hyperfine interaction between a singlet-triplet state and a spin-1/2 state. In addition, in the ST qubit case, the frequency fluctuations of the qubit come from the difference of the nuclear spin magnetic field at two different quantum dot locations while for the spin-1/2 qubit, they come from the nuclear spin magnetic field in a single quantum dot. This difference does not appear to have any consequence on the variance correlator of the noise. One difference between this set of experiments and the experiments done by other groups is the presence of the micromagnet which induces magnetic field gradients ∆B z MM (y) and ∆B x MM (z) as described in subsection 1.1.1, with the the x-axis out of plane, the y-axis oriented along the quantum dot alignment and the z-axis in-plane as in Figure 2.1. However, taking the micromagnet field into account would not lift the difficulty to infer the fluctuation mechanism without assuming normal diffusion. [START_REF] Noiri | A fast quantum interface between different spin qubit encodings[END_REF]) and in a different wafer with a spin-1/2 qubit (blue dots, taken from Ref. [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF]).

We now turn to the qubit phase stability. The standard way is to fit the qubit state estimation evolution to oscillations with a Gaussian decay

s(τ c ) fit -→ cos(2πf 0 τ c ) exp - τ c T * 2 2 , (2.18)
and define the dephasing time as the fitted decay parameter. If τ c is much smaller than the acquisition time, always fulfilled, the frequency change during the time τ c is negligible and we get

s(τ c ) = 1 N R r∈R cos (2πf c,r τ c ) , (2.19)
with f c,r the qubit frequency during the c-th cycle of the r-th record. From here it follows that the frequency and dephasing extracted from the fit in Eq. (2.18) are given, respectively, as the average and the variance of the set of frequencies {f c,r }. These statistical properties in turn depend on how the set R is chosen.

The standard way is to choose R as a single block of N consecutive records. Doing so defines T * 2,φ , for which is observed a gradual increase of T * 2,φ ∼ 120, 220 and 570 ns upon decreasing N , for respective acquisition times ∆t ∼ 1.6, 0.4 and 0.1 s as shown in Figure 2.4(a). Since each of these qubit evolutions results from a particular noise realization, T * 2 becomes a stochastic variable itself. It is possible to extract its probability distribution for various acquisition times, as shown in Figure 2.4(b). It is always well fitted 2.3. FIGHTING AGAINST NUCLEAR SPINS DEPHASING by a Gamma distribution3 whose skewness does not significantly change for ∆t varying from 38 ms to 7.6 s. The shape parameter of the fitted Gamma distribution is k = 7.25 ± 1 over this range. This robustness can be interpreted as a signature that the nature of the underlying dynamics of nuclei does not change within this timespan. A Simulation of the whole measurement sequence with an Ornstein-Uhlenbeck normal diffusion process at all time scales similarly gives T * 2,φ distributed according to a Gamma distribution with a larger skewness corresponding to k = 5.5 ± 0.6 over the same time span. The conclusion of this statistical analysis is that a single trace is not sufficient to reliably estimate the phase decay, as the most probable T * 2,φ is smaller than the mean T * 2,φ , whereas occurrences of T * 2,φ several times larger than T * 2,φ are common.

Is it therefore enough to measure as fast as possible with respect to the decorrelation time of the environment noise to improve the dephasing time? Of course not as the full variance of the noise σ 2 B (∞) is still hidden in the unknown value of the Overhauser field, hence the actual qubit frequency, at the beginning of the qubit operation. Therefore the dephasing time T * 2,φ , although much larger than T * 2 (∞) ≈ 10 ns, is useless for quantum information manipulation. For practical quantum computation (QC) the qubit frequency must be known in advance. This can be done using the Bayesian approach described here on a first record before the actual qubit operations. The corresponding dephasing time T * 2,QC is found to increase to ∼ 600 ns on average for an acquisition time of a single record (N = 1) corresponding to 3.8 ms. While for large N a Gamma distribution similar as the one for T * 2,φ was observed as expected, statistics of T * 2,QC for acquisition times shorter than 10 × 3.8 ms could not be estimated due to the lack of total number of records, limited by the physical memory of the digitizer acquisition board. With at least 10 times more records, it could have been possible to obtain statistics at such short time scale. We will now turn in the beginning of the next section to a practical implementation of T * 2,QC with a spin-1/2 spin qubit in GaAs.

Fighting against nuclear spins dephasing

This section discusses results published in Ref. [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF][START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF][START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF][START_REF] Cubaynes | Highly coherent spin states in carbon nanotubes coupled to cavity photons[END_REF].

In the previous two sections we have seen how the nuclear spins environment influences in a detrimental way the dephasing time of the spin qubit and how it can induce non trivial statistics of the dephasing time itself. As the fluctuations we have investigated are slow with respect to the qubit actual operation time, such noise is classified as quasi-static. We have thus seen that operating fast with respect to the noise decorrelation time can in principle significantly improve the dephasing time, according that the environment is monitored just before operation, doing a so-called Hamiltonian estimation. This is discussed in the following subsection. Reducing the number of nuclear spins by changing the host material and performing isotopic purification is the second approach as discussed in section 2.1 which will be the subject of the following subsections. Typically using both approaches, the remaining dominant source of noise becomes charge noise, larger than the high frequency component of the nuclear spin noise. While the improvement of the dephasing time is the primary target, one should not forget that the actual goal is to develop a qubit useful for quantum computation. A good figure of merit to quantify how good a qubit is for this purpose is the qubit quality factor, defined as Q = 2f Rabi T Rabi 2 which simply tells how many of the simplest quantum gate (a π-rotation gate) can be done during the dephasing time. Here f Rabi is the Rabi frequency and T Rabi 2 is the Rabi dephasing time, usually larger than the free induction decay time T * 2 discussed in the previous section due to the qubit being actively driven [START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF]. Importantly, the quality factor sets an upper limit on the fidelity F of the simplest single quantum gate as F ≤ e -1/Q . As fault tolerance requires gate fidelities of at least about 99% [START_REF] Fowler | High-threshold universal quantum computation on the surface code[END_REF], this imposes a lower bound of 100 on the qubit quality factor for practical quantum computation. Therefore it is also essential to not forget about the qubit frequency. This is why in GaAs and silicon based devices a micromagnet is used to exploit EDSR which enhances single spin-1/2 qubit drive frequencies. GaAs is for once naturally better than silicon due to the presence of an intrinsic not so small spin-orbit interaction which naturally induces larger qubit operational frequencies than in silicon where the spin-orbit interaction is weak. We shall keep those elements of thoughts in mind throughout the rest of this section.

Hamiltonian estimation in GaAs

All figures of this section are taken and adapted from Ref. [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF].

The prolongation of the dephasing time by exploiting the low frequency quasi-static noise has been demonstrated with a ST qubit in GaAs, using Hamiltonian estimation as described in the previous section, giving an impressive enhancement of T * 2 ∼ 2 µs [START_REF] Shulman | Suppressing qubit dephasing using real-time Hamiltonian estimation[END_REF]. In this work, the Hamiltonian estimation was used to evaluate the stability of the idle qubit frequency through the measurement of T * 2 . Here "idle" means that only the free induction 2.3. FIGHTING AGAINST NUCLEAR SPINS DEPHASING decay of the qubit is measured with a Ramsey sequence. This was highly instructive on the effective enhancement of the dephasing time using Hamiltonian estimation technique. However, for practical quantum computation, one needs to actively control the qubit, typically performing Rabi control. This is why we implemented a similar approach with a single spin-1/2 qubit. The differences with the work of Ref. [START_REF] Shulman | Suppressing qubit dephasing using real-time Hamiltonian estimation[END_REF] are threefold : (1) the feedback protocol has to be changed with respect to ST qubit. ( 2), the Overhauser field couples differently to the spin qubit, through its local value rather than to its spatial gradient as discussed in the previous section. (3), as the single spin-1/2 qubit will be driven, we will investigate the enhancement of the driven qubit performances through T Rabi 2 and be able to access the qubit noise at the Rabi frequencies, orders of magnitude larger than the typical Overhauser field frequency range we have discussed so far.

Regarding point (1), a ST qubit is naturally precessing at the qubit frequency f qubit set by the local gradient of the external magnetic field and the Overhauser field. Estimation of f qubit with Bayesian inference is done by leaving the qubit evolve for a time t R . For a spin-1/2 qubit, it is necessary to drive the qubit to the Bloch sphere equator where it evolves under the qubit frequency f qubit = gµ B (B ext + B nuc )/h set by the total external magnetic field B ext plus the Overhauser field B nuc . Therefore, estimating the qubit frequency necessitates to perform a Ramsey measurement, whose protocol and sequence is described in Figure 2.5(b). Ramsey oscillations are measured by doing two π/2 Rabi pulses separated by a time t R . The Rabi pulses are done by driving the qubit at f qubit for a time 1/f Rabi . To observe Ramsey oscillations at frequency ∆ in the qubit rotating frame, it is necessary to actually drive the qubit off resonant at f MW = f qubit + ∆. We therefore see how the feedback protocol can be implemented alternating "probe" and "target" steps (see Figure 2.5(c)). In the former, the qubit frequency is probed by sampling 150 qubit measurement outcomes of a Ramsey oscillation with t R = 2, 4, . . . , 300 ns using f n-1 MW (∆) = f est,n-1 qubit + ∆ at the (n -1) repetition of the feedback sequence. Bayesian inference algorithm is applied in real time using a Field Programmable Gated Array (FPGA) board to estimate f est,n qubit setting f n MW (∆) = f est,n qubit +∆. The subsequent "target" step begins afterwards using f n MW (∆) with either keeping ∆ = 0 to perform again Ramsey oscillations or setting ∆ = 0 to perform a Rabi drive of the qubit. Setting ∆ = 0, Rabi oscillations can be performed, a typical measurement of which is presented in Figure 2.6(c). It shows an exponential decay, which is common for silicon spin qubits [START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF][START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF] but atypical for GaAs [START_REF] Yoneda | Fast Electrical Control of Single Electron Spins in Quantum Dots with Vanishing Influence from Nuclear Spins[END_REF] as it implies 1/f noise (usually charge noise) rather than 1/f 2 noise associated with nuclear spin "diffusion". T Rabi 2 ≈ 1.5 µs is found for the trace shown. From this point, the quality factor Q = 2f Rabi T Rabi 2 is optimized, adjusting f Rabi through the microwave drive amplitude as well as the strength of the spin-electric An electron spin qubit in the middle QD (red arrow with a circle) is controlled by the EDSR where the spin is coupled to a microwave (MW) electric field via a stray magnetic field of the micromagnet deposited on the wafer surface. The right QD hosts an electron spin (blue arrow with a circle) used as a readout ancilla while the left QD hosts another electron which is unused and decoupled from the two spins. The energy detuning between the middle and the right QDs is gate-tunable and the QD electron occupancies are probed by a proximal single-electron transistor (SET) [START_REF] Barthel | Fast sensing of double-dot charge arrangement and spin state with a radio-frequency sensor quantum dot[END_REF]. (b) Schematic of the Ramsey measurement. Two electrons (qubit and ancilla) are initialized to a doubly-occupied singlet state in the right QD and an up-spin qubit is prepared by adiabatically loading one of the electrons to the middle QD [START_REF] Noiri | Coherent electron-spinresonance manipulation of three individual spins in a triple quantum dot[END_REF]. Two π/2 microwave bursts, separated by time t R , are applied (before and during these, off-resonant microwave bursts are optionally applied). The ancilla-spin state is not affected by the microwave bursts. The final state is read out by unloading an up-spin (anti-parallel to the ancilla) state from the middle QD while a down-spin (parallel to the ancilla) state remains blocked. (c) Schematic of the feedback control loop for a spin qubit. Data of a Ramsey oscillation are processed in a digital signal processing hardware with programmable logic (FPGA) to estimate the frequency detuning δf = f qubit -f est qubit between the current qubit frequency fqubit and its previous estimate f est qubit ("probe" step). The value of f est qubit is updated to f est qubit → f est qubit + δf ("update" step), after which the target experiment follows ("target"step). In the ideal case, the subsequent qubit algorithms can be executed with a microwave frequency f MW matching fqubit exactly (by choosing ∆ = 0). coupling through the detuning energy between the two neighbouring quantum dots [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF]. A highest Q = 85 ± 8 was achieved in this device, setting an upper single qubit gate fidelity F ≤ 98.8 ± 0.1%. This prediction was tested using randomized benchmarking [START_REF] Muhonen | Quantifying the quantum gate fidelity of single-atom spin qubits in silicon by randomized benchmarking[END_REF], finding an X π gate fidelity of 99.04 ± 0.23% close to the Q-factor limited value, and setting the highest fidelity for single spin qubits in GaAs to date, above the fault tolerant threshold of 99% [START_REF] Fowler | High-threshold universal quantum computation on the surface code[END_REF]. The other single qubit gates show a lower fidelity of 97.5% on average, most likely limited by systematic unitary errors due to the microwave setup. We note however that these still fall above the fault tolerance threshold of 95% of new quantum error correction codes, assuming that dephasing noise is significantly larger than relaxation noise [START_REF] Tuckett | Fault-tolerant thresholds for the surface code in excess of 5% under biased noise[END_REF], which is most probably the case with this type of spin qubits.

FIGHTING AGAINST NUCLEAR SPINS DEPHASING

Interestingly, the dependence of T Rabi 2 with f Rabi offers a direct way to investigate the dephasing noise at frequencies order of magnitude larger than for the quasi-static Overhauser noise. The power spectral density S(f ) in the low frequency range is computed by Fourier transform of the data in Figure 2.6(a) while it is calculated from the exponential decay rate of the Rabi oscillations for higher frequencies [START_REF] Yan | Rotating-frame relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution[END_REF][START_REF] Yoshihara | Flux qubit noise spectroscopy using Rabi oscillations under strong driving conditions[END_REF]. Both are plotted on the same graph in Figure 2.7(a). When no feedback is active, the low frequency S(f ) shows a 1/f 1.7 dependence, as expected from the noise correlator variance dependence in time, as the exponent of the former is half the exponent of the latter (σ 2

N (t) ∝ t α ↔ S(f ) ∝ 1/f 2α ) [82].
As the feedback is turned on, the spectral density is significantly suppressed with a flat noise spectrum region suggesting uncorrelated white noise turning to a 1/f noise more visible as the spin-electric-coupling is increased (red curve), typical of charge noise [START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF]. The noise power spectral density extracted from the Rabi measurements at f ≈ 10 MHz strikingly falls on the same 1/f noise line, highly suggestive of a charge noise dominated dephasing of the qubit over 6 orders of magnitude, which was again a first in a 100% nuclear spins material like GaAs. Despite the charge noise character of S(f ) at high frequencies, nuclear spins are still lurking around. The noise power spectral density extracted from the Rabi measurements is displayed in Figure 2.7(b). For f < 20 MHz, S(f ) shows three prominent peaks, on top of the 1/f baseline, at the nuclear Larmor precession frequencies of 75 As, 69 Ga and 71 Ga. It clearly suggests that such high frequency noise sources indeed CHAPTER 2. SPIN QUBITS IN A NUCLEAR SPIN BATH ENVIRONMENT influence the Rabi decay of the spin qubit. The qubit spin quantization axis is defined by the external magnetic field along the z-axis. The Overhauser field aligns along the same z-axis and when nuclear spins precess around it, it remains constant and there is no noise at the Larmor frequencies. In the present device, the stray field of the micromagnet induces field inhomogeneity, making each nuclear spin (indexed by k) at position r k precess around a local magnetic field vector B(r k ) slightly off the z-axis (see the inset of Figure 2.7(b)). The inhomogeneity of the nuclear spin polarization leads to small but finite residual oscillations of Overhauser field at the nuclear precession frequencies.

Natural and isotopically purified silicon

All figures of this section are taken and adapted from Ref. [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF][START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF][START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF] We will now discuss spin qubit dephasing performances as the proportion of nuclear spins is decreased in order to reduce the Overhauser noise, using as host materials natural silicon [START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF] and isotopically purified silicon [START_REF] Yoneda | A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%[END_REF]. As the qubit operations are similar to the ones already presented and no complex feedback protocol is needed, we will keep the description of the results short and focus on the comparison between the different implementations of spin qubits in regard of the various figure of merits. Previous implementations of single spin qubits in natural silicon had shown a significant improvement over GaAs spin qubits, with T * 2 ∼ 800 ns [START_REF] Kawakami | Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot[END_REF]. This value is however relatively less than the expected value presented in Table 2.2. This rather large dephasing could be explained by a rather large charge noise in this particular device. In addition, the spin qubit suffered from the presence of the silicon valley degree of freedom whose degeneracy was not lifted enough, impeding the qubit operations. Even though they used a micromagnet to enhance the control frequency of the qubit to f Rabi ∼ 5 MHz, they could only achieve a quality factor of Q ∼ 9. This indicated that fully exploiting the potential of the lower concentration of 2.3. FIGHTING AGAINST NUCLEAR SPINS DEPHASING nuclear spins of silicon is more difficult than anticipated. This amounts principally to a more delicate fabrication process, the quality of the two-dimensional electron gas and the presence of the valley degree of freedom when compared to GaAs. The latter can be lifted simply by designing smaller quantum dots to sets the energy spacing between the two valleys at values much larger than temperature and qubit energy (we should recall however from Eq. (2.16) that smaller dots at constant nuclear spin concentration is detrimental for the dephasing time). This is what was implemented in our natural, industrial standard, silicon spin qubit in RIKEN (see Figure 2.8(a)) [START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF], with an emphasis on properly designing the micromagnet to optimize the qubit control frequency [START_REF] Yoneda | Robust micromagnet design for fast electrical manipulations of single spins in quantum dots[END_REF]. Low charge noise and the absence of the spurious valley degree of freedom allowed for observing dephasing times in line with expectations T * 2 ∼ 2 µs (see Figure 2.8(b)), two orders of magnitude larger than in GaAs. Consequently, Rabi decoherence times of T Rabi 2 ∼ 10 µs where observed, which coupled to an optimized large control frequency of the qubit f Rabi ∼ 5 MHz allowed to reach quality factors of Q ∼ 140. This in turn allowed for reaching fault tolerance single qubit gates with an average fidelity of 99.59%. Remarkably, this is similar to quality factors (Q ∼ 80 -140) and fidelities (F = 99.57%) of previously demonstrated single spin qubits in isotopically purified silicon [START_REF] Veldhorst | An addressable quantum dot qubit with fault-tolerant control-fidelity[END_REF][START_REF] Veldhorst | A two-qubit logic gate in silicon[END_REF]. While the isotopic purification allowed for reaching the expected nuclear spin limited dephasing times of T * 2 = 120 µs, these works suffered from low qubit control frequency of f Rabi ∼ 100 kHz due to the absence of an efficient spin to charge conversion mechanism. It is therefore natural to use a similarly optimized micromagnet with istopically purified silicon to try to obtain a better qubit than with natural silicon. The isotopically purified silicon wafer was provided by the group of Kohei Itoh from Keio university who also provided previous wafers to New South Wales University [START_REF] Veldhorst | An addressable quantum dot qubit with fault-tolerant control-fidelity[END_REF][START_REF] Veldhorst | A two-qubit logic gate in silicon[END_REF]. With the micromagnet, we could reach Rabi frequencies about two orders of magnitude larger f Rabi ∼ 30 MHz. However at these frequencies, the Rabi coherence time degrades (this will be discussed in a few paragraphs) and f Rabi ∼ 3 MHz has to be chosen for keeping long coherence times of T Rabi 2 ∼ 110 µs giving an optimized qubit quality factor of Q ∼ 888 and the corresponding single qubit gate fidelities of F > 99.9%. As a matter of comparison, these record metrics for a single spin qubit compare with Google's superconducting qubits (Q ∼ 1000, f Qubit ∼ 20 MHz) with which "quantum supremacy" was demonstrated in 2019 [START_REF] Arute | Quantum supremacy using a programmable superconducting processor[END_REF].

An interesting aspect of the enhanced spin to charge coupling in the isotopically purified silicon device is that now charge noise dominates at all frequencies as shown in Figure 2.9, effectively completely removing nuclear spin noise. This is first observed in the dephasing time T * 2 = 20 µs extracted from Ramsey measurement (see Figure 2.8(e)) which is well below the expected theoretical value. Varying the total acquisition time of the Ramsey measurement allows to extract the noise power spectrum over two orders of magnitude in the range 0.01 -1Hz, as in the previous sections. Higher frequency spectral noise density is extracted from Carr-Purcell-Meiboom-Gill (CPMG) protocols [START_REF] Medford | Scaling of dynamical decoupling for spin qubits[END_REF][START_REF] Cywiński | How to enhance dephasing time in superconducting qubits[END_REF]. Such dynamical decoupling sequences 4 can partially cancel the dephasing effect, with efficacy strongly dependent on the qubit noise spectral density S(f ). The CPMG inferred noise spectral density lies in the frequency range 13 -320kHz with a 1/f dependence which is in line with the low frequency part. This demonstrates a 1/f charge noise over seven orders of magnitude and a purely charge noise limited spin qubit.

From the results discussed so far, we can conclude that "simply" removing nuclear 4 The CPMG two pulses sequence, nπ = 2, is the Hahn or spin echo sequence. spins to enhance the dephasing time is not enough to make a good practical qubit 5 . The qubit operational frequency is also a very important aspect that was somehow eluded in the community despite the crucial importance of the qubit quality factor which sets an elementary upper bound on the quantum gates fidelities. Improving the qubit frequency is done at the expense of the coherence time as it requires to increase the coupling to the qubit by some means. A compromise has to be taken. This is illustrated in Figure 2.10 where the qubit Rabi frequency, Rabi coherence time and quality factors are presented as a function of the drive amplitude for the natural silicon spin qubit [START_REF] Takeda | A fault-tolerant addressable spin qubit in a natural silicon quantum dot[END_REF] and the feedback controlled GaAs spin qubit [START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF]. For both, f Rabi grows linearly with the drive amplitude as expected, before departing from this behaviour, most probably due to anharmonicity of the confinement potential of the quantum dot [START_REF] Tokura | Coherent Single Electron Spin Control in a Slanting Zeeman Field[END_REF]. Again for both, the coherence time T Rabi 2 shows a non monotonic behaviour with a maximum value before a decrease at large drive amplitude, most probably due to heating. For the natural silicon spin qubit, as the coherence time is still nuclear spin limited, the Rabi coherence time is roughly saturating at small frequencies, in contrast to the clearly increasing charge noise limited Rabi coherence time in the feedback controlled GaAs spin qubit. As such a careful optimization has to be done to reach the highest quality factor and qubit fidelity.

An important point is that in all the three qubits discussed here, the single qubit fault tolerance threshold F ≥ 0.99 based on surface code quantum error correction (QEC) [START_REF] Fowler | High-threshold universal quantum computation on the surface code[END_REF] has been demonstrated. QEC relies on scaling errors exponentially with polynomial number of operations (gates). The isotopically purified spin qubit has a quality factor ten times larger than the feedback controlled GaAs spin qubit Q28 Si ∼ 10Q GaAs so its base error rate is ten times smaller. However, we have also seen that the operation speed of the 28 Si spin qubit is ten times smaller f Rabi GaAs = 33 MHz ∼ 10f Rabi 28 Si with f Rabi 28 Si = 3 MHz. Therefore it is not completely clear that within the same operation time (including the time needed to perform QEC), the feedback controlled GaAs spin qubit does not reach comparable error rates as the 28 Si spin qubit due to the exponential scaling. One could argue that the feedback operation adds complexity but this would probably be at most (and probably less) comparable with the computational requirements of performing QEC. Ultimately, QEC as it is envisioned at the moment cannot be handled with the current classical technology (due to to constrains on the amount of data to be processed on the short coherence times of the qubits). Most probably, it would have to be dealt with on-chip cryogenic CMOS-like components [START_REF] Bohuslavskyi | 28nm Fully-depleted SOI technology: Cryogenic control electronics for quantum computing[END_REF]. And as of now, none of the spin qubits discussed so far are actually CMOS compatible, even the silicon ones. Regarding this technological barrier, CMOS silicon spin qubits are promising although their metrics need to be boosted to reach the fault tolerance threshold despite promising qubit operation frequencies of several tens of MHz [START_REF] Maurand | A CMOS silicon spin qubit[END_REF]. We will now put an end to our small venture on the slippery slopes of practical quantum computation and get back to more basic concerns regarding the development of spin qubits with carbon nanotubes.

Natural carbon CNT in a microwave cavity

This section discusses results published in Ref. [START_REF] Cubaynes | Highly coherent spin states in carbon nanotubes coupled to cavity photons[END_REF] In section 2.1 we saw that carbon nanotubes (CNT) hold great promises in terms of nuclear spin dephasing, even better than silicon. However, realizations of valley-spin qubits6 in CNT have revealed surprisingly short dephasing times of the order of ten nanoseconds [START_REF] Laird | A valley-spin qubit in a carbon nanotube[END_REF][START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF]. The conclusion was that the hyperfine interaction in CNT would be two orders of magnitude larger than what is theoretically expected [START_REF] Yazyev | Hyperfine Interactions in Graphene and Related Carbon Nanostructures[END_REF][START_REF] Bulaev | Spin-orbit interaction and anomalous spin relaxation in carbon nanotube quantum dots[END_REF] and what has been measured with fullerenes [START_REF] Pennington | Nuclear magnetic resonance of C 60 and fulleride superconductors[END_REF]. One possible explanation for this important dephasing could be large charge noise mediated by the valley degree of freedom, and interdot exchange [START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF]. However, these qubits showed promising operation frequencies in the range of several tens 2.3. FIGHTING AGAINST NUCLEAR SPINS DEPHASING of MHz. It is thus important to develop other types of CNT spin qubits, not based on valley-spin coupling, to try to observe and exploit their potential long coherence times. The approach we take is based on non-collinear ferromagnetic leads double quantum dot, as discussed in subsection 1.1.2, embedded in a microwave cavity [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF]. One reason to use a microwave cavity in a circuit quantum electrodynamics (cQED) architecture [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF][START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF] is because CNTs are inherently one dimensional. While it is possible to build several spin qubit devices in a row on a single CNT, it is quite unrealistic to envision a two-dimensional network built out of CNTs. However, such a two-dimensional array of qubits is required for scalability, both for space occupation and the implementation of QEC codes like the surface code [START_REF] Fowler | High-threshold universal quantum computation on the surface code[END_REF]. Another interest of using cavities is that they can enhance qubits lifetime, acting as narrow bandpass filters, through inverse Purcell effect [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF][START_REF] Goy | Observation of Cavity-Enhanced Single-Atom Spontaneous Emission[END_REF]. We will not go into the details of mesoscopic cQED or "mQED" in this short manuscript as it will not be necessary for the coming discussions (there are now several in-depth reviews that describe the field [START_REF] Cottet | Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena[END_REF][START_REF] Burkard | Superconductor-semiconductor hybrid-circuit quantum electrodynamics[END_REF]).

Spin qubits in microwave cavities have been demonstrated in various materials, first in CNT [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF] with the demonstration of a coherent coupling of the qubit to the cavity photons, then in Si [START_REF] Mi | A coherent spin-photon interface in silicon[END_REF][START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Borjans | Resonant microwavemediated interactions between distant electron spins[END_REF] and GaAs [START_REF] Landig | Coherent spin-photon coupling using a resonant exchange qubit[END_REF], all reaching the so-called strong coupling regime where the spin-photon coupling g s exceeds the qubit and photon loss rates γ and κ respectively. The strong coupling regime consisted in a milestone in the field as it is required to coherently manipulate the qubit with the photons (the slightly weaker coherent coupling of Ref. [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF] is actually already enough). The natural (or magnetic) single spin-photon coupling in a cavity is of the order of 10 Hz [START_REF] Kubo | Strong Coupling of a Spin Ensemble to a Superconducting Resonator[END_REF]. As the spin qubit decoherence rate obtained in these architecture is of about 1 MHz, it means that the effective spin-photon coupling has been boosted by five orders of magnitude using spin-electric-coupling as described in section 1.1 (the work of Ref. [START_REF] Landig | Coherent spin-photon coupling using a resonant exchange qubit[END_REF] used another type of qubit, an exchange-only qubit [START_REF] Medford | Self-consistent measurement and state tomography of an exchange-only spin qubit[END_REF], which presents a very large electric dipole and a qubit control purely relying on exchange interaction which does not require local magnetic fields). We will come back in the following to the interplay between the charge and spin character of the qubit in this architecture. Concerning the spin qubit dephasing times in these various implementations, they are found to be significantly smaller in Si devices compared to non cQED architectures, with T * 2 ∼ 60 -120ns when the qubit is resonant with the cavity [START_REF] Mi | A coherent spin-photon interface in silicon[END_REF][START_REF] Samkharadze | Strong spin-photon coupling in silicon[END_REF][START_REF] Borjans | Resonant microwavemediated interactions between distant electron spins[END_REF]] and T * 2 ∼ 600 ns when the qubit is off-resonant with the cavity [START_REF] Mi | A coherent spin-photon interface in silicon[END_REF]. With the CNT device, T * 2 ∼ 60 ns with the qubit on resonance with the cavity was observed [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF], about an order of magnitude higher than previously reported [START_REF] Laird | A valley-spin qubit in a carbon nanotube[END_REF] but still orders of magnitude smaller than expected. The origin of the larger dephasing rate in the cQED architecture compared to non cQED architecture is not understood yet but could arise from the electromagnetic environment of the cavity whose fundamental mode is much larger than the spin qubit device. This is why it is reasonable to think that there is room for improvement for the CNT device as it was not an ultra clean CNT which would have much less defects that could couple to the electromagnetic field of the cavity and induce dephasing.

With this in mind, we developed a high vacuum stapling technique that allows for the realization of ultra-clean CNT devices in microwave cavities during the PhD work of Tino Cubaynes. The development of this stapling technique started in the group of Z. Zhong [START_REF] Wu | One-Step Direct Transfer of Pristine Single-Walled Carbon Nanotubes for Functional Nanoelectronics[END_REF], and inspired several other groups in Delft, Basel, Regensburg, Barcelona and at the Weizmann institute. It allows for realizing a priori arbitrary complex circuits with CNTs. The principle of the stapling technique is depicted in Figure 2.11 and follows closely from the technique developed at the Weizmann institute which was the most successful [START_REF] Waissman | Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes[END_REF].

CHAPTER 2. SPIN QUBITS IN A NUCLEAR SPIN BATH ENVIRONMENT

Contrarily to previous fabrication techniques used in our team, the CNT is deposited on the nanocircuit electrodes at the last step, just before being transferred in the cryostat for measurement as depicted on Figure 2.11. Therefore the CNT is never exposed to any resists nor to any possibly strongly damaging electronic beams during the lithography process. The CNTs are grown on a comb of cantilevers (30 µm wide, 5 µm thick with a pitch of 60 µm). Then they are aligned with the nanocircuit electrodes thanks to micro-and nano-manipulators (piezo-electric actuators) with direct optical access in a high vacuum chamber around 5 × 10 -7 mbar. The actual contact between the CNT and the electrodes of the circuit is monitored electrically as the circuit is voltage biased and becomes closed when the CNT is deposited. A "touch-down" contact resistance of ∼ 10 MΩ -100 GΩ is typically observed. Then the CNT is cut on both sides of the electrical circuit by passing a strong current of the order of 10 µA. During this electrical cut step, an annealing of the contact between the CNT and the electrode occurs, which lowers the contact resistance to values of ∼ 100 kΩ-2 MΩ at room temperature, compatible with transport and microwave experiments in the milli-Kelvin range afterwards. We note that by adjusting the relative height of the electrodes and their pitch, it is possible to realize either suspended CNT device which offer high controllability or devices with the CNT lying on the electrodes. The latter choice is typically useful with magnetic gates because it maximizes the magnetic stray field at the CNT location, as will be discussed in the following chapter 3. In the resulting ultra-clean CNT devices, that are now routinely fabricated in the laboratory, we observe much cleaner, closer to ideal, electronic spectrum with a high controllability. A spin qubit device compatible with the CNT stapling technique is presented in Fig- 2.3. FIGHTING AGAINST NUCLEAR SPINS DEPHASING ure 2.12. Trenches for the cantilever stapling are visible in the cavity ground plane and the non-collinear ferromagnetic electrodes are fabricated as zig-zag to ensure having an angle of about 30 degrees between the two magnetization axis at the quantum dots locations, wherever the CNT is deposited. The working principle of this spin qubit based on artificial two-sites spin-orbit interaction [START_REF] Cottet | Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED[END_REF] is recalled on Figure 2.12(d,e). The local Zeeman field in each dot (achieved in our case by exchange interaction) lifts the spin degeneracy and the coupling between the two dots orbitals with different spin quantization axis create avoiding crossings giving the spectrum shown in Figure 2.12(d). Each K/K' valley of CNT generates a similar spectrum and we omit the valley index in this spectrum, for clarity. At large detuning energy between the two dots (outside the grey shaded area), the spectrum becomes one of a pure spin transition while it is a pure charge qubit spectrum at zero detuning (in the middle of the grey shaded area). The qubit is then manipulated by cavity photons (on or off resonance) and the qubit state population is read-out through the transmitted cavity field. In the following, only the phase of the cavity field will be measured for this matter. In this work the double quantum dot is operated in a regime of a few electrons occupancy, typically with around 10 electrons. This is to be compared to previously discussed spin qubits which were operated in the single electron regime as is standard. It is widely admitted that the single electron regime is more suitable, in particular because the spectrum is the simplest possible, but we can mention that it was also reported that multi-electron quantum dot spin qubits could be favourable in some particular electron occupancies [START_REF] Higginbotham | Coherent Operations and Screening in Multielectron Spin Qubits[END_REF]. While we could not play with the electron number (the reason being that the coupling of the spin qubit to the cavity field was not large enough with other low electron occupancy configurations [START_REF] Cottet | Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena[END_REF]), it is actually not a problem as it is typically only the last electron added to the nanotube which interacts electrically with the cavity.

The microwave spectroscopy of the ferromagnetic spin qubit is conveniently done by reading out in the dispersive regime the phase of the cavity signal when a second tone is applied through the cavity and its frequency is swept. At large detuning δ , the phase is mainly sensitive to the expectation value σ z of the spin projection along the quantization axis of the left (right) dot. In the dispersive regime, the expression of the phase φ reads

φ = 2g 2 s κ∆ σ z + φ 0 , (2.20)
where φ 0 is a constant which only depends on the microwave setup, κ is the linewidth of the cavity, g s is the spin-photon coupling strength and ∆ = f cav -f spin is the detuning between the cavity frequency f cav and the spin qubit frequency f spin . Such a measurement is shown in Figure 2.13(a) which displays the phase contrast ∆φ as a function of the tone frequency f pump and δ . In order to avoid cavity photon back-action on the spin qubit, we used a pulsed microwave spectroscopy : the qubit is first driven for t = 3 µs, then the cavity is filled after 90 ns and finally read-out using a fast data acquisition board for t = 700 ns. Apart from the frequency independent vertical blue stripe which simply signals the left/right degeneracy line at zero detuning, we observe three resonances which disperse close to zero detuning and saturate at 6.506, 6.530, and 6.540 GHz, respectively. The dispersion of each of these transitions with a minimum at zero detuning and a saturation at large detuning is characteristic of a transition which becomes a pure spin transition in the large detuning limit due to the perfect localization of the electron in one quantum dot (see Figure 2.12(d)). 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 can be attributed to the lifting of the K/K' valley degeneracy of the nanotube as well as from the fact that we are not in the single electron regime. A cut along the lowest resonance at CHAPTER 2. SPIN QUBITS IN A NUCLEAR SPIN BATH ENVIRONMENT large detuning is shown in Figure 2. 13 (b). This measurement, fitted by a Lorentzian, has a full width at half maximum of γ FWHM /(2π) = (498 ± 80)kHz which sets an upper bound of the decoherence rate γ s /(2π) ≤ γ FWHM /2/(2π) = 249 kHz. This is the main result of this work as such a narrow line width is two orders of magnitude lower than what was observed in the previous works with CNT and valley-spin qubit devices as discussed at the beginning of this section. Interestingly, the qubit is here off-resonance from the cavity and the measured spectroscopic line width compares to the ones mentioned in Si spin qubit devices in cavity. This is therefore the first time that a CNT shows its potential as a host for highly coherent spin qubits.

In order to specify the decoherence mechanism explaining the linewidth found for our spin transition, the dependence of the decoherence rate as a function of the detuning δ was measured. Such a measurement is displayed in Figure 2.14 (a). The two main decoherence sources for a spin qubit in double quantum dot are again charge noise and nuclear spin noise. The charge noise is related to the fact that the qubit 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 qubit transition frequency 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 residual constant of about 500 kHz to the derivative in Figure 2.14(a). The linear behavior of the decoherence rate γ s as a function of the derivative ∂ω/∂ δ , displayed in inset, shows that our spin-photon interface is dominated by charge noise at small detuning. Interestingly, it allows us from the slope of the linear behaviour to extract a charge noise detuning variance of about 34 µeV. While this noise is larger than in the previous work in carbon nanotubes [START_REF] Viennot | Outof-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF] and could be in principle easily lowered, it is interesting to see that we can completely reduce its influence by going at large detuning while keeping a large spin-photon coupling strength with respect to γ s . The shaded gray corresponds to the residual decoherence mechanisms with a decoherence rate in the range γ s /(2π) ≈ 500 kHz. Note that this value corresponds to twice as much as the lowest decoherence rate presented in Figure 2.13 (b) (250 kHz), probably because it corresponds to a lower detuning. This value of residual decoherence rate allows us to estimate the contribution of the nuclear spin noise due to the 1.1% 13 C present in CNT and therefore of the hyperfine coupling constant A CNT n . In Ref. [START_REF] Cubaynes | Highly coherent spin states in carbon nanotubes coupled to cavity photons[END_REF], we claimed that this number was for the first time compatible with the tabulated expected value of A CNT n = 0.1 -0.5µeV. What is actually correct is that this number is indeed compatible with the tabulated expected value when compared to previous experimental estimations in CNT. What it means is this : the correct formula for the dephasing time that we derived in section 2.1, Eq.(2.16), was not used because "unknown" to all of these works. Therefore the relative comparison between them is correct and we indeed observed a two orders of magnitude improvement of the spin decoherence rate in CNT, as observed from the resonance line width, with respect to previous works [START_REF] Laird | A valley-spin qubit in a carbon nanotube[END_REF][START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF], therefore reducing by about two orders of magnitude the upper bound on A CNT n . Let us now recalculate A CNT n of these works using the correct formulas Eqs. (2.16) and (2.17). We find for Ref. [START_REF] Pei | Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube[END_REF], A CNT n = 800 µeV instead of 400 µeV reported7 , and for Ref. [START_REF] Cubaynes | Highly coherent spin states in carbon nanotubes coupled to cavity photons[END_REF] A CNT n = 20 µeV instead of 0.1 µeV reported. The two orders of magnitude improvement in dephasing rate does not fully translate to a two orders of 2.4. PERSPECTIVES magnitude decrease in the estimation of the hyperfine constant due to the quantum dots not being of the same size, hence not having the same number of atoms, in both works. Looking at Figure 2.14(a) with fresh eyes, we can re-interpret the charge noise saturation by a still slightly decreasing contribution, probably with a quadratic dependence as the dot is in the deep Coulomb blockade regime and co-tunnelling process are dominating. With a tenfold decrease of the charge noise as already realized in the lab [START_REF] Viennot | Outof-equilibrium charge dynamics in a hybrid circuit quantum electrodynamics architecture[END_REF][START_REF] Bruhat | Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs[END_REF], we can reasonably expect to obtain a tenfold decrease in the observed nuclear spin noise and consequently the of hyperfine coupling constant.

To finish with the CNT spin qubit, we observed an optimal operation point due to the spin-electric-coupling, as for non cQED spin qubit discussed in the previous sections. The trade-off works as follows; the electron-photon coupling decreases with increasing detuning (and departure from pure charge qubit) as shown in Figure 2.14 (b) while the decoherence rate decreases with increasing detuning as the qubit becomes more spin-like as shown in Figure 2.14 (a). While in this work we could not performe time resolved manipulations of the qubit, Rabi oscillations, we can still characterize the optimal qubit working point. Namely, we look at the cooperativity of the spin-photon interface, defined as C = (2g s ) 2 /(κγ s ) which tells that the coupling between the two systems (spin and photon) is stronger than the geometric mean of their respective losses8 . Cooperativity is shown in Figure 2.14(c) as a function of detuning. Clearly, the decoherence rate decreases faster than the spin-photon coupling in the beginning, increasing C until it reaches a maximum at C = 12.9 before decreasing, meaning that the enhancement in the coherence time does not compensate for the decrease of the spin-photon coupling. We note here that C = 12.9 is the largest cooperativity reported in any spin-photon interface in cQED architecture so far.

Perspectives

The field of spin qubits in semiconducting host material has reached maturity in the last years, efficiently fighting nuclear spin noise and reaching beyond the fault tolerance threshold in the two most developed materials, GaAs and Si. The next challenge is now the one common to all qubit platforms, scaling-up the demonstrated single or two-qubits devices to many qubits (several tens first, then hundreds, thousands and millions). This task requires the assistance of companies, as is already the case with Google, IBM, Microsoft, Intel and many others already engaged in the race. There is still room for basic research but a significant part of the job has become what is now called "quantum engineering".

My personal perspectives for the coming years regarding spin qubits are all regarding CNT based spin qubits as CNT is now my main working host material. They can be divided into two directions, which have been discussed in this chapter. First there is the motivation to perform manipulations of spin qubits as described in subsection 2.3.3. With the stapling technique that allows to embed multiple devices in a single cavity and the development of cQED for superconducting qubits, CNT based spin qubits could be a credible alternative to Si based spin qubits with coherence times that are starting to show comparable values. Besides, coupling multiple spins in a cavity through microwave photons opens possibilities to tackle several fundamental problems. Indeed it would enable to go beyond first neighbour spin-spin coupling and could possibly allow one to investigate spin fluctuations in spin-spin coupling engineered diagrams that could have implication for high-Tc superconductivity for example. With this in mind, the next target is to realize a 3-spin qubit device which would allow for demonstrating the suitability of this architecture and play with quantum information manipulation (3 qubits Toffoli gate for example) and more condensed matter situation of long distance spin-spin coupling.

Second there is the desire to investigate more deeply the nuclear spins in CNT. Obviously the first target is to reduce the charge noise in future devices to measure nuclear spin noise and re-estimate the upper bound on the hyperfine coupling constant in CNT, hoping to reduce it closer to its expected value. Then there is nuclear spins dynamics in CNT. Is it also sub-diffusive as in GaAs? If yes, will it show a similar exponent α = 0.8? We can expect it will not be similar as "diffusion" in the CNT should be one dimensional compared to being two dimensional in 2DEG. Then it will be exciting to grow 12 C purified CNTs to further increase the dephasing time of spin qubits. A new chemical vapour deposition setup dedicated for isotopically purified carbon CNT growth has been developed in the laboratory and is expected to deliver nuclear spin free CNT in the coming year. The first use of such CNT will of course be for quantum information manipulation with the hope of reaching several tens of microsecond dephasing times similar to Si spin qubits. Another aspect is that with the typical quantum dots size we use and a residual concentration of 800 ppm of 13 C, we would get of the order of 20 nuclear spins only interacting with the electron spin qubit (comparatively, in purified 28 Si spin qubits, there are still about 100 -200 nuclear spins interacting with the electron spin). This limit gets very close to the purely quantum noise regime with typically around 10 nuclear spins [START_REF] Khaetskii | Electron spin decoherence in quantum dots due to interaction with nuclei[END_REF][START_REF] Schliemann | Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei[END_REF][START_REF] Klauser | Nuclear spin dynamics and Zeno effect in quantum dots and defect centers[END_REF][START_REF] Coish | Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics[END_REF][START_REF] Deng | Analytical solution of electron spin decoherence through hyperfine interaction in a quantum dot[END_REF]. In this limit, it is predicted that the quantum limited coherence time is smaller than the spin echo coherence time because spin echo would also compensate for quantum spin flips [START_REF] Yao | Theory of electron spin decoherence by interacting nuclear spins in a quantum dot[END_REF], which we could directly investigate.

Spin textures for topological excitations

In the current race for building a quantum information processor, there are various types of qubits in condensed matter that are actively developed. We have seen in the previous chapter that electron spin qubits in semiconductors are good candidates, still lacking the demonstration of their potential scalability compared to superconducting qubits which are slightly ahead, with the demonstration of "quantum supremacy" with a 53-qubit chips in 2019 [START_REF] Arute | Quantum supremacy using a programmable superconducting processor[END_REF]. At this stage, both these implementations are still very far from delivering a full scale quantum processor comprising millions of physical qubits.

One major reason is the need to perform quantum error correction which is very demanding in the number of physical qubits. An alternative approach is to use topologically protected qubits which possess an autonomous exponential protection to errors [START_REF] Kitaev | Unpaired Majorana fermions in quantum wires[END_REF] that would significantly reduce the hardware overhead. These topologically protected qubits rely on the existence of so-called Majorana zero modes (MZM) which are the condensed matter counterparts of Majorana fermions in particle physics [START_REF] Majorana | Teoria simmetrica dell'elettrone e del positrone[END_REF]. The promises of MZM based topologically protected qubits arise from their intrinsic exotic properties; they are their own antiparticle (they are said to be self-adjoint) and they possess non-abelian exchange statistics1 . The latter property says that the result of the exchange of two localized particles depends on the path taken during the exchange (either in real space or in phase space) and is named braiding as it is an operation of the braid group [START_REF] Nayak | Non-Abelian anyons and topological quantum computation[END_REF]. Braiding of MZM, beyond being a resource for topologically protected quantum computation, is a fascinating quantum operation by itself and is worth experimental demonstration on its own.

Majorana zero modes in 1D

While braiding can only be done with two dimensions (it was first proposed to braid vortices [START_REF] Ivanov | Non-abelian statistics of half-quantum vortices in p-wave superconductors[END_REF] in 2D topological superconductors where MZM should exist at the core of the vortices [START_REF] Moore | Nonabelions in the fractional quantum hall effect[END_REF]), these two dimensions do not need to be both, or either, in real space so that a 1D support is also suitable. In 2010, two theoretical proposals found a way to realize a non-trivial topological phase that can host MZM in 1D without relying on exotic materials [START_REF] Oreg | Helical Liquids and Majorana Bound States in Quantum Wires[END_REF][START_REF] Lutchyn | Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures[END_REF]. While the question of braiding MZM in 1D systems was not clearly settled theoretically at the time, 1D systems had several advantages that put them forward. Indeed the 1D systems envisioned at the time (that are still the main investigated systems as of now) were semiconducting nanowires which benefited from many technological advances in nanodevices fabrication. Indeed, the degree of control of the microscopic systems in semiconducting nanodevices (as the ones of the previous chapter) is unprecedented compared with 2D material science where the strengths lie more in spectroscopy than in manipulation. Still we note here that several experimental implementations of the latter have been recently realized in atomic chains [START_REF] Nadj-Perge | Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor[END_REF][START_REF] Pawlak | Probing atomic structure and Majorana wavefunctions in mono-atomic Fe chains on superconducting Pb surface[END_REF][START_REF] Jeon | Distinguishing a Majorana zero mode using spin-resolved measurements[END_REF][START_REF] Jäck | Observation of a Majorana zero mode in a topologically protected edge channel[END_REF] or islands [START_REF] Ménard | Two-dimensional topological superconductivity in Pb/Co/Si(111)[END_REF] where signature of MZM have been observed. The common ground between these implementations is that the non-trivial topological phase hosting the MZM is not intrinsic to an exotic material but rather engineered by combining the right ingredients from more conventional materials.

To understand these ingredients, we should first quickly come back to the origin. The fact that MZM are the counterparts of Majorana fermions arises from the equivalence of the Bogoliubov-De Gennes (BdG) Hamiltonian describing superconductivity with the Dirac equation. The superconducting gap and the particle-hole symmetry of the former are the equivalent of the mass term and the particle-antiparticle symmetry of the latter. Searching for eigenstates of the BdG Hamiltonian satisfying the self-adjointness condition of "Majorana fermions" γ = γ † with γ the annihilation operator of a MZM, it appears that either p-wave superconductivity is required, as initially envisioned [START_REF] Kitaev | Unpaired Majorana fermions in quantum wires[END_REF][START_REF] Read | Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect[END_REF], or spin rotation symmetry needs to be broken in a s-wave superconductor [START_REF] Fu | Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator[END_REF][START_REF] Sato | Non-Abelian Topological Order in ¡math display="inline"¿ ¡mi¿s¡/mi¿ ¡/math¿ -Wave Superfluids of Ultracold Fermionic Atoms[END_REF][START_REF] Sau | Generic New Platform for Topological Quantum Computation Using Semiconductor Heterostructures[END_REF][START_REF] Alicea | Majorana fermions in a tunable semiconductor device[END_REF]. p-wave superconductivity, sometimes referred to as topological superconductivity is an exotic type of superconducting pairing that is still difficult to realize experimentally. On the contrary, s-wave superconducting pairing is standard and spin-rotation symmetry can be broken if such pairing correlation is induced in a helical electron fluid. A helical electron fluid is found in systems with Rashba spin-orbit interaction and a Zeeman field perpendicular to it, which can be either realized intrinsically in edge states of topological insulators or in 1D systems with spin-orbit interaction under an external magnetic field (see Figure 1.1). The latter choice, being more simple in terms of material and benefiting from years of nanofabrication techniques development, is the one that was massively adopted by experimentalists. Therefore the recipe to engineer MZM in 1D systems requires the following ingredients to be combined : 1D semiconductor, strong spin-orbit coupling, homogeneous magnetic field perpendicular to it and s-wave superconductivity.

The most advanced devices made for that purpose are those combining superconductors and semiconducting nanowires [START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF][START_REF] Deng | Majorana bound state in a coupled quantum-dot hybrid-nanowire system[END_REF][START_REF] Zhang | Quantized Majorana conductance[END_REF] or 2DEGs [START_REF] Suominen | Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform[END_REF]. In these systems, the engineering of the topological phase is done using the large intrinsic spin-orbit interaction of the nanowires combined with superconductivity under a high magnetic field [START_REF] Kitaev | Unpaired Majorana fermions in quantum wires[END_REF][START_REF] Oreg | Helical Liquids and Majorana Bound States in Quantum Wires[END_REF][START_REF] Lutchyn | Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures[END_REF][START_REF] Lutchyn | Majorana zero modes in superconductor-semiconductor heterostructures[END_REF]. Superconductivity is now induced in the nanowire using epitaxial aluminium directly deposited at its surface, which provides hard superconducting gaps [START_REF] Higginbotham | Parity lifetime of bound states in a proximitized semiconductor nanowire[END_REF]. The readout of the emerging excitations is done by conventional transport measurements searching for robust zero bias conductance peaks (ZPCB), a signature of MZMs [START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF][START_REF] Deng | Majorana bound state in a coupled quantum-dot hybrid-nanowire system[END_REF][START_REF] Zhang | Quantized Majorana conductance[END_REF]. Manipulation of the MZM is expected to be performed by making 2D networks with nearest neighbour tunnel couplings [START_REF] Suominen | Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform[END_REF][START_REF] Lutchyn | Majorana zero modes in superconductor-semiconductor heterostructures[END_REF][START_REF] Aasen | Milestones Toward Majorana-Based Quantum Computing[END_REF][START_REF] Vijay | Teleportation-based quantum information processing with Majorana zero modes[END_REF]. This scheme was expected to have bear fruits two 3.1. MAJORANA ZERO MODES IN 1D years ago, which shows that this approach is very challenging.

It is important to stress at this point that, as of now, it has become widely accepted in the community that none of the observed ZBCP constitute an irrefutable proof of existence of MZMs in these systems, as they can be attributed to topologically trivial, Andreev-like, states [START_REF] Liu | Zero-Bias Peaks in the Tunneling Conductance of Spin-Orbit-Coupled Superconducting Wires with and without Majorana End-States[END_REF][START_REF] Lee | Zero-Bias Anomaly in a Nanowire Quantum Dot Coupled to Superconductors[END_REF][START_REF] Rainis | Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions[END_REF]. Recently, a ZBCP with quantized conductance (another expected characteristics of MZMs) was observed for the first time [START_REF] Zhang | Quantized Majorana conductance[END_REF], but the same team reported shortly after, that such a signature was actually not sufficient as it was not observed on large enough ranges of magnetic field and gate voltages [START_REF] De Moor | Electric field tunable superconductor-semiconductor coupling in Majorana nanowires[END_REF]. It was even theoretically shown that quantized ZBCP can be due to trivial Andreev states [START_REF] Moore | Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Majorana zero modes[END_REF], which now seems to have been confirmed experimentally [START_REF] Yu | Non-Majorana states yield nearly quantized conductance in superconductor-semiconductor nanowire devices[END_REF]. More generally, experiments now find ubiquitous ZBCP due to trivial Andreev states mimicking MZM [START_REF] Chen | Ubiquitous Non-Majorana Zero-Bias Conductance Peaks in Nanowire Devices[END_REF] which are well supported by several theoretical works [START_REF] Woods | Zero-energy pinning of topologically trivial bound states in multiband semiconductor-superconductor nanowires[END_REF][START_REF] Pan | Zero-bias conductance peaks in Majorana nanowires: the good, the bad, and the ugly[END_REF]. A well established theory group in the field even states that most, if not all, of the observed ZBCP in superconductor-semiconductor nanowire structures are actually not MZM due to, in particular, large disorder in the devices [START_REF] Pan | Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures[END_REF]. Another transport signature of MZM is the splitting oscillation of the ZBCP with magnetic field, wire length or chemical potential [START_REF] Cheng | Splitting of Majorana-Fermion Modes due to Intervortex Tunneling in a p x +ip y Superconductor[END_REF][START_REF] Sarma | Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire[END_REF][START_REF] Lin | Zero-bias conductance peak in Majorana wires made of semiconductor/superconductor hybrid structures[END_REF]. The amplitude of the oscillations should increase with increasing magnetic field or decreasing length, but the opposite was systematically observed [START_REF] Albrecht | Exponential protection of zero modes in Majorana islands[END_REF][START_REF] Albrecht | Transport Signatures of Quasiparticle Poisoning in a Majorana Island[END_REF][START_REF] Sherman | Normal, superconducting and topological regimes of hybrid double quantum dots[END_REF][START_REF] O'farrell | Hybridization of Subgap States in One-Dimensional Superconductor-Semiconductor Coulomb Islands[END_REF][START_REF] Shen | Parity transitions in the superconducting ground state of hybrid InSb-Al Coulomb islands[END_REF][START_REF] Vaitiekėnas | Selective-Area-Grown Semiconductor-Superconductor Hybrids: A Basis for Topological Networks[END_REF]. A possible explanation for this behaviour, compatible with MZM, was found using a spatial step-like spin-orbit interaction in the system [START_REF] Cao | Decays of Majorana or Andreev Oscillations Induced by Steplike Spin-Orbit Coupling[END_REF]. However a recent work shows that such a behaviour arises mostly in a region of the parameter space where the excitations are trivial Andreev states [START_REF] Sharma | Majorana versus Andreev bound state energy oscillations in a 1D semiconductor-superconductor heterostructure[END_REF]. In addition, the epitaxial superconductor at the surface of the nanowire is now also considered potentially detrimental to the emergence of the topological phase despite the induced superconducting hard gap, because amongst other things, it strongly affects the chemical potential in the nanowire [START_REF] Reeg | Metallization of a Rashba wire by a superconducting layer in the strong-proximity regime[END_REF][START_REF] Woods | Effective theory approach to the Schrödinger-Poisson problem in semiconductor Majorana devices[END_REF][START_REF] Mikkelsen | Hybridization at Superconductor-Semiconductor Interfaces[END_REF][START_REF] Antipov | Effects of Gate-Induced Electric Fields on Semiconductor Majorana Nanowires[END_REF]. Finally, the latest devices use a quantum dot in the normal state at the end of the 1D topological section to read out the ZBCP (this quantum dot actually forms on the side of the epitaxial superconductor not covering the whole nanowire and is thus a "side effect"). It has recently been theoretically shown that in such systems, the electron g-factor of the quantum dot is strongly renormalized and can decrease exponentially to zero for some orbitals, which would explain why some states appear to be pinned at zero energy in magnetic field [START_REF] Dmytruk | Renormalization of the quantum dot g -factor in superconducting Rashba nanowires[END_REF][START_REF] Jünger | Magnetic field independent sub-gap states in hybrid Rashba nanowires[END_REF]. It would therefore always be possible to find an apparently ZBCP robust to magnetic field in this architecture by looking for the right dot orbitals.

The previous paragraph summarizes well the current state of the field in what is now called the "zero bias peak controversy". What the debate points to is not that the observed transport signatures are not due to MZM, but that they could as well be due to topologically trivial states. The reasons for doubting the nature of the observed ZBCP seem to arise from the materials and device design as well as from the readout scheme. Disorder in the nanowires which are not 1D enough (typically nanowire devices show of the order of about ten conduction channels) and disorder in the spin-orbit interaction and in the chemical potential which are affected by the epitaxial superconducting material. Ultimately, it appears clearly that transport measurements can hardly distinguish between trivial and non-trivial topological states. This analysis calls for the need of developing alternative platforms that could host MZM as well as using other experimental tools to evidence and manipulate their exotic properties. Carbon nanotubes are good candidates for the host material as they are near ideal 1D conductors2 that can be ultra-clean (without defects) as discussed in subsection 2.3.3. Carbon nanotubes have a relatively large spin-orbit interaction due to their curvature, theoretically predicted and measured to be of the order of several hundreds of µeV (see Ref. [START_REF] Laird | Quantum transport in carbon nanotubes[END_REF] table. II), with even a few meV value reported [START_REF] Steele | Large spin-orbit coupling in carbon nanotubes[END_REF], one order of magnitude larger than expected. While the spin-orbit strength in CNT is comparable to those found in semiconducting nanowires like InAs or InSb (respectively 0.015 -0.135meV and 0.25 -1meV [START_REF] Yokoyama | Josephson Current through Semiconductor Nanowire with Spin-Orbit Interaction in Magnetic Field[END_REF]), it is not of Rashba type and thus cannot induce a helical fluid in the presence of a perpendicular magnetic field. This is where the proposals to induce a synthetic Rashba spin-orbit interaction with a periodic spatially varying magnetic field [START_REF] Klinovaja | Topological Superconductivity and Majorana Fermions in RKKY Systems[END_REF][START_REF] Braunecker | Nuclear magnetism and electronic order in C13 nanotubes[END_REF][START_REF] Klinovaja | Transition from Fractional to Majorana Fermions in Rashba Nanowires[END_REF][START_REF] Kjaergaard | Majorana fermions in superconducting nanowires without spin-orbit coupling[END_REF][START_REF] Egger | Emerging Dirac and Majorana fermions for carbon nanotubes with proximity-induced pairing and spiral magnetic field[END_REF][START_REF] Klinovaja | Giant Spin-Orbit Interaction Due to Rotating Magnetic Fields in Graphene Nanoribbons[END_REF] discussed in section 1.2 becomes very attractive. In addition to the synthetic spin-orbit interaction, the cycloidal magnetic field generated by a magnetic texture also induce autonomously a Zeeman field perpendicular to it. This aspect is particularly interesting as it could allow for MZM to emerge without the use of a global external magnetic field. It would therefore make the devices more compatible with complex geometries (where it is basically impossible to always satisfy the global magnetic field to be perpendicular to the spin-orbit interaction of all topological parts of the device) as well as with superconductivity whose order is destroyed by strong magnetic fields. We will come back to these points in section 3.3. We will now discuss in the coming section the experimental realization and observation of the synthetic spin-orbit interaction in a carbon nanotube.

Revealing synthetic spin-orbit interaction with magnetic field

This section discusses results published in Ref. [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF] The device that we designed differs in several points from now conventional devices. The device geometry is schematically depicted in Figure 3.1(a) and Figure 3.2(a). As said in the introduction of this chapter, we use a carbon nanotube as it is the closest possible to an ideal 1D conductor. It is deposited on top of a magnetic texture with alternating magnetic domains pointing out of plane to generate a synthetic Rashba spin-orbit interaction in the the conductor. Finally, compared to all current devices implementations for generating MZM, we will induce superconducting correlations from one lead (or both), meaning from the side of the 1D conductor and not along the 1D portion with either epitaxial superconductor [START_REF] Higginbotham | Parity lifetime of bound states in a proximitized semiconductor nanowire[END_REF] or the device deposited on a superconducting layer [START_REF] Nadj-Perge | Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor[END_REF]. This last point is critical as many believe that the spin-orbit region and the superconductor need to spatially overlap in order to induce the topological phase which can host the MZM. However as we have seen from the short literature review above, the "conventional" geometry seems to lead to many parasitic side effects that cast doubts on the actual possibility to induce the topological phase. We therefore investigated theoretically our non-conventional geometry using tight-binding calculations on a Kitaev chain, a portion of which has superconducting correlations and another portion being normal with a spatially varying magnetic field, both being connected through a tunnelling element, as shown in Figure 3.1(b). Calculating the density of states (DOS) along the chain as a function of energy, shown in Figure 3.1(c), we observe that it is possible to find two zero energy modes, localized at both ends of the normal region, with the one at the normal-superconductor interface being more strongly 3.2. REVEALING SYNTHETIC SPIN-ORBIT INTERACTION WITH MAGNETIC FIELD localized. The Majorana character of these zero energy modes is evidenced by calculating the spin singlet and spin triplet correlations of the DOS independently, which shows that these zero energy modes are mainly triplet-like, as expected for MZM while the higher delocalized modes are mainly singlet-like (details can be found in the supplementary material of Ref. [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF] or in the thesis of L. Contamin [START_REF] Contamin | Mise en évidence de textures de spin synthétiques par des mesures de transport et de champ microonde[END_REF]). One important aspect though is that it is needed to have at least one site of overlap between the oscillating magnetic field and the superconductor to observe the appearance of the localized zero energy modes. We can understand this as the necessity that the rotating field must induce triplet correlations right in the superconductor, at least at the interface. This is experimentally relevant as the stray field of the magnetic texture will extend laterally beyond its edges. These preliminary calculations are thus encouraging that the device geometry we chose is well suitable for the emergence of a topological phase that can host MZM. The device is shown in Figure 3.2(b) and (e). A single wall nanotube is stamped3 onto a magnetic Co/Pt bottom gate bottom gate which is capacitively coupled to two gate electrodes, Gate 1 and Gate 2. The Co/Pt is expected to have a small pitch and an out of plane anisotropy, giving rise to several domains over the length of the nanotube, with a strong stray field of about 0.4 T as supported by magnetic characterization and simulations. The Magnetic Force Microscope (MFM) picture shown in Figure 3.2(e) evidences magnetic domains in the bottom gate, which have a typical size of about 100 -150nm. An external magnetic field B ext changes the magnetic structure and can therefore reveal the existence of the synthetic spin orbit interaction. Superconducting correlations are induced by connecting the nanotube to two Nb/Pd superconducting electrodes. We address the discrete spectrum induced by the superconductor by transport spectroscopy. The typical measurement of the differential conductance G as a function of source-drain bias V sd is shown in figure Figure 3.2(c). The conductance displays a well-defined energy gap of about 550 µeV containing two peaks, symmetric with respect to zero bias. These two peaks signal Andreev-like states (ALS) arising from superconducting correlations. As sketched in Figure 3.2(d), our measurements are equivalent at low energy to conventional tunnel experiments as a consequence of the finite density of states at the Fermi energy in one of the two superconducting contacts (contact 2). Such a residual density of states in the superconducting leads is systematically observed in our devices and has also been reported by other groups (see for example Ref. [START_REF] Su | Mirage Andreev Spectra Generated by Mesoscopic Leads in Nanowire Quantum Dots[END_REF]). This is why the device was sketched with a normal contact and a superconducting one in Figure 3.1(a). The global shape of the conductance curve is well accounted for by the quasi-classical dethin film deposition.

N S

REVEALING SYNTHETIC SPIN-ORBIT INTERACTION WITH MAGNETIC FIELD

scription of superconductivity in the electrodes, based on Usadel equations and reveals that contact 1 displays a well-defined superconducting hard gap. The large subgap slope is shown to arise mainly from a residual pair-breaking in one of the superconductor (contact 2 in Figure 3.2(d)). The ratio between the high bias conductance and the zero bias conductance which measures the "hardness" of the gap is of about 45 which compares favourably with the recently reported figures in semiconducting nanowires [START_REF] Albrecht | Exponential protection of zero modes in Majorana islands[END_REF].

One of the main findings of the experiment is displayed in figure Figure 3.3(c). In this colour scale map of G as a function of V sd and the external magnetic field B ext , we observe the evolution of the ALSs under an external magnetic field. They display oscillations with a period of about 0.6 T (±10% from one magnetic sweep to another). We can resolve up to three oscillations around the mean energy of 220 µeV, together with the expected slow reduction of the superconducting gap. Such a behaviour is unusual for ALSs and has not been observed in any other system. It stems from the progressive alignment of the magnetic domains with the global magnetic field as depicted in Figure 3.3(a) which shows how the magnetic domains evolve from pointing out of plane with alternate orientations to all aligning in plane at a saturation magnetic field. The evolution of the magnetic texture domains leads inevitably to an evolution of the underlying helical band structure it induces. We can therefore get insights on the observed oscillations of the ALSs from the energy dispersion of electrons subject to a rotating magnetic field, E(K) with K the wave vector, shown in figure Figure 3.3(b). The interference conditions defining the energies of the ALSs are set by the wave vectors difference ∆K between right-moving and left-moving electrons with non-orthogonal spins eigenstates. A variation of the magnetic domains induces a shift k in the wave vectors K. Near the helical gap, where the spin states are not orthogonal, it adds a term 2kL to the interference condition :

E ALS ≈ ±E ALS,0 (1 + a cos [2∆K(B ext )L]) , (3.1)
with ∆K(B ext ) = ∆K(B ext = 0)+2k(Bext), E ALS,0 the ALS energy at B ext = 0 T and a the relative amplitude of the oscillations. The ALSs magnetic field dependence is well accounted for by this formula, under the assumption that the spin-orbit strength decays linearly as the field increases, up to a saturation field of about 1 T. Such an evolution of the synthetic spin orbit energy is supported by magnetic measurements as well as micromagnetic simulations.

The number of oscillations N sets the range of modulation of k(B ext ) and therefore allows us to give a lower bound for the induced spin-orbit energy at zero magnetic field: E so > δN/2. From the number of oscillations in Figure 3.3(c) for B ext > 0 (N ∼ 1.5) and the extracted level spacing δ ∼ 1.5 meV, we deduce E so > 1.1 meV. This is of the order of the simple estimate for a linear spectrum [START_REF] Kjaergaard | Majorana fermions in superconducting nanowires without spin-orbit coupling[END_REF][START_REF] Egger | Emerging Dirac and Majorana fermions for carbon nanotubes with proximity-induced pairing and spiral magnetic field[END_REF] E so = hv F /(2λ) = δ(L/λ) with L/λ ∼ 2 corresponding to about 5 domains, inferred from the MFM picture in Figure 3.2(e). Strikingly, this spin-orbit energy is larger than the ones found in InSb or InAs nanowires. Moreover, we can reproduce the ALSs oscillations with simulations based on the scattering theory, with δ and ∆ ∼ 0.6 meV extracted from the data, an amplitude of the stray field B osc of 400 mT extracted from the magnetic simulations and a chemical potential close to the helical regime. These oscillations are robust to disorder in the magnetic texture, as studied numerically in detail in the supplementary material of Ref. [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF]. They can also be qualitatively reproduced from the spatial field evolution inferred from the MFM data of Figure 3.2(e). We point out here that the method we employed to estimate the spin-orbit interaction in our system is novel. Previous estimations in semiconducting nanowire devices comprises notably transport measurements, looking for a dip in the conductance as the chemical potential goes through the helical gap [START_REF] Kammhuber | Conductance through a helical state in an Indium antimonide nanowire[END_REF][START_REF] Sun | Helical Hole State in Multiple Conduction Modes in Ge/Si Core/Shell Nanowire[END_REF]. However in these setup, no superconducting correlations are induced in the nanowire and especially no superconductor is epitaxially deposited on the nanowire, which could have detrimental effects for the spin-orbit interaction as reported. Therefore, our new method is the first that can estimate the spin-orbit energy in a nanowire device combining all the ingredients to host the topological phase of MZM.

The large measured value of spin-orbit interaction is an important prerequisite for driving a hybrid device into the topological regime, where zero energy Majorana modes can emerge. In all the devices experimentally investigated so far, this has only been pursued by applying a large external magnetic field, with severe constraints on network designs, Majorana mode lifetimes, and coupling to superconducting quantum circuits. In contrast, 3.3. PERSPECTIVES our magnetic texture is equivalent to both a finite and large spin-orbit interaction and an external magnetic field: our device could host Majorana modes without any external magnetic field, thus lifting these constraints. In Figure 3.4, at zero external field, a ZBCP emerges, simply upon tuning Gate 2 at V g > 0.5 -0.6V. We note that this gate does not affect the ALSs nor the superconducting gap but only the appearance of the ZBCP along with a slight increase of the conductance background. The ZBCP has a width of about 150 µeV as shown in Figure 3.4(b), and a height of about 0.05e 2 /h, comparable to the recent findings in semiconducting nanowires (see e.g. Ref [START_REF] Deng | Majorana bound state in a coupled quantum-dot hybrid-nanowire system[END_REF]). In addition, in our case the finite slope of the probe contact density of states affects the conductance height which cannot be mapped directly on the spectral weight of the states in the nanotube. In Figure 3.4(d), we measure a large magnetoresistance of 20% for this ZBCP, accompanied by a hysteretic behaviour which is a signature of the effect of the magnetic texture. This strong dependence at small magnetic field could come from local reconfiguration of the magnetic domains, consistent with the expected spatial localization of the state corresponding to a Majorana peak, contrary to the finite energy ALSs which are not affected by a small magnetic field. Finally, Figure 3.4(c) displays a conductance map where the ZBCP is robustly pinned at zero energy at large external magnetic field. These features are compatible with the ZBCP indicating the presence of a MZM, however they most probably still fall in the grey area of the zero bias peak controversy and more advanced manipulations of these states are needed to confirm their nature.

As a conclusion, we have demonstrated a device with a synthetic spin-orbit interaction induced by a proximal ferromagnetic multilayer producing an inhomogeneous local magnetic field. This spin-orbit interaction deeply modifies the superconducting correlations induced by superconducting contacts and allows us to observe a zero bias peak suggestive of a Majorana mode without any external magnetic field. By relaxing the constraint of an external magnetic field, our setup is suitable for advanced experiments that would unambiguously characterize Majorana modes with the tools of cQED circuits [START_REF] Cottet | Squeezing light with Majorana fermions[END_REF][START_REF] Virtanen | Microwave spectroscopy of Josephson junctions in topological superconductors[END_REF][START_REF] Müller | Detection and manipulation of Majorana fermions in circuit QED[END_REF][START_REF] Dartiailh | Direct Cavity Detection of Majorana Pairs[END_REF][START_REF] Väyrynen | Microwave signatures of Majorana states in a topological Josephson junction[END_REF]. The use of a magnetic texture also enables obtaining Majorana modes in any conductor, such as CNTs but also graphene, Si/SiGe 2DEG. . . The built-in 2D pattern of our magnetic textures could also be interesting for braiding schemes [START_REF] Fatin | Wireless Majorana Bound States: From Magnetic Tunability to Braiding[END_REF] which could require networks of Majorana modes with local and autonomous generation of topological superconductivity.

Perspectives

As outlined throughout the previous section, the next move in the community should be to go to more advanced experiments. The motivation to realize topologically protected qubits inevitably requires to achieve manipulation of individual MZM, which in turn seems now to be the only way to unambiguously determine if the observed ZBCP signatures are indeed due to MZM. My perspectives in this field are therefore to exploit the synthetic spin-orbit interaction and device geometry discussed in the previous section to go beyond transport experiment to reveal the existence and exotic properties of MZM. As outlined, one of the advantages of the magnetic texture is that it induces autonomously both a spin-orbit interaction and a perpendicular Zeeman field which makes it possible to enter the topological phase at zero external magnetic field. The magnetic texture technology is thus fully compatible with conventional superconductor microwave resonators with which I plan to use the tools of cQED to probe and manipulate MZMs (current experiments with MZM devices in cavities were limited to zero external magnetic field and the investigation of trivial Andreev bound states [START_REF] Hays | Direct Microwave Measurement of Andreev-Bound-State Dynamics in a Semiconductor-Nanowire Josephson Junction[END_REF]). The first objective will be to evidence the self-adjoint character of MZM. The self-adjoint property γ = γ † with γ the Majorana fermionic operator of a MZM induces a non standard longitudinal coupling of the MZM pair to the cavity mode [START_REF] Cottet | Squeezing light with Majorana fermions[END_REF]. Such a longitudinal coupling has been considered as a hindrance or at best an indirect way to detect the selfadjointness of MZM [START_REF] Dartiailh | Direct Cavity Detection of Majorana Pairs[END_REF]. I propose to directly exploit the longitudinal coupling to read out the state of the MZM topological qubit, a scheme recently proposed in cQED with superconducting qubits [START_REF] Didier | Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction[END_REF] and with spin qubits [START_REF] Lambert | Amplified and tunable transverse and longitudinal spin-photon coupling in hybrid circuit-QED[END_REF]. As this protocol only works with systems longitudinally coupled to the microwave photons, it will unambiguously reveal the self-adjointness, hence the non-trivial character, of the MZM. This is in contrast to the more common, and widely used in the cQED community, transverse coupling readout scheme which will be established for trivial excitations only, typically Andreev-like states.

In short, a MZM pair γ1 γ2 , possibly realized in the experiment of the previous section, would be coupled to the microwave field of a cavity with a coupling g. This coupling is then modulated at the cavity frequency (using a local radio-frequency electrostatic gate, the coupling g being proportional to the overlap of the MZM electronic wave functions) to turn on the longitudinal readout [START_REF] Didier | Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction[END_REF]. It actually performs a parity measurement P12 = iγ 1 γ2 = ±1 of the MZM pair, directly measurable as a displacement of the The longitudinal readout scheme therefore essentially realizes a parity measurement of a MZM pair, which is at the heart of measurement-only manipulation of MZM. It relies on a sequence of parity projective measurements instead of actually physically displacing the MZM around each others [START_REF] Bonderson | Measurement-Only Topological Quantum Computation[END_REF]. Such a protocol is sometimes referred to as "braiding without braiding" or braiding with quantum teleportation. A technically very challenging implementation of this protocol, based on tunneling transport measurement, was recently proposed [START_REF] Vijay | Teleportation-based quantum information processing with Majorana zero modes[END_REF]. In the longer run I plan to exploit the longitudinal coupling scheme to perform the successive projective parity measurements in a device hosting four MZM. The extension of number of MZM can readily be done by adding a second magnetic texture to a device similar to the one previously described. A schematics representation of this proposed device is shown in Figure 3.5 (a) with the expected localization of the four MZM γ i for i ∈ {1, 2, 3, 4}. We have already started to develop this geometry in the lab with two magnetic textures on the side of a central electrode as shown in Figure 3.5(b,c). We actually embedded such a device, with a tunnel barrier instead of the central superconducting lead thus forming a double quantum dot, inside a microwave cavity. The double quantum dot geometry allowed us to probe the internal transitions between orbitals of the two dots. The magnetic textures induce a spin texture to each dot subject to their rotational stray field. This in turn give the double quantum dot transitions a non-trivial spin component that can be probed with the microwave field, in a similar manner to the two-site artificial spin-orbit coupling scheme of subsection 1.1.2 and subsection 2.3.3. The microwave signal of one of these transitions as a function of the energy detuning between the two dots and an external longitudinal magnetic field is shown in Figure 3.6(a) and (b). We observed a non monotonous dispersion of the transition with slopes corresponding to a electron g-factor of up to 60. It is important to stress that what is measured along the detuning axis is the difference in energy between the energy levels of the two orbitals. Therefore a naive interpretation of the dispersion slope is that it corresponds to the difference of g-factors of each dot, thus implying that one or both g-factors are actually larger than 60.

The naive model consists in a spin qubit with non-collinear magnetization as discussed in previous chapters, only with significantly stronger electron g-factors, dependent on the orbital due to the overlap of the electron wave function with the magnetic texture field. We see in Figure 3.6(c) and (d) that with this model we can quantitatively account for the shape of the transition dispersion and qualitatively for the contrast of the signal. This model however does not yield a microscopic understanding and we still need to investigate how the synthetic spin-orbit interaction induced in each dot can lead to such large g-factors. An important follow-up experiment with this device is to reach in one (or ideally both) dot the helical gap by tuning the chemical potential. There we would get more insights on how the magnetic texture shapes the spin component of the dot levels. This preliminary experiment combined with the experimental results of previous section as well as the theoretical calculations showing that our geometry can in principle host MZMs is thus encouraging that we can couple a MZM pair to a microwave cavity and probe their self-adjoint character. Finally, another perspective is to go back upstream and investigate devices with less combined ingredients. Indeed, in the experimental work presented in this chapter, we found a way to probe the synthetic spin-orbit interaction in a carbon nanotube device induced by a magnetic texture in the presence of superconductivity. This was absolutely relevant by itself and the observation of a ZBCP at zero magnetic field, however controversial it might be or appear, confirmed that this new device geometry is promising. Nevertheless, adding all those ingredients together can make it difficult to finely understand the role of each. This is why we devised the device previously mentioned with magnetic textures but no superconductivity. As we use an external magnetic field to probe the spin-orbit interaction from the oscillations of Andreev-like states, it would be very instructive to perform the same measurements on a device without a magnetic texture. This would probably be more than just a control experiment. Indeed, Andreev states in 1D systems are mostly not investigated with magnetic field but rather by controlling the superconducting phase, the chemical potential of the normal region where they exist or temperature (see for example Refs. [START_REF] Franceschi | Hybrid superconductor-quantum dot devices[END_REF][START_REF] Pillet | Andreev bound states in supercurrent-carrying carbon nanotubes revealed[END_REF][START_REF] Lee | Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures[END_REF][START_REF] Kumar | Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot[END_REF][START_REF] Gramich | Resonant and Inelastic Andreev Tunneling Observed on a Carbon Nanotube Quantum Dot[END_REF][START_REF] Gramich | Andreev bound states probed in three-terminal quantum dots[END_REF][START_REF] Murani | Microwave Signature of Topological Andreev level Crossings in a Bismuth-based Josephson Junction[END_REF]). In this kind of experiments a magnetic field is typically applied to destroy 3.3. PERSPECTIVES superconductivity and perform a control measurement. To our knowledge, only few works reported on the evolution of Andreev states outside the topological phase under continuous tuning of an external magnetic field, in InAs nanowires with Al contacts [START_REF] Lee | Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures[END_REF] and InSb nanowires with Al or NbTiN contacts [START_REF] Su | Mirage Andreev Spectra Generated by Mesoscopic Leads in Nanowire Quantum Dots[END_REF]. The conclusion of both experiments is that great care should be taken when interpreting MZM experiments. There can be a ZBCP without topological superconductivity due to a competition with the Kondo effect [START_REF] Lee | Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures[END_REF] and the tunnelling probe can be more complex than anticipated because of confined quantum states adjacent to the nanowire section [START_REF] Su | Mirage Andreev Spectra Generated by Mesoscopic Leads in Nanowire Quantum Dots[END_REF]. Carbon nanotubes being much closer to ideal 1D conductors than semiconducting nanowires, and being also potentially much cleaner, are therefore very attractive to investigate the behaviour of Andreev states with magnetic field. Will such a simple and clean system without topological superconductivity yield the simple expected behaviour of oscillating Andreev states with magnetic field on a period larger than the level spacing as theoretically predicted [START_REF] Desjardins | Synthetic spin-orbit interaction for Majorana devices[END_REF]? Or will we find more complex signatures than expected that would impact the interpretations of MZM experiments? Any way, it is clear that we still lack such kind of experimental investigations in lower complexity devices that are crucial to properly understand the features observed in more complex MZM devices.

Conclusion

We have seen through the studies presented in this manuscript that the quantum property of the spin, although relatively simple in essence, still brings many attractive prospects in "modern" condensed matter, despite being one of the first quantum property discovered a century ago. Manipulation of a single spin or the shaping of the spin texture of electronic bands can be efficiently achieved with spin-orbit coupling. While nature can provide spin-orbit coupling intrinsically in materials, it is not always of the right form or of magnitude large enough. Hybrid circuits which combine materials of various dimensionalities and with different electronic degrees of freedom make it possible to engineer the desired Hamiltonian with the right spin-orbit coupling term. This is clearly far more a starting point than the final achievement.

The realization of good spin qubits reaching the fault tolerance threshold could only be done by actively fighting against the nuclear spin noise of the environment, either by exploiting the slowness of its fluctuations or by removing the nuclear spins, in combination with a strong spin-electric coupling. This is the first milestone on the road to quantum computation as it proves that these qubits are viable. The next move is to deal with the challenging task of scalability, which requires more man power and technological efforts. Or it can also be "insert a coin and play again", trying to find yet another alternative platform with promising properties. This is the way I want to pursue as it suits more my personal approach to experimental physics. Spin qubits in carbon nanotubes are quite close in essence to the semiconductor spin qubits aforementioned, with possibly better performances in terms of coherence time. And of course there remains many fundamental aspects to explore, in particular with regards to nuclear spins.

The other path is the one of topologically protected qubits based on Majorana zero modes for which spin-orbit interaction is a key ingredient and which are still far from the fault tolerance threshold milestone. Indeed, demonstration of a qubit is still missing, let alone if its fidelity can hold to the promise. At this stage, as underlined with the "zero bias peak controversy", it even remains to be experimentally demonstrated if all the ingredients of the 1D Majorana recipe are actually combining well in the devices, in the sense of understanding if they survive and behave as expected. In this respect, we demonstrated the first observation of strong enough a spin-orbit interaction in a device combining all the ingredients. The helical gap remains elusive though, and the behaviour of non topological states with external parameters such as magnetic field or chemical potential seems more complex than anticipated and thus would require some more in-depth investigations. However, it is also important to try to directly evidence the existence of the topological and exotic nature of MZMs, with probes other than transport measurements. Coupling pairs of MZMs (forming a qubit) to the photon field of a microwave cavity seems a promising way of testing their self-adjoint character and even perform braiding. While eminently challenging technologically, we already showed preliminary results hinting at the possibility to realize the required devices and measurements.

Finally, an important message from chapter 2 is that for a qubit to have long coherence times is not necessarily enough to make it a good practical qubit. There needs also to be a good controllability with sufficiently large operation speed to reach the fault tolerance threshold. While the first step for a qubit to prove its viability is obviously to reach this stage, it arises right after the question of the operational frequency of the future quantum processor on which will run quantum error correction codes. We briefly outlined that the apparently worst of the qubits having reached fault tolerance could turn out to be better than the others depending on how quantum error correction codes are actually implemented. Until practical implementations are realized, it would therefore seem unreasonable to discard any qubit that could meet the fault tolerance threshold. As of now, superconducting qubits look to be quite ahead of other types of qubits but I honestly think that looking at the entire quantum computer road, almost all implementations are somehow at the same point. We are at an interesting stage where both quantum technology is developing fast to reach the holy grail and there is still plenty of room for fundamental condensed matter physics phenomena to explore. Both are important and needs one another.

Figure 1 . 1 :

 11 Figure 1.1: Evolution of the electronic bands of a 1D semiconductor (a) when adding first a Rashba type spin-orbit energy (b) and an additional magnetic field (c). The colorcode represent the spin eigenvalue, except when the bands are degenerate in spin with no favored orientation (black). Figure taken from Ref. [17].

Figure 1 . 2 :

 12 Figure 1.2: (a) Micromagnet spin-electric-coupling fields. The magnet is designed to induce a spatially inhomogeneous stray field B MM at the quantum-dot position when magnetized along B ext z. The transverse coupling is produced by the inhomogeneous component perpendicular to B ext and is proportional to the field slope b t = (u • ∇)B ⊥ MM where u is the unit vector along an in-plane (yz) electric field, ∇ denotes the vector differential operator and ⊥ indicates the component perpendicular to B ext . The longitudinal coupling is, in contrast, mediated by the gradient of the parallel component b l = (u • ∇)B z MM . The quantum-dot confinement is assumed to be strong vertically (along x) and symmetric laterally. (b) Typical design of a micro magnet which is optimizing the slanted field b l at the location of the quantum dots for the situation of a modulation-doped GaAs/AlGaAs wafer with a 2DEG 57 nm below the surface with a 100 nm insulating layer on top. Figures are taken and adapted from Refs. [4, 19].

CHAPTER 1 .Figure 1 . 3 :

 113 Figure 1.3: (a) Micromagnet magnetic stray fields B x MM ≡ B x and B z MM ≡ B z as a function of z at the position of a quantum dot, calculated for a micromagnet with a design similar to the one of Figure 1.2(b). (b) Micromagnet magnetic stray fields at the position of a quantum dot as a function of time for a cosine displacement of the quantum dot of amplitude 50 nm centred around z = 0 nm and period T 0 . In typical experiments, the displacement is of the order of 20 nm, which is the confinement length. (c) Angle of the magnetic field at the quantum dot position as a function of time.

1. 1 .Figure 1 . 4 :

 114 Figure 1.4: (a) General principle of the two-site spin-orbit coupling mechanism with noncollinear ferromagnetic electrodes activated by cavity photons. The proximity of the noncollinear ferromagnets induces a different equilibrium spin orientation if an electron is localized in the left or in the right dot. Photons are coupled to transitions changing the localization of the wave function ψ and hence coupled to transitions changing the spin orientation. (b) Principle of the confinement-induced exchange field arising from tunnel coupling to a ferromagnetic lead. φ σ is a spin-dependent interfacial phase shift and η is the electronic path acquired by the electronic phase when propagating through the coherent conductor over a length L. (c) and (d), principle of the two-site spin-orbit coupling mechanism with a micromagnet slanted magnetic field. Despite the continuous character of the slanted field, the two specific locations of the quantum dots turn it to a discrete two-site spin-orbit coupling with equivalent non-collinear magnetic fields composed by B x being different for each dot and B z being the same. Figures are taken and adapted respectively from Refs. [24, 27-29].

Figure 1 . 6 :

 16 Figure 1.6: Magnetic stray field vector fields as a function of space for the case of (a) the magnetic texture of Figure 1.5 and as a function of time for the case of (b) the micromagnet slanted field with periodic displacement of the quantum dot of Figure 1.3.

  Figure 2.1(a). A micromagnet as described in subsection 1.1.1 generates a magnetic field difference ∆B z M M between the dots 2 , setting the average frequency of the qubit f = |g|µ B ∆B z /2π with ∆B z = ∆B z MM + ∆B z nuc and g = -0.44 the electron g-factor in GaAs. The leftmost spin qubit is left idle during all operations in this work.

Figure 2 . 1 :

 21 Figure 2.1: (a) SEM micrograph of a similar device to the one measured. Lateral gates defining quantum dots (bottom) and charge sensors (top) are shown in light grey on the dark grey surface of the GaAs substrate. The three leftmost quantum dots are formed and manipulated while the upper left charge sensor, connected to an rf-reflectometry circuit is used. The "C-shaped" light colored area denotes the micromagnet providing inhomogeneous magnetic field. An external magnetic field B ext z = 0.7 T is applied. (b) Charge stability diagram in the plane defined by plunger gates P 1 and P 3 . The positions for initialization (I), operation (O) and measurement (M) configurations are denoted.

  Figure 2.2(a) is presented on Figure 2.2(b), showing how the probability distribution of the qubit frequency during that record is narrowing as more observations are acquired. The nuclear spin noise correlator C ∆B (∆t) = ∆B z nuc (t + ∆t) -∆B z nuc (t), shown in Figure 2.3(a), displays a clear Gaussian probability distribution which broadens as the acquisition time ∆t increases. As shown in Figure 2.3(b), its variance grows as σ 2 B (∆t) = D(∆t) α over more than three orders of magnitude of timespan, with α = 0.8 and D = 0.048 MHz 2 /ms 0.8 . Though such long times are not reached in the measurement, the growth has to saturate, at σ 2 B (∞), since the fluctuating Overhauser field is bounded. Taking a value σ B (∞) corresponding to T * 2 = 10 ns typical for dots comparable to ours [61], we can roughly estimate the nuclear decorrelation time as (σ 2 B (∞)/D) 1/α ≈ 107 s.

2. 2 .Figure 2

 22 Figure 2.2: (a) Typical record showing the results of the projective measurements P S of the qubit as a function of the qubit evolution time. (b) Probability distribution of the qubit frequency calculated with Bayesian inference from the record in (a). The color code shows how the probability distribution narrows down as the qubit evolution time increases and more observations are used. (c) Nuclear field gradient ∆B z (t) extracted from the qubit frequency as a function of time.

CHAPTER 2 .Figure 2

 22 Figure 2.3: (a) The probability distribution of the nuclear field gradient time correlator C ∆B (∆t) for acquisition time ∆t from 3.8 ms (dark green) to 7.6 s (yellow). Data (dots) are fitted with a Gaussian distribution (line). Variance of the nuclear field gradient correlator as a function of the acquisition time ∆t. The solid line is a fit showing a growth with a power law exponent α = 0.8. The dashed line shows a power law behavior with α = 1 for comparison. The variance of the nuclear field gradient correlator as a function of the acquisition time ∆t measured with a ST spin qubit in a different wafer (c) (taken from the supplementary material of Ref.[START_REF] Noiri | A fast quantum interface between different spin qubit encodings[END_REF]) and in a different wafer with a spin-1/2 qubit (blue dots, taken from Ref.[START_REF] Nakajima | Coherence of a Driven Electron Spin Qubit Actively Decoupled from Quasistatic Noise[END_REF]).

Figure 2 . 4 :

 24 Figure 2.4: (a) Typical qubit evolution traces for different acquisition times. Solid lines are fit to decaying oscillations giving T * 2,φ = 120, 220 and 570 ns respectively. (b) Probability density distributions of T * 2,φ corresponding to the same acquisition times as for (a). The red solid line is a fit to a Gamma distribution resulting in skewness γ 1 ≈ 0.75, and T * 2,φ ≈ as given.

Keeping ∆ = 50

 50 MHz for performing Ramsey oscillations allows for validating the suppression of the electron spin qubit dephasing. Time traces of the frequency detuning between the applied qubit frequency f qubit and the estimated qubit frequency f est qubit with and without feedback are shown in Figure 2.6(a) demonstrating a clear stabilization with feedback control on. The variance of the nuclear field induced fluctuations falls from σ 2 B,off = (7.92 MHz) 2 to σ 2 B,on = (0.29 MHz) 2 . Equivalently, this can be measured by evaluating T * 2 from Ramsey oscillations which increases from T * 2,off = 28.4 ns to T * 2,on = 766.7 ns. Interestingly the 27-fold enhancement of T *2 is in line with the 29-fold enhancement that was measured with the ST qubit[START_REF] Shulman | Suppressing qubit dephasing using real-time Hamiltonian estimation[END_REF]. Finally, it seems to be only limited by the feedback hardware as the value of σ 2 B,on starts to saturate for short timescales (see Figure2.3(d)) at a value close to the bin width of the frequency discretization (0.25 MHz) of the Bayesian estimation algorithm.

Figure 2 . 5 :

 25 Figure 2.5: (a) False-colored scanning electron micrograph image of the TQD device.An electron spin qubit in the middle QD (red arrow with a circle) is controlled by the EDSR where the spin is coupled to a microwave (MW) electric field via a stray magnetic field of the micromagnet deposited on the wafer surface. The right QD hosts an electron spin (blue arrow with a circle) used as a readout ancilla while the left QD hosts another electron which is unused and decoupled from the two spins. The energy detuning between the middle and the right QDs is gate-tunable and the QD electron occupancies are probed by a proximal single-electron transistor (SET)[START_REF] Barthel | Fast sensing of double-dot charge arrangement and spin state with a radio-frequency sensor quantum dot[END_REF]. (b) Schematic of the Ramsey measurement. Two electrons (qubit and ancilla) are initialized to a doubly-occupied singlet state in the right QD and an up-spin qubit is prepared by adiabatically loading one of the electrons to the middle QD[START_REF] Noiri | Coherent electron-spinresonance manipulation of three individual spins in a triple quantum dot[END_REF]. Two π/2 microwave bursts, separated by time t R , are applied (before and during these, off-resonant microwave bursts are optionally applied). The ancilla-spin state is not affected by the microwave bursts. The final state is read out by unloading an up-spin (anti-parallel to the ancilla) state from the middle QD while a down-spin (parallel to the ancilla) state remains blocked. (c) Schematic of the feedback control loop for a spin qubit. Data of a Ramsey oscillation are processed in a digital signal processing hardware with programmable logic (FPGA) to estimate the frequency detuning δf = f qubit -f est qubit between the current qubit frequency fqubit and its previous estimate f est qubit ("probe" step). The value of f est qubit is updated to f est qubit → f est qubit + δf ("update" step), after which the target experiment follows ("target"step). In the ideal case, the subsequent qubit algorithms can be executed with a microwave frequency f MW matching fqubit exactly (by choosing ∆ = 0).

Figure 2 . 6 :

 26 Figure 2.6: (a) Time dependence of the frequency detuning δf = f qubit -f est qubit extracted from Ramsey measurements. The blue trace is taken with a fixed f MW (no feedback) and the red trace is taken with a feedback controlled f MW . (b) Up-spin probability P ↑ as a function of t R . The upper panel shows the trace obtained when the feedback is off. The decay envelope gives the dephasing time of T * 2 = 28.4 ns. The upper panel shows the trace obtained with the feedback on (∆ = 50 MHz). The envelope of the oscillation is a Gaussian decay function with T * 2 = 767 ns. (c) Rabi oscillations obtained at zero detuning upon compensating for the induced shift ∆f qubit showing an exponential decay function with T Rabi 2

Figure 2 . 7 :

 27 Figure 2.7: (a) Power spectral density S(f ) of the longitudinal noise in f qubit from Ramsey measurements of Figure 2.6(a) for f < 10 3 Hz and from Rabi measurements for f > 10 6 Hz (black boxed red curve). (b) Zoom on the power spectral density S(f ) of the longitudinal noise extracted with Rabi drive. Vertical grey lines show Larmor precession frequencies for the three nuclear species, 75 As, 69 Ga and 71 Ga calculated with the micromagnet -induced field component of B MM z = 70 mT. The inset illustrates electron-nuclear spin coupling in an inhomogeneous magnetic field. Each nuclear spin is randomly oriented and precesses around a local magnetic field vector B(r k ), leading to an oscillatory Overhauser field in the z direction.

Figure 2 . 8 : 2 ∼ 8

 2828 Figure 2.8: (a)-(c) (first row) Natural silicon spin qubit results (from Ref. [3]) and (d)-(f) (second row) isotopically purified silicon spin qubit results (from Ref. [4]). The two devices are very similar, except for the host material, and are presented in (a) and (c) as false colored micrographs. The square boxes in orange (a) or white (b) represent the ohmic contacts. The small circles show the approximate position of the quantum dot spin qubits while the large circles show the approximate position of the sensor quantum dot. (b) Ramsey fringe envelope of the natural silicon spin qubit driven on resonance showing a T * 2 = 1.84 µs Gaussian decay. (e) Ramsey fringe oscillations of the isotopically purified silicon qubit driven off-resonance showing a T * 2 = 20.4 µs Gaussian decay. (c) Rabi oscillations of the natural silicon qubit with T Rabi 2 ∼ 8 µs and f Rabi ∼ 9 MHz. (e) Rabi oscillations of the isotopically purified silicon qubit with T Rabi 2 too long ( 100 µs) to be estimated from the trace and f Rabi ∼ 16.6 MHz.

Figure 2 . 9 :

 29 Figure 2.9: Noise power spectral density of the isotopically purified silicon spin qubit. The high frequency data points are extracted from dynamical decoupling sequences and the low frequency points are extracted from Ramsey oscillations measurement over 24 minutes, similarly to the feedback controlled GaAs spin qubit. The solid line corresponds to S(f ) ∝ 1/f 1.01 . The inset shows typical fluctuations of the qubit frequency detuning over 2 minutes.

Figure 2 . 10 :

 210 Figure 2.10: Evolution of Rabi frequency f Rabi (a) and dephasing time T Rabi 2 (b) as a function of the driven amplitude for the natural silicon spin qubit. In (b), the dashed line is a linear fit to the data a small driving amplitude. (c) Natural silicon spin qubit quality factor showing an optimum at small driving amplitude. (d) Evolution of Rabi frequency f Rabi and dephasing time T Rabi 2 as a function of the driven amplitude for the feedback controlled GaAs spin qubit of subsection 2.3.1.

Figure 2 .

 2 Figure 2.11: (a) Schematics of the stapling technique showing the comb of cantilever with a carbon nanotube approaching the electrodes (yellow) of a circuit embedded in a microwave cavity in coplanar waveguide geometry. A trench in the substrate is etched (dark blue) to allow for bringing down the cantilevers. (b) Side view of the device and the cantilever after CNT "touch-down" with CNT lying on the contact electrodes (yellow) and suspended over the electrostatic gates (grey). The inner contacts are the leads of the circuit and the outer contacts are the cutting electrodes. The electrical setup to cut the CNT between an outer and inner contacts is shown.

Figure 2 .Figure 2 .

 22 Figure 2.12: (a) Large scale view of the circuit QED setup. Only one of two circuit areas is used. Scale bar: 1 mm. (b) Scanning electron microscope (SEM) image of the pedestal structure on which the device is made with the zig-zag magnetic contacts. Scale bar: 10 µm. (c) Zoom on the device showing the bottom gates and the non-collinear ferromagnetic contacts. Scale bar: 200 nm. (d) Spectrum of our spin quantum bit. The spin transition addressed in this work, the 01 transition, saturates to a value defined by the effective Zeeman splitting of each dot at large detuning. In the shaded gray region, spin and charge are not good quantum numbers anymore. (e) Schematics of our device showing the concept of spin-photon coupling.

Figure 2 .

 2 Figure 2.14: (a) Linewidth and derivative of the dispersion relation of the spin transition as a function of detuning (the derivative is obtained by fitting the dispersion relation of Figure 2.13a and calculating the derivative from the fit). A constant corresponding to a decoherence rate of 560 kHz is added to the derivative. Inset: Linewidth as a function of derivative. (b) Spin-photon coupling strength as a function of detuning. (c) Spin-photon cooperativity as a function of detuning. The error bars are the standard deviation extracted from least square fitting of the spin spectroscopic line in Figure 2.13a

Figure 3 . 1 :

 31 Figure 3.1: (a) Schematic picture of the device measured in Ref. [6] with a CNT contacted by a superconducting lead and a normal lead, lying above a magnetic texture gate. In such a device MZM (red stars) can localize at the boundaries of the region of topological phase generated by the cycloidal magnetic field. (b) Cycloidal magnetic field profile as used for Kitaev chain tight binding simulations, showing the normal and superconducting (SC) parts of the device. The envelope of the oscillations is an artefact due to undersampling because of the small number of sites (N = 20). (c) Density of states (DOS) as a function of site position and energy for the the situation in (b). Two localized MZM at zero energy are observed simultaneously with delocalized Andreev bound states at higher energy. The vertical black line indicates the boundary of the superconducting contact.

Figure 3 . 2 :

 32 Figure 3.2: Hybrid superconductor-nanotube-magnetic texture setup. (a) Schematic picture of the multilayer magnetic texture with up and down domains (white and black arrows) inducing the rotating magnetic field in space (Bosc, red line) leading to the synthetic spin-orbit interaction. (b) Zoom on the device showing the single wall carbon nanotube (in red). The bottom gate is made from a multilayer of Co/Pt. The source and drain superconducting electrodes are made out of Pd/Nb. (c) Conductance of the device as a function of source-drain bias displaying a well-defined gap with two symmetric ALSs at energy E, shown again in the inset. The "hardness" of the gap is measured by the ratio of the conductance values marked by the star and the circle. (d) Density of states of the probe contact and of the nanotube, with the ALS arising from the coupling between the nanotube and the left superconductor, as fitted by Usadel equations. The right superconductor has a residual density of states at zero bias allowing for a direct spectroscopy of the ALSs. (e) Magnetic Force Microscope (MFM) micrograph of the device showing the magnetic texture of the bottom gate. The cut of the magnetic signal indicating field modulations (yellow and grey) along the nanotube on a scale of about 200 nm is shown at the bottom. Figures taken from Ref. [6].

Figure 3 . 3 :

 33 Figure 3.3: Oscillations of the subgap states and synthetic spin-orbit interaction. (a) Magnetic domains of the magnetic texture in the absence of external magnetic field (left) and in the presence of a strong external magnetic field B ext ≥ B sat with B sat the saturation field of the magnetic texture. (b) Left panel: Band structure arising from the synthetic spin-orbit interaction with N domains. The allowed interferences in the finite length system are represented with arrows. Right panel: Schematics of how the band structure can be tuned by changing the spin-orbit energy, here decreasing (with N domains, the bands are shifted by k). (c) Low bias differential conductance G map in the V sd -B ext plane showing the oscillations of the ALSs (indicated by purple arrows) as a function of the magnetic field. The black lines are the fit to the theory. Figures adapted from Ref. [6].

Figure 3 . 4 :

 34 Figure 3.4: (a) Map of G in the V sd -V g plane showing the appearance of a ZBCP when gate G is tuned. The ZBCP position is indicated by a green arrow and the ALSs by purple arrows. (b) G profiles for V g = 0, -1, -2, -3 V. (c) Map of G at V g = -3 V in the V sd -B ext plane showing the evolution of the ZBCP as a function of the in-plane magnetic field. The overall background G gap arising from the superconducting gap has been subtracted for clarity. The black lines correspond to the same fit as Figure 3.3(c). The orange and black arrows indicate the magnetic field range of panel d. (d) Low magnetic field conductance G diff profile map in the V sd -B ext plane for V g = -3 V displaying the large magnetoresistance of the zero-bias peak. The orange and black arrows represent the direction of the magnetic field sweep. Figures taken from Ref. [6].

3. 3 .

 3 PERSPECTIVES coherent field of the cavity.

Figure 3 . 5 :

 35 Figure 3.5: (a) Schematics of a device that can host four MZM in a carbon nanotube contacted by two normal conductors (N), one central superconducting lead (S) and above two magnetic textures inducing synthetic spin-orbit. The two MZM pairs are indicated by red and blue stars. (b) Magnetic force microscope image of two magnetic texture gates as required for (a). (c) Cut along the dashed line in (b) showing the modulation of the magnetic field. The color code is the same as in (a) to illustrate the feasibility of the device.

Figure 3 . 6 :

 36 Figure 3.6: Cavity signal of the dispersion of the double magnetic texture quantum dot for one set of dots orbitals in the detuning and magnetic field B ext plane. Experimental phase variation ∆φ (a) and amplitude relative variation ∆A/A (b). Reciprocal simulations in (c) and (d) using a simple spin qubit model with non collinear magnetization similar to the one discussed in subsection 1.1.2 and subsection 2.3.3.

Table 2 .

 2 1: Table presenting atomic quantities related to the hyperfine contact useful for our purpose. The values are taken from Refs.

	.1

Table 2 .

 2

			Si	Si	CNT	CNT
			natural purified	natural	purified
	p	1	0.047	8×10 -4	0.011	8 × 10 -4
	N	∼ 2 × 10 6 ∼ 10 5	∼ 10 5	∼ 5 × 10 4 ∼ 5 × 10 4
	T * 2 (µs)	∼ 0.01	∼ 2	∼ 120	∼ 25	∼ 350

2: Table summarizing the dephasing times for quantum dot spin qubits in various materials and with different nuclear spins isotope concentrations.

The more "classical" dipole-dipole interaction with a 1/r 3 dependence is valid only if the distance between the dipoles is much larger than the dipole dimension, which is not the case here.

We do not use EDSR provided by the micromagnet for the ST qubit. Rather, we exploit the gradient of B z MM (y) conceived for scalability to generate the inhomogeneous Zeeman field which sets the ST qubit energy.

The Gamma distribution of a variable X is defined by GX (x; k, h)= (2k/h) k 2 k Γ(k) x k-1 e -k hx with Γ the Euler function, h the mean and k is the shape parameter related to the distribution skewness γ1 = 2/ √ k. Details of the fit procedure can be found in the supplementary material of Ref.[START_REF] Delbecq | Quantum Dephasing in a Gated GaAs Triple Quantum Dot due to Nonergodic Noise[END_REF]

Or a good memory as well because a good memory needs to be written and read fast enough also!

These qubits exploit the valley degree of freedom of CNT for spin-to-charge coupling and a bend in the CNT to get effectively two non collinear magnetic field axis, in a similar fashion to the non-collinear ferromagnet leads discussed in subsection 1.1.2

There was also an error in the estimation of the number of atoms which partially compensated for the misuse of Eq.(2.16).

Being in the strong coupling regime implies having C ≥ 1 however it is possible to have C ≥ 1 without being in the strong coupling regime, for example with a very small photon loss and a large loss of the other system, typical with lasers for example.

A particularity of localized MZM in 1D or

2D systems is their non-abelian exchange statistics. Indeed, while in 3D particles can only follow the bosonic or fermionic statistics (when exchanging two undistinguishable particles, the global wave function stays identical or picks up a minus sign), in 2D a richer behaviour can exist. For example, the wave function can pick up a phase (anyonic statistic). When the ground state is degenerate, particle exchange can lead to a change in the state of the system, represented by a unitary operation on the wave function (non-abelian statistics). The state of a system after several particle exchanges then depends on the order of the exchanges, since the corresponding unitary operations do not necessarily commute.

the Fermi wavelength is of about 10 nm, larger than the typical nanotube radius of about 2 nm so that the transverse confinement leads to only one conduction channel with spin and valley degeneracy.

the "stamping" technique is the ancestor of the "stapling" technique discussed in subsection 2.3.3. In the stamping technique, the carbon nanotube is stamped or deposited on the wafer surface after being grown on quartz pillars. It then undergoes nanofabrication techniques like electron beam lithography and