Ramón Aguado

Manuel Houzet

Hélène Bouchiat

Jérôme Cayssol

Ryosuke

Raphaëlle

Anil

Alesya

Emilie Anaïs

Quantum Transport in a Correlated Nanostructure Coupled to a Microwave Cavity

Keywords: quantum transport, cavity QED, Majorana fermion

vii 7.1 The system and model Hamiltonian 107 7.2 Electronic susceptibility and cavity response 112 7.2.1 Theoretical approach to susceptibility 112 7.2.2 Flux dependence of the cavity response 115 7.2.3 Parity dependence 119 7.2.4 Frequency dependence 119 7.3 Conclusions 120 8 conclusions and perspectives 123 a appendix a 127 b appendix b 133 b.1 Scattering matrix approach for a non-interacting quantum conductor 133 b.2 Calculation of the quantum dot susceptibility 135 b.2.1 Expressions of the intermediate F γγ αβ functions 135 b.2.2 Expressions of the intermediate K αβ functions 139 b.2.3 Calculation of the real part of the susceptibility 140 b.2.4 General expression for the quantum dot susceptibility 142 c appendix c 149 c.1 Derivation of the effective Kitaev Hamiltonian in the presence of the cavity field 149 c.2 The susceptibility for the Kitaev model in case of periodic boundary conditions 155 c.3 The susceptibility of the nanowire in case of periodic boundary conditions 158 c.4 The correlation function for open boundary conditions 163 d appendix d 169 e synthèse en français 175 e.1 La théorie des entrées-sorties pour les cavités micro-ondes 177 e.2 Boîte quantique hors équilibre couplée à une cavité micro-ondes 180 e.3 Fermions de Majorana dans des supraconducteurs topologiques couplés à une cavité micro-ondes 182 viii contents e.4 Effet Josephson dans des anneaux topologiques supraconducteurs couplé à une cavité micro-ondes 186 e.5 Conclusions et perspectives

Titre : Transport quantique dans une nanostructure corrélée, couplée à une cavité micro-ondes Mots-clés : transport quantique, électrodynamique quantique des cavités, fermion de Majorana Dans cette thèse, nous étudions d'un point de vue théorique les propriétés physiques de nanostructures couplées à des cavités micro-ondes. L'électrodynamique quantique (QED) en cavité en présence d'une boîte quantique s'est révélée être une technique expérimentale puissante, permettant d'étudier cette dernière par des mesures photoniques en plus des mesures de transport électronique conventionnelles. Dans cette thèse, nous proposons d'utiliser le champ micro-ondes de la cavité afin d'extraire des informations supplémentaires sur les propriétés des conducteurs quantiques : le coefficient de transmission optique est directement lié à la susceptibilité électronique de ces conducteurs quantiques. Nous appliquons ce cadre général à différents systèmes mésoscopiques couplés à une cavité supraconductrice micro-ondes comme une jonction tunnel, une boîte quantique couplée à des réservoirs, un fil topologique et un anneau supraconducteur. La QED en cavité peut être utilisée pour son-der, par l'intermédiaire de mesures photoniques, la dépendance en fréquence de l'admittance du puits quantique couplé à la cavité micro-ondes. En ce qui concerne le fil topologique, nous avons montré que la cavité permet de caractériser la transition de phase topologique, l'émergence de fermions de Majorana, ainsi que la parité de l'état fondamental. Pour l'anneau supraconducteur, nous étudions par l'intermédiaire de la réponse optique de la cavité l'effet Josephson et le passage à l'effet Josephson fractionnaire, qui est associé à l'apparition de fermions de Majorana dans le système. Le cadre théorique proposé dans cette permet de sonder de manière non-invasive un large éventail de nanostructures, des boîtes quantiques aux supraconducteurs topologiques. En outre, il donne de nouvelles informations sur les propriétés de ces conducteurs quantiques, informations non accessibles via des expériences de transport.

In this thesis, we study theoretically various physical properties of nanostructures that are coupled to microwave cavities. Cavity quantum electrodynamics (QED) with a quantum dot has been proven to be a powerful experimental technique that allows to study the latter by photonic measurements in addition to electronic transport measurements. In this thesis, we propose to use the cavity microwave field to extract additional information on the properties of quantum conductors: optical transmission coefficient gives direct access to electronic susceptibilities of these quantum conductors. We apply this general framework to different mesoscopic systems coupled to a superconducting microwave cavity, such as a tunnel junction, a quantum dot coupled to the leads, a topological wire and a superconducting ring. Cavity QED can be used to probe the finite frequency admittance of the quan-tum dot coupled to the microwave cavity via photonic measurements. Concerning the topological wire, we found that the cavity allows for determining the topological phase transition, the emergence of Majorana fermions, and also the parity of the ground state. For the superconducting ring, we propose to study the Josephson effect and the transition from the latter to the fractional Josephson effect, which is associated with the emergence of the Majorana fermions in the system, via the optical response of the cavity. The proposed framework allows to probe a broad range of nanostructures, including quantum dots and topological superconductors, in a non-invasive manner. Furthermore, it gives new information on the properties of these quantum conductors, which was not available in transport experiments.

I N T R O D U C T I O N

Light-matter interaction is one of the fundamental physics phenomena that has seen a major research interest recently. Discovery of the photoelectric effect by Einstein and the breakthrough of quantum mechanics with its further rapid development considerably improved our understanding of light-matter interaction. This interaction is the cornerstone of the entire field of quantum electrodynamics (QED). At the most elementary level, the interaction between light and matter takes place when a single photon interacts with a single atom. Serge Haroche et.al. and David Wineland et.al. independently performed experiments on atom-photon coupling in their strive to prove quantum decoherence. Both of them were jointly awarded the Nobel Prize in Physics for "ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems" related to these experiments. Haroche et.al. studied properties of photons trapped in a 3D cavity by sending Rydberg atoms through the cavity [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]. On the other hand, David Wineland et.al. used an opposite approach and studied trapped ions by sending photons through the cavity [START_REF] Leibfried | Quantum dynamics of single trapped ions[END_REF]. However, the main idea of these experiments remains the same: studying the interaction between a single atom, or an ion, and a photon. These experiments allowed to confirm quantum decoherence effects and demonstrated a possibility to create non-classical states of light. Also, Ref. [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] provides a first implementation of a qubit -a two-level system can be used as a building block for a quantum computerthat can be probed in atomic QED experiments. However, in these pioneering experiments, the coupling constant between the atom and cavity photons is fixed by the dipole moment of the atom, restricting the potential application area in quantum computing. Moreover, Rydberg atoms in a 3D cavity are not scalable to many-qubit architecture required for implementing quantum information protocols.

The idea of a tunable atom-cavity coupling constant as well as the need for implementing scalable qubits led to the theoretical proposal of circuit cavity QED. A real atom was substituted with an arificial one and a 3D cavity was replaced by a 1D transmission line resonator to enable large coupling constant and ensure scalability. In [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF], an artificial atom, or a two-level system, was proposed to be implemented as a Cooper pair box. It was achieved experimentally by the same authors in [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF]. In circuit QED experiments, qubit-cavity coupling constant depends on the parameters of the superconducting circuit and the resonator, and the geometry of a 1D transmission line resonator allows for embedding many qubits in the cavity. Having multiple qubits implies entanglement, which is essential for implementing the quantum computing.

However, further progress in qubit implementation requires these qubits to be situated far away from each other and being highly tunable. This can be achieved in mesoscopic cavity QED [START_REF] Childress | Mesoscopic cavity quantum electrodynamics with quantum dots[END_REF] by using quantum dots as qubits. Quantum dots can be tuned by coupling them to gate electrodes. The electrons in quantum dots are not paired, allowing access to the electron spin. Also quantum dots can be tunnel coupled to leads and studied by transport measurement in addition to photonic probe in a microwave cavity.

Another direction for application of mesoscopic cavity QED is to use it to probe properties of the nanostructures embedded in microwave resonator. One of the systems that has been largely studied by transport measurements is a topological wire that can host Majorana fermions. Majorana fermions, which are particles that are their own antiparticles, have been first predicted by Ettore Majorana in 1937 as a real solution to Dirac equation.

It was shown later that Majorana fermions can also appear in condensed matter physics, for example, in superconductors [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. Due to its degenerate ground state and protection against local perturbation, Majorana fermions can be also used as qubits. In this thesis, we demonstrate that the methods of cavity QED can be used to probe Majorana fermions in topological superconductors coupled to a microwave cavity.

in this thesis

In this thesis, we apply the methods of cavity QED to study quantum conductors and topological wires coupled to a microwave cavity. We start with a theoretical description of the nanostructure-cavity coupling. Then we derive the cavity transmission in a weak electron-photon coupling limit and its relation to the electronic susceptibility of the quantum conductor (see chapter 2). This susceptibility gives information about the nanostructure embedded in a microwave cavity. We apply cavity QED approach to study theoretically four specific mesoscopic conductors: a tunnel junction, a quantum dot coupled to the leads, a one-dimensional topological wire and a superconducting ring.

In chapter 4, we use the photonic response of the cavity to study a tunnel junction and a quantum dot tunnel coupled to metallic leads. We consider both capacitive coupling of the quantum dot to the cavity as well as coupling of the metallic leads to the cavity. We demonstrate that these two types of couplings result in a totally different cavity response and thus probe different observables associated with the quantum conductor. We find some relations between optical observables, such as the cavity transmission, and transport quantities, such as the differential conductance dI/dV. These relations are valid up to high frequency and voltage (therefore in the out-of-equilibrium regime) for quantum conductors far from resonance and hold only at low frequency and voltage for the quantum dot near resonance. Beyond these regimes, measurements of the cavity photonic field provide new information about the quantum conductor.

In chapter 5, we apply the cavity QED approach to find signatures of Majorana fermions in topological wires. We consider a p-wave superconductor capacitively coupled to a microwave cavity. The phase shift and frequency broadening due to the presence of the p-wave superconductor in the cavity are proportional to the electronic susceptibility of the wire. Photonic observables can be used to obtain signatures of the Majorana fermions, such as a topological phase transition and oscillations of the electronic susceptibility in the topological phase. Moreover, cavity QED experiments should allow to probe the parity of the Majorana state, which is not accessible in transport experiments.

In chapter 6, we study a topological wire that can host multiple Majorana fermions at its ends by coupling it to a microwave cavity. This situation is possible when there is longrange hopping in the wire. We demonstrate that the photonic field of the cavity allows to probe the topological phase transition between phases with different number of Majorana fermions as well as oscillations of the susceptibility. The period of the oscillations is dif-ferent when there are two or one Majorana fermion at each end of the chain, while these oscillations are absent in the trivial phase.

In chapter 7, we apply methods of cavity QED to study the fractional Josephson effect associated with Majorana fermions physics. We consider a superconducting ring with a weak link threaded by a dc magnetic flux. An AC flux created by the microwave cavity induces the coupling between the ring and the cavity photonic field. We propose to study the Josephson effect and the transition from the latter to the fractional Josephson effect, which is associated with the emergence of the Majorana fermions in the system, via the response of the cavity. We take fully into account the interplay between the low-energy Majorana modes and the gaped bulks states, which we demonstrate is crucial for visualizing the evolution of the Josephson effect during the transition from the topological to the trivial phase.

In this chapter, we discuss the field of cavity quantum electrodynamics (QED). We start with atomic cavity QED that studies properties of atoms or ions coupled to discrete modes of the electromagnetic field [START_REF] Leibfried | Quantum dynamics of single trapped ions[END_REF][START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]. In [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] the interaction between a single photon and a single atom was observed by measuring the latter as it has passed through a 3D optical cavity. Also these experiments allowed to analyze decoherence effects as well as to demonstrate the implementation of quantum gates required for quantum computation. But atomic cavity QED with real atoms is characterized by a constant atom-cavity coupling and is not scalable to multiple qubits architecture. In order to overcome these constraints, a real atom was substituted with an artificial atom and a 3D cavity was substituted with a 1D transmission line resonator, which led to the development of circuit quantum electrodynamics [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF][START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF]. Circuit version of cavity quantum electrodynamics ensures tuning the coupling between an artificial atom and a superconducting resonator and implementing multiple qubits on a single chip, thus allowing for their entanglement and future application in solid-state quantum computation [START_REF] Hur | Many-Body Quantum Electrodynamics Networks: Non-Equilibrium Condensed Matter Physics with Light[END_REF]. Moreover, in mesoscopic version of circuit QED connecting metallic leads to a mesoscopic conductor allows to study the latter by transport measurement in addition to optical probe. The combination of the different techniques allows to probe all properties of the mesoscopic system. Coupling closed systems to superconducting stripline resonators was first performed in [START_REF] Reulet | Dynamic response of isolated aharonov-bohm rings coupled to an electromagnetic resonator[END_REF] as a probe of the ac conductance related to the susceptibility. In section 2.4, we derive the relation between the transmission of the superconducting resonator and charge susceptibility of the electronic system embedded in the resonator. cavity quantum electrodynamics 2.1 from optical cavity to transmission line resonator 2.1.1 Single atom coupled to a single photon Interaction between light and matter at the level of coupling a single atom or ion to a single photon has been first experimentally studied in [START_REF] Leibfried | Quantum dynamics of single trapped ions[END_REF][START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]. In the first set of experiments [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF], photons trapped in a cavity have been controled and measured by sending prepared atoms through the cavity. While in the second set of experiments [START_REF] Leibfried | Quantum dynamics of single trapped ions[END_REF], electrically charged atoms, or ions, have been measured and manipulated with light. These experiments are studying the same quantum systems such as a single atom and a single photon, but using opposite approaches. Coupling atoms with photons in a cavity allows to study entanglement of the combined system, and thus to scrutinize decoherence effects.

Entanglement of qubits is also essential for implementation of quantum information protocols required for building a quantum computer in the future.

Atomic cavity quantum electrodynamics studies the properties of atoms coupled to the quantized electromagnetic modes in a cavity. In [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF], Rydberg atoms are sent through the cavity in order to probe the properties of a photon trapped in this cavity. Rydberg atom is an excited atom with one or more electrons that has a large electric dipole moment d (due to the large separation between the electron and ion-core). That is why such atoms have a large response to an electric field and consequently are suitable for studying light-atom interaction. The cavity in these experiments is an open Fabry-Perot resonator, made of two mirrors. The geometry of such resonator allows to apply an electric field along the cavity axis.

A two-level atom could be used as a qubit with two states |0 and |1 . It allows to perform logic operations required for implementing quantum information protocols. When the state of the atom-cavity system changes from |0, e , in which the atom is in the excited state and there is no photon in the cavity to |1, g , in which the atom is in the ground state and there is one photon in the cavity, the phase of the atom changes. The photonic field is coupled to Rydberg atom by dipole interaction. Measurement of the state of the atom after it has passed through the cavity allows to obtain information about the properties of the photon trapped inside the cavity. Two-level systems, such as Rydberg atoms, coupled to the photonic modes of the cavity can be described by Jaynes-Cummings Hamiltonian H = hω eg σ z + hω c a † a + 1/2 + hg a † σ -+ aσ + , (2.1) where ω eg is the transition frequency between the excited and ground state of the atom, ω c is the cavity frequency, which in resonant cavity is tuned to be equal to ω eg , g denotes the atom-photonic field coupling constant, σ z , σ + and σ -are the Pauli matrices of the atomic pseudospin, a(a † ) is the photon annihilation (creation) operator. The coupling constant g = rms d/ h, where rms is the root mean square (r.m.s.) vacuum field at the cavity center and d is the dipole matrix element for the excited to ground state transition in the atom. The dipole moment of the atom is fixed for a given atom, that is why the atom-photon coupling cannot be tuned in cavity QED with nature-given atoms.

It was shown [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] that Rydberg atoms, coupled to a superconducting cavity containing a few photons, could be used to test quantum decoherence and realize quantum logic operations necessary for building a quantum computer.

Though Ref. [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] were the first experiments to demonstrate the atom-photon interaction and implementation of qubits based on two-level atoms, further refinements of the setup were required. In cavity QED with real atoms, the coupling between an atom and a photon is fixed by the dipole moment of the atom, as well as the resonant cavity frequency is fixed by the transition frequency between excited and ground states of the atom. Moreover, nature-given atoms coupled to a 3D cavity are not scalable to many qubit setups required for implementation of quantum computer.

Cavity QED with superconducting circuits

Realization of cavity QED that allows to overcome the fixed atom-photon coupling and scalability issues using superconducting circuits has been proposed in [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF]. In this theoretical proposal, the cavity is implemented as a 1D transmission line resonator consisting of a full-wave section of superconducting coplanar waveguide and an atom is implemented as a superconducting qubit, or more precisely a Cooper pair box. Scheme of the proposed implementation of cavity QED based on transmission line resonator is presented in Fig. 2.1. Already at that time, lithographic technique allowed to create a Cooper pair box coupled to a 1D transmission line resonator on a single solid-state chip. Cooper pair box, which is a mesoscopic superconducting island which is coupled via two Josephson tunnel junctions to a superconducting reservoir, plays a role of a qubit in this proposal. In the proposed geometry, this qubit can be isolated from the electromagnetic environment in controllable manner thus being suitable for solid-state quantum computing. Since in this case the cavity is a 1D transmission line resonator, and not a 3D cavity as in [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF], the zeropoint energy rms is distributed over a very small effective volume (< 10 -5 cubic wavelengths) leading to large rms voltage V rms ≈ hω c cL between the center conductor and the adjacent ground plane at the antinodal positions, where L is the length of the resonator and c is the capacitance per unit length of the transmission line. Resonant frequency of the transmission line resonator is determined by a fixed geometry thus allowing to perform reproducable experiments. Also, the artificial atom could be controled by external fields, for example, by threading a flux Φ ext in the loop formed by the pair of junctions and changing the gate voltage V g (see Fig. 2.1). By tuning the capacitance and the Josephson energy of the superconducting circuit, one can obtain an effective two-level system, similar to Rydberg atom studied in [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]. Moreover, it was demonstrated that such a two-level system could be described by the Jaynes-Cummings Hamiltonian [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF] with the atom-cavity coupling constant g = eC g h (C g + C J )

V rms , where C g is the the capacitance between the center conductor of the resonator and the Cooper pair box and C J is capacitance of the Josephson junction. Considering the cavity resonance frequency ω c = 10

GHz leads to the coupling constant g ≈ 100 MHz, which is three orders of magnitude larger than in atomic cavity QED experiments with Rydberg atoms. Moreover, the position of the Cooper pair box in transmission line resonator is fixed thus ensuring that the coupling constant g does not depend on position of the atom as in 3D cavities with Rydberg atoms. Thus, cavity QED with superconducting circuits should allow to reach easily the strong-coupling limit (the coupling between the qubit and the cavity is larger than both the damping rates of the qubit and the cavity). As for the practical application, two or more qubits can be placed along the length of the transmission line resonator. In case of two qubits, they can be separated by the length of the resonator ∝ 1cm and tuned by capacitive coupling to the left and right center conductors of the transmission line. Thus, a 1D transmission line resonator allows for entanglement of qubits, making it a suitable platform for quantum computation. Reproduced from [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF].

The theoretical proposal [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF] of a 1D transmission line resonator with an artificial atom has been experimentally implemented by the same authors in [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF]. This work [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF] also demonstrates the first experimental observation of a strong coupling between a two-level system and a single photon in a cavity. A two-level system in this experiment has been implemented as a Cooper pair box as was suggested in [START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF], since its dipole moment is at least one order of magnitude larger than in Rydberg atoms [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF]. As for the transmission line resonator that consists of a narrow center conductor and two nearby lateral ground planes (see Fig. 2.3), it provides a large electric field between the center conductor and the ground plane which is hundred times larger than in 3D microwave resonators. Combination of large dipole moment and the large vacuum field strength ensures the possibility of entering a strong-coupling limit in superconducting circuit QED. The resonator is coupled to the transmission lines (see Fig. Thus, the properties of the qubit can be obtained by performing photonic measurements of the cavity.

But superconducting stripline resonators were first coupled to isolated mesoscopic systems such as sets of isolated rings [START_REF] Reulet | Dynamic response of isolated aharonov-bohm rings coupled to an electromagnetic resonator[END_REF] as a probe to investigate the susceptibility of these systems without any connection to an invasive probe. The magnetic susceptibility of the rings χ(ω) is connected to ac conductance Y(ω) as χ(ω) ≈ iωY(ω). The measure of the modification of the resonance frequency f n and the quality factor Q n due to the presence of the rings in the superconducting resonator gives access to the complex ac conductance of the rings [START_REF] Deblock | Diamagnetic orbital response of mesoscopic silver rings[END_REF][START_REF] Deblock | ac electric and magnetic responses of nonconnected Aharonov-Bohm rings[END_REF][START_REF] Reulet | Dynamic response of isolated aharonov-bohm rings coupled to an electromagnetic resonator[END_REF]]

δf n f n ∝ [Y(f n )] , δ δQ n Q 2 n ∝ Re [Y(f n )] . (2.2)
Though this experiment [START_REF] Reulet | Dynamic response of isolated aharonov-bohm rings coupled to an electromagnetic resonator[END_REF] was performed in closed system and in the classical regime, it was a first demonstration that properties of the electronic systems can be obtained by non-invasive measurements by coupling them to superconducting resonators. Later on, this technique was used to obtain the conductance of the non-connected Aharonov-Bohm rings [START_REF] Deblock | ac electric and magnetic responses of nonconnected Aharonov-Bohm rings[END_REF] and silver rings [START_REF] Deblock | Diamagnetic orbital response of mesoscopic silver rings[END_REF], as well as the susceptibility of the SNS junctions [START_REF] Dassonneville | Dissipation and Supercurrent Fluctuations in a Diffusive Normal-Metal-Superconductor Ring[END_REF][START_REF] Ferrier | Phase-dependent Andreev spectrum in a diffusive SNS junction: Static and dynamic current response[END_REF].

We compare the electronic susceptibility of the ring obtained by the classical approach and the quantum one in chapter 7.

Pioneering theoretical proposal of mesoscopic cavity QED with quantum dots instead of superconducting qubits has been reported in [START_REF] Childress | Mesoscopic cavity quantum electrodynamics with quantum dots[END_REF]. This proposal is based on capacitive coupling between the electron charge of the quantum dot and a superconducting transmission line resonator. Compared to cavity QED with superconducting qubits, cav- ity QED with quantum dots allows for strong coupling between distant quantum dots and thus for implementing a scalable quantum computer. Also quantum dots are highly tunable and give access to the electron spin. Since quantum dots can be tunnel coupled to metallic leads, mesoscopic cavity QED allows to study the properties of the quantum dots by electronic transport measurements in addition to photonic measurements.

Later on, it was demonstrated that for open mesoscopic conductors, such as quantum dots, the transmission coefficient at energy near the cavity frequency is sensitive to the charge susceptibility of the quantum dot circuit [START_REF] Cottet | Mesoscopic admittance of a double quantum dot[END_REF].

coupling a quantum dot to a microwave cavity

Quantum dots are nanoscopic systems that emulate atoms by showing discrete energy levels and atomiclike electronic filling shell structure. Thus, they could be measured by optical means. But contrary to atoms, quantum dots can be coupled to metallic leads and probed in electronic transport experiments similarly to large metallic conductors [START_REF] Childress | Mesoscopic cavity quantum electrodynamics with quantum dots[END_REF]. By embedding quantum dots in superconducting transmission line resonators, it is possible to study quantum dots by photonic measurements in addition to electronic transport.

Some recent experiments probed and manipulated a single or double quantum dot with resonators [START_REF] Basset | Single-electron double quantum dot dipolecoupled to a single photonic mode[END_REF][START_REF] Bruhat | Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua[END_REF][START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF][START_REF] Deng | A quantum electrodynamics Kondo circuit: probing orbital and spin entanglement[END_REF][START_REF] Frey | Dipole coupling of a double quantum dot to a microwave resonator[END_REF][START_REF] Frey | Quantum dot admittance probed at microwave frequencies with an on-chip resonator[END_REF][START_REF] Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF][START_REF] Schroer | Radio frequency charge parity meter[END_REF][START_REF] Stockklauser | Microwave Emission from Hybridized States in a Semiconductor Charge Qubit[END_REF][START_REF] Toida | Vacuum Rabi splitting in a semiconductor circuit QED system[END_REF][START_REF] Wang | Charge relaxation in a single-electron Si/SiGe double quantum dot[END_REF]. Theoretical study of the quantum dots coupled to transmission line resonators have been performed in [START_REF] Bergenfeldt | Microwave quantum optics and electron transport through a metallic dot strongly coupled to a transmission line cavity[END_REF][START_REF] Childress | Mesoscopic cavity quantum electrodynamics with quantum dots[END_REF][START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF][START_REF] Cottet | Mesoscopic admittance of a double quantum dot[END_REF][START_REF] Schiró | Tunable hybrid quantum electrodynamics from nonlinear electron transport[END_REF], and we discuss it in the next section.

One of the pioneering experiments, in which a quantum dot has been coupled to a single mode of electromagnetic field in a superconducting resonator, has been reported in [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF]. In this experiment, a single wall carbon nanotube is embedded in the coplanar waveguide resonator (see Fig. 2.3) in order to implement a single quantum dot coupled to a cavity situation. A new aspect of this experiment is that there are two wires, source and drain, that go inside the cavity in order to drive a DC current through the quantum dot.

This allows to measure a differential conductance dI/dV of the quantum dot while changing the source-drain voltage V sd , which is a quantity typically measured in transport experiments. The quantum dot is also connected to gate electrodes that allows to change the position of the energy level inside the quantum dot by varying the gate voltage V g .

At the same time, a continuous microwave signal is transmitted through the cavity. The signal is sent through one side of the cavity and measured at the other one, thus allowing to measure the cavity transmission. The variation of the phase and the amplitude of the microwave signal is due to the presence of the quantum dot circuit inside the cavity. The differential conductance dI/dV of the quantum dot tunnel coupled to metallic leads as a function of the source-drain V sd and gate V g that has been measured in [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF] and is pre- Mesoscopic cavity QED has been proven to be a powerful measurement technique that allows to detect extremely small variations of the phase of the signal transmitted through the cavity (up to δφ ∝ 10 -3 ) [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF]. The recent progress in engineering superconducting cavities makes it possible to implement on-chip mesoscopic circuits coupled to high finesse resonators [START_REF] Samkharadze | High-Kinetic-Inductance Superconducting Nanowire Resonators for Circuit QED in a Magnetic Field[END_REF]. Thus, properties of different mesoscopic systems can be studied by coupling them to a cavity. Cavity QED with a quantum dot has been extended to experimental study of interaction between two distant quantum dots [START_REF] Delbecq | Photon-mediated interaction between distant quantum dot circuits[END_REF][START_REF] Deng | Coupling Two Distant Double Quantum Dots with a Microwave Resonator[END_REF], spin-photon coupling [START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF] and Kondo physics [START_REF] Deng | A quantum electrodynamics Kondo circuit: probing orbital and spin entanglement[END_REF]. On the theory side, mesoscopic QED has been used

sented in
to investigate tunnel junctions [START_REF] Dmytruk | Out-of-equilibrium quantum dot coupled to a microwave cavity[END_REF][START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF] and quantum point contacts [START_REF] Mendes | Electron-photon interaction in a quantum point contact coupled to a microwave resonator[END_REF], as well as Majorana fermions in topological superconductors (see chapter 3). Also nonlocal electronic transport through two quantum dots coupled to a microwave cavity was studied in [START_REF] Bergenfeldt | Nonlocal transport properties of nanoscale conductor-microwave cavity systems[END_REF][START_REF] Contreras-Pulido | Non-equilibrium correlations and entanglement in a semiconductor hybrid circuit-QED system[END_REF][START_REF] Lambert | Photon-mediated electron transport in hybrid circuit-QED[END_REF].

electron-photon coupling in cavity qed

In this section, we discuss how to describe theoretically the coupling between the quantum conductor (for example, a quantum dot tunnel coupled to metallic leads) and photonic field inside the cavity. For simplicity, we consider a single mode of a microwave cavity field, but this description can be generalized to multimode cavity by introducing a sum over all cavity modes. The detailed derivation using photonic pseudopotential to account for electric coupling between electrons in a nanostructure and cavity photons has been recently done in [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF]. Here we demonstrate how to obtain the capacitive coupling between the quantum conductor and photonic field inside the cavity. In a continuum description, we can describe the interaction between the electrons in the nanostructure and the electromagnetic field via the minimal coupling, i.e., p → p -eA in the electronic Hamiltonian, where A is the electromagnetic field vector potential and p is the momentum of the electrons.

The combined Hamiltonian that describes the electronic system and cavity photons can be written as

H tot = d 3 rψ † ( r)h( r)ψ( r) + hω c a † a with h( r) = 1 2m -i h∇ r + e Â( r) 2 , (2.3)
where ψ( r) is the field operator, a(a † ) is photon annihilation (creation) operator and h( r) is a single-electron Hamiltonian that includes coupling to the electromagnetic field with

Â( r) = A( r)i a -a † (2.4)
being a photonic vector potential. hω c a † a describes a single-mode electromagnetic field with frequency ω c . e > 0 is elementary charge.

The electron-photon coupling in Eq.2.3 contains creation or annihilation of one photon as well as two photons transitions via Â2 term. In order to eliminate two-photon processes in Hamiltonian Eq.2.3, let us perform a unitary transformation. But let us first introduce a photonic pseudopotential that account for possible spatial variations of A( r)

V ⊥ ( r) = ω c C( r, r 0 ) A( r )d l, (2.5)
where C( r, r 0 ) is a continuous functional path inside the circuit relating a reference point r 0 and any point r. After performing a unitary transformation

Htot = U † H tot U with U = exp e a -a † hω c d 3 rV ⊥ ( r)ψ † ( r)ψ( r) , (2.6)
the combined Hamiltonian of the electron-photon system Eq.2.3 becomes

Htot = d 3 rψ † ( r) h( r)ψ( r) + hω c a † a + V a + a † + V 2 / hω c , (2.7)
where

V = -e d 3 rV ⊥ ( r)ψ † ( r)ψ( r) and h( r) = - h 2 2m ∆ r . (2.8)
Now the electron-photon coupling is included in Htot only by V a + a † term, which is a single-photon process. When the spacial dependence of the photonic vector potential A( r) ≡ A 0 can be disregarded, the photonic vector potential

V ⊥ ( r) = ω c A 0 r with A 0 = E 0 /ω c .
In the following, we use the second-quantized formalism to describe an electronic system coupled to photonic field in the cavity. As an example of the electronic system, let us consider a quantum dot tunnel coupled to metallic leads. In the absence of the cavity, the electronic Hamiltonian can be written as

H el = o,j oj c † oj c oj + o,j =o ,j t oj,o j c † oj c o j + h.c. , (2.9)
where c oj (c † oj ) is the electron annihilation (creation) operator with o = d, l 1 , l 2 corresponding to the quantum dot and metallic leads, respectively. j accounts for orbitals in the dot or the leads. oj is the energy of the quantum dot (o = d) or the leads (o = l 1 , l 2 ) that corresponds to wavefunction φ oj ( r). In order to make the connection with Eq.2.7, let us write the field operator as

ψ( r) = o,j φ oj ( r)c oj .
(2.10)

In second-quantized description Eq.2.7 reads

H QD tot = H el + hω c a † a + h int a + a † , (2.11)
where the electron-photon Hamiltonian has the form

h int = o,j α oj c † oj c oj + o,j =o ,j γ oj,o j c † oj c o j + h.c. (2.12)
where we have neglected the terms V 2 / hω c . Using Eq.2.8 we can find the electronphoton couplings α oj and γ oj,o j .

α oj = -e d 3 rV ⊥ ( r) φ oj ( r) 2 (2.13)
acts as to shift the energy oj . And

γ oj,o j = -e d 3 rV ⊥ ( r)φ * oj ( r)φ o j ( r) with oj = o j (2.14)
accounts for photon-induced tunneling coupling between the dot and the leads.

If photonic vector potential varies slowly inside the quantum conductor V ⊥ ( r) = const, then γ oj,o j ≡ 0 since the wavefunctions with oj = o j are orthogonal. And we are only left with capacitive coupling between the quantum dot (and metallic leads) and the cavity central conductor.

In most of the papers [START_REF] Cottet | Mesoscopic admittance of a double quantum dot[END_REF][START_REF] Schiró | Tunable hybrid quantum electrodynamics from nonlinear electron transport[END_REF], the coupling between the quantum dot circuit and the cavity is described as the capacitive coupling between number of electrons in the dot and the photonic field

H el-ph = αd † d(a + a † ), (2.15)
where d(d † ) is the electron annihilation (creation) in the quantum dot and α is the electron-photon coupling constant. While the coupling between the metallic leads and the cavity is disregarded. It could be done assuming that the leads-cavity coupling is symmetric, in which case it just shifts the dot-cavity coupling constant. In chapter 4 of this thesis, we consider fully the coupling between the metallic leads and a microwave cavity.

The derivation for the electron-photon coupling for topological superconductor discussed in chapter 5 is presented in appendix C.1.

input-output theory for microwave cavities

In this section, we present details on the input-output theory [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF] for the cavity in the presence of the coupling to a quantum conductor (tunnel junction, quantum dot, topological wire, etc.). We demonstrate that the transmission of the cavity depends on the charge susceptibility of the electronic system. We concentrate on only single resonance of the cav-

Ê(x) ∝ (a + a † ) H el b in b out c out c in κ 1 κ 2
| 2 ≡ f 1(2) f * 1(2)
being the average coupling between the cavity and the bath modes on the right (left).

In the first part, for simplicity, we assume that κ 2 = 0, which corresponds to a one-side cavity.

ity with frequency ω c and, for simplicity, we assume for the moment a single-sided cavity (for more details see appendix A). The total Hamiltonian describing the system reads [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF][START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF][START_REF] Dmytruk | Out-of-equilibrium quantum dot coupled to a microwave cavity[END_REF]]

H = H b + H c-b + H el + H c + H el-c H sys . (2.16)
The bath Hamiltonian is

H b = q ω q b † q b q , ( 2.17)
where b q (b † q ) are the annihilation (creation) operators for the bath modes with energy ω q , with q labeling their quantum numbers. The cavity Hamiltonian is

H c = ω c a † a, (2.18)
where a (a † ) is the annihilation (creation) operator for the cavity mode with energy ω c .

The cavity-bath Hamiltonian is [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF] H

c-b = -i q f q a † b q -f * q b † q a , (2.19)
where the complex coefficients f q are coupling parameters between the cavity and the external bath. The electron-cavity Hamiltonian is

H el-c = α(a + a † )n, (2.20)
where α is the coupling strength between the cavity field and the total number operator in the system n ≡ N j=1 c † j c j , with c j (c † j ) being the annihilation (creation) operator for the electrons (fermionic degrees of freedom) at site j in the electronic system. And H el is the electronic Hamiltonian only.

The idea of the input-output theory is to find the output photons (or field) in terms of the input ones, as shown schematically in Fig. 2.5. Following Ref. [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF], we obtain for the cavity equation of motion for a one-side cavity (κ 2 = 0 in Fig. 2.5):

ȧ = i [H sys , a] - κ 2 a - √ κb in , (2.21)
for the input field, and

ȧ = i [H sys , a] + κ 2 a - √ κb out , (2.22)
for the output field, where κ = 2πρ|f| 2 is the cavity decay rate, with ρ the bath density of states and f being the average coupling between the cavity and the bath modes. Subtracting Eq. (A.22) from Eq. (A. [START_REF] Churchill | Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover[END_REF]) we obtain that

b out (t) = b in (t) + √ κa(t), (2.23)
a result which holds for any general cavity Hamiltonian.

In the following, we establish the relationship between b out and b in in the presence of the electronic system, as depicted in Fig. 2.5. For that, we first evaluate the commutator:

[H sys , a] = -ω c a -αn , (2.24)
where n is the time-dependent electronic particle number (see below). In order to utilize this contribution to the equation of motion of the cavity field, we need to evaluate the time-dependent particle number n(t), which itself depends on the coupling to the cavity.

We can represent n H (t) in the Heisenberg picture as follows:

n H (t) = U † (t, t 0 )n I (t)U(t, t 0 ), (2.25)
where

U(t, t 0 ) = T c exp -i t t 0 dt H el-c (t ) (2.26)
is the evolution operator with T c the time-ordering operator that puts operators with later times to the left of the ones with earlier times. We can then write Eq. (2.25) up to leading order in the coupling constant α. Thus, the time-evolution of the electronic particle number contains, besides the electronic component, a contribution that arises because of the coupling to the cavity. Introducing Eq. (2.27) into Eq. (2.24) and assuming t 0 → -∞ we obtain:

[H sys , a] = -ω c a -αn I (t) -iα 2 t -∞ dt a(t)e -iω c (t -t) + a † (t)e iω c (t -t) n I (t ), n I (t) , (2.28)
where a(t) ≈ ae -iω c t and a † (t) ≈ a † e iω c t in zeroth order in α (because the expression is already multiplied by α 2 we can utilize the bare time dependence in this expression). In the following, we switch to the Fourier space in order to solve the equation for a(t), and take into account all contributions that affect its time-dependence, namely the external modes too. We obtain:

-iωa(ω) = -iω c a(ω) - κ 2 a(ω) - √ κb in (ω) -iαn I (ω) + α 2 ∞ -∞ dte iωt t -∞
dt a(t)e -iω c (t -t) + a † (t)e iω c (t -t) n I (t ), n I (t) . (2.29)

Before continuing with the derivation, let us describe each term in the above expression.

The first term describes the free cavity evolution, the second term the leaking into the continuum of modes (the external bath) at rate κ/2, the third term is the input field supplied from the right side, the fourth term correspond to another "input" contribution to the cavity from the electronic system (a noise term), while the last term leads to both a shift in the cavity frequency as well as to frequency broadening. We can now average over the electronic system. Moreover, we can neglect the highly oscillating term a † (t) ∝ e iω c t , namely we perform the so called Rotating Wave Approximation (RWA). Under all these assumptions, the last term in Eq. (2.29) becomes:

α 2 ∞ -∞ dte iωt t -∞ dt a(t)e -iω c (t -t) [n I (t ), n I (t)] 0 = -ia(ω)Π(ω c ), (2.30)
where

Π(t -t ) = -iθ(t -t )α 2 [n I (t), n I (t )] 0 , ( 2.31)
is the retarded density-density electronic correlation function or electronic susceptibility, and . . . 0 means the expectation value of the unperturbed electronic system. Note that for deriving the above expression, we assumed that the electronic system is in a stationary state and thus time translational invariant.

We are now in position to find the cavity field a(t) and the output field b out (t) in terms of the input field b in (t). Introducing Eq. (2.30) into Eq. (2.29) we obtain

a(ω) = - √ κb in (ω) + iα n I (ω) 0 -i(ω -ω c ) + κ/2 + iΠ(ω c ) . (2.32)
We thus have two contributions to the cavity field: the external input and the input from the electronic system. Our aim is to relate in fact the output and input fields, which can be done easily via the expression in Eq. (2.23):

b out (ω) = [-i(ω -ω c ) -κ/2 + iΠ(ω c )]b in (ω) -i √ κα n I (ω) 0 -i(ω -ω c ) + κ/2 + iΠ(ω c ) . ( 2.33)
In the limit of large number of photons in the input beam, we can neglect the contribution from the electronic system so that we obtain:

b out (ω) ≈ -i(ω -ω c ) -κ/2 + iΠ(ω c ) -i(ω -ω c ) + κ/2 + iΠ(ω c ) b in (ω) . (2.34)
In experiments, one actually encounters a two-sided cavity (see Fig. 2.5 for the nomenclature), in which case the expression for the cavity equation of motion reads:

ȧ = i [H sys , a] - κ 1 2 + κ 2 2 a - √ κ 1 b in - √ κ 2 c in , (2.35)
so that for the output fields we get:

b out = √ κ 1 a + b in (2.36) c out = √ κ 2 a + c in . (2.37)
By following the same reasoning as for the one-sided cavity and assuming the two mirrors are the same κ 1 = κ 2 ≡ κ, we obtain: .38) Assuming again that the input flux is much larger than the electronic contribution, we can write:

a(ω) = - √ κ[b in (ω) + c in (ω)] + iα n I (ω) 0 -i(ω -ω c ) + κ + iΠ(ω c ) . ( 2
c out (ω) = -τb in (ω) + (. . .)c in (ω) (2.39) with τ = κ -i(ω -ω c ) + κ + iΠ(ω c ) ≡ Ae iφ (2.40)
being the transmission of the cavity, which is a complex number, and which depends on the electronic susceptibility Π(ω c ) at cavity frequency ω c .

The phase and amplitude response of the cavity close to resonance ω ≈ ω c are related to the susceptibility Π(ω c ) as follows:

δφ = Π (ω c )/κ, (2.41
)

δA/A in = Π (ω c )/κ, (2.42)
where δφ = φ out -φ in , δA = A in -A out , and

Π (ω) = Re[Π(ω)] (Π (ω) = Im[Π(ω)])
is the real (imaginary) part of the susceptibility. The frequency shift δφ (or the modification of the resonance frequency ω c ) is proportional to the real part of the electronic susceptibility Π (ω c ) while the frequency broadening δA/A in (or the modification of the quality factor Q) is proportional to the imaginary part of the susceptibility Π (ω c ).

summary

In this chapter, we summarize theoretical and experimental studies of atoms (real or artificial) coupled to superconducting resonators. In subsection 2.1.1, we demonstrate that the properties of a single photon in a 3D resonator can be probed by atoms passing through the cavity. Then we discuss how the fixed atom-cavity coupling and scalability problems present in atomic cavity QED can be fixed by going from 3D cavity to 1D transmission line resonator and from a real atom to an artificial one. The main advantages of circuit QED:

1. The large dipole moment of the superconducting qubit and rms voltage of the resonator allow to reach easily the strong-coupling limit 2. Qubits can be separated by centimeter distances and entangling of qubits can be realized to implement two-qubits gates.

The idea to study quantum conductors embedded in superconducting resonators by coupling them to the wires and measuring differential conductance comes from mesoscopic cavity QED. In section 2.2, we discuss experimental and theoretical study of quantum dots with cavity QED. We provide the derivation of the electron-photon coupling

Hamiltonian for mesoscopic cavity QED in section 2.3. And in section 2.4, we derive the optical transmission coefficient of the cavity

τ = κ -i(ω -ω c ) + κ + iΠ(ω c ) (2.43)
and demonstrate that if the frequency is close to the cavity resonance, the phase shift and frequency broadening are related to the electronic susceptibility as

δφ = Π (ω c )/κ, (2.44
)

δA/A in = Π (ω c )/κ. (2.45)
In this thesis, we apply methods from cavity QED to study different systems coupled to superconducting microwave cavity, such as a tunnel junction, a quantum dot tunnel coupled to metallic leads, a one-dimensional topological semiconducting wire and a superconducting ring.

M A J O R A N A F E R M I O N S
Majorana fermion is a particle that is its own antiparticle. Majorana fermions have been predicted by Ettore Majorana in 1937 [START_REF] Majorana | Teoria simmetrica dell'elettrone e del positrone[END_REF]. Dirac equation

(iγ µ ∂ µ -m) φ = 0, (3.1)
where φ is a four-component complex field, and γ matrices are required to obey the rules of the Clifford algebra, describes charged particles with spin 1/2, such as electrons.

The idea of Ettore Majorana was to find an equation similar to Dirac equation that will describe a particle with spin 1/2 that is its on antiparticle [START_REF] Wilczek | Majorana returns[END_REF]. If a particle is associated with a complex filed φ (see Eq.3.1), then the corresponding antiparticle will be associated with the conjugated field φ * . For example, the antiparticle of an electron with charge e is a positron with charge -e that appears as a solution of conjugated Dirac equation. Particles and their antiparticles have opposite charges, which means that the particle proposed by Ettore Majorana should have zero charge. Also, if the particle is associated with the field φ M and its antiparticle is associated with the field φ * M , the Majorana particle should be associated with the real field φ M ≡ φ * M . Under this condition, Ettore Majorana found the equation (later called Majorana equation)

iγ µ M ∂ µ -m φ M = 0, (3.2)
where γ µ M = -γ µ M * are modified purely imaginary γ matrices that obey the rules of the Clifford algebra. One of the possibles forms of γ M matrices is

29 γ 0 M =         0 0 0 -i 0 0 -i 0 0 i 0 0 i 0 0 0         , ( 3.3
)

γ 1 M =         0 0 i 0 0 0 0 i i 0 0 0 0 i 0 0         , ( 3.4
)

γ 2 M =         i 0 0 0 0 i 0 0 0 0 -i 0 0 0 0 -i         , ( 3.5
)

γ 3 M =         0 0 0 -i 0 0 i 0 0 i 0 0 -i 0 0 0         . ( 3.6)
Although Majorana fermions have been initially predicted as particles in high energy physics (the only elementary particle that could be Majorana fermion is neutrino), they could also emerge as quasiparticle excitations in condensed matter (solid-state) physics.

The special property of being its own antiparticle could be conveniently written in the second quantization formalism as

γ j = γ † j , (3.7)
where γ j is the Majorana operator. Conventional fermions are described by creation c † j and annihilation c j operators. In order to build the Majorana operator γ j , one should take a linear combination of conventional fermionic operators c † j and c j , or electrons and holes

γ j = uc † j + vc j . (3.8)
In superconductors, the particle-hole symmetry relates the creation operator γ † E at energy E to the annihilation operator γ -E at energy -E [START_REF] Beenakker | Random-matrix theory of Majorana fermions and topological superconductors[END_REF]. At zero energy E = 0, a double degenerate level emerges in superconductors. This level has a special property

γ † E=0 = γ E=0 . (3.9)
The zero-energy state in superconductors is the Majorana fermion.

Majorana fermions could be also used to build a topological quantum computer since they obey the non-abelian statistics in 2D [START_REF] Ivanov | Non-Abelian statistics of half-quantum vortices in p-wave superconductors[END_REF]. In 3D, particles obey either the Fermi-Dirac statics and are thus called fermions or the Bose-Einstein statistics and are subsequently called bosons. Observing the behavior of multiparticle states during the exchange of particles allows to determine the statistics of the particles. For example, for a two-particle state

|ψ 1 ψ 2 = ±|ψ 2 ψ 1 , (3.10)
where sign +(-) corresponds to particles being bosons (fermions). But in 2D systems, quasiparticles could obey statistics ranging continuously between Fermi-Dirac and Bose-Einstein statistics

|ψ 1 ψ 2 = e iθ |ψ 2 ψ 1 , (3.11)
where θ is an arbitrary real number. If θ = 0(π), we recover the Bose-Einstein (Fermi-Dirac) statistics. Otherwise, particles obtain an arbitrary phase when they are interchanged.

Such particles are called anyons. When anyon 1 and anyon 2 are interchanged twice, their wavefunction is not the same as the initial one, but rather e i2θ |ψ 1 ψ 2 . This means that the exchange of particles changes the state of the system with the same particle configuration (see Fig. 3.1). The interchange of two Majorana operators is called braiding. And the result of the braiding depends on the order in which it is performed (U 12 = U 21 ).

γ 1 γ 2 Studying Majorana fermions in the context of condensed matter physics is in focus of tremendous theoretical and experimental research activity. On one hand, this scrutiny is driven by the fundamental interest in particles that are their own antiparticles. On the other hand, such particles may have numerous applications in quantum computing.

conventional fermionic and majorana operators

In condensed matter physics, Majorana fermions emerge as zero-energy and chargeneutral quasiparticles in superconductors.

Usual fermionic creation c † j and annihilation c j operators obey anticommutation relations

{c i , c † j } = δ i,j , {c i , c j } = 0. (3.12)
In order to create a neutral excitation, one needs to take a linear combination of fermionic creation and annihilation operators c † j , c j . Let us define Majorana operators as

γ A,j = -i c j -c † j , γ B,j = c j + c † j . (3.13)
These Majorana operators still satisfy the anticomutation relations

{γ α,j , γ α ,j } = 2δ αα δ jj with α = A, B and j = 1, ..., N, (3.14)
but also have a special property

γ α,j = γ † α,j . (3.15)
Eq.3.15 demonstrates that Majorana fermion γ α,j is identical to its antiparticle γ † α,j . In the following sections, we discuss several theoretical models of the systems that support Majorana fermions as well as the experimental implementation of these systems. In section 3.2, we provide details on the Kitaev chain model as we use it in chapters 5,7. The Kitaev model is the simplest model that takes into account both Majorana states and bulk states of the superconductor. In section 3.3, we describe the spin-orbit coupled nanowire model, a realistic model that describes emergence of the Majorana fermions and has been already implemented experimentally. We use this model in chapter 5 to demonstrate that our findings for Kitaev model remain valid for a more realistic system. In section 3.4, we discuss first experimental signatures of Majorana fermions in condensed matter systems, e.g. zero-bias peak. We continue by presenting the fractional Josephson effect and its connection to the Majorana fermions physics in section 3.5 before a conclusive discussion in section 3.6.

prototype model for 1d p-wave superconductor: kitaev model

In this section, we introduce a toy model for a 1D topological superconductor that supports Majorana fermions, namely the Kitaev model for a spinless p-wave superconductor [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. This model describes a chain of N sites with nearest-neighbor p-wave superconducting pairing. The Kitaev chain Hamiltonian reads

H K = -µ N j=1 c † j c j - 1 2 N-1 j=1 tc † j c j+1 + ∆c j c j+1 + h.c. , (3.16)
where µ is the chemical potential, t is the nearest-neighbor hopping amplitude and ∆ is the p-wave superconducting pairing potential. It is instructive to first study the bulk properties of the Kitaev chain by imposing periodic boundary conditions on the system, c N+1 ≡ c 1 . For periodic boundary conditions, we can use the Fourier description for the electronic operators: .17) where c k is the fermionic annihilation operator with momentum k = 2πn/N (assuming the lattice spacing d ≡ 1 from now on), with n = 1, ..., N. By doing so, we can write the Kitaev Hamiltonian Eq. 5.4 in momentum space

t µ γ A ,1 γ B, 1 γ A ,2 γ B, 2 γ B ,N γ A ,N γ A ,N -1 γ B ,N -1 t t t µ µ µ
c j = 1 √ N k c k e ikj . ( 3
H K = k>0 H K BdG (k) , (3.18)
with

H k BdG (k) = ξ k (c † k c k -c -k c † -k ) -i∆ sin k(c -k c k -c † k c † -k ), (3.19)
is the Bogoliubov-de Gennes Hamiltonian and ξ k = -t cos k -µ. We can simply diagonalize the H K in the k-space and write:

H K = k>0 E k γ † k γ k + γ † -k γ -k , (3.20)
when expressed in terms of quasiparticle operators

c k = u * k γ k + v k γ † -k , (3.21) c † -k = -v * k γ k + u k γ † -k . (3.22)
Here, the functions u k and v k are given by

|u k | 2 = 1 2 1 + ξ k E k , (3.23
)

|v k | 2 = 1 2 1 - ξ k E k , ( 3.24)
and

E k = ± (-t cos k -µ) 2 + (∆ sin k) 2 , ( 3.25)
is the bulk energy spectrum for the Kitaev chain (see Fig. 3.3).

- The topological phase transition happens when the gap closes, E k = 0. From Eq.3.25

3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k Energy (a) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k (b) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k (c)
we notice that the gap closes at k = 0 for µ = -t and k = π for µ = t. Thus |µ| = t is the topological phase transition point. In the following, we demonstrate that the Kitaev chain is in the topological phase when |µ| < t, and in the non-topological phase when |µ| > t.

In the topological phase, the Majorana fermions emerge at both ends of the Kitaev chain with open boundary conditions.

Let us determine a topological phase of the system by introducing an integer-valued topological invariant -a winding number [START_REF] Niu | Majorana zero modes in a quantum Ising chain with longer-ranged interactions[END_REF]. It is convenient to define the invariant in the pseudospin basis h(k), so let us rewrite the BdG Hamiltonian Eq. C.30 and set ∆ = t as

H k BdG (k) = h(k) • τ (3.26)
where τ = τ k x , τ k y , τ k z are Pauli matrices acting on the particle-hole basis

c k , c † -k , h y (k) = -t sin (k) and h z (k) = (-t cos (k) -µ). The pseudospin h(k) is defined in the yz-plane.
Let us introduce a unit vector ĥ

(k) = h(k) h(k) ≡ cos (θ k ) τ k y + sin (θ k ) τ k z , (3.27)
where

h(k) = h 2 y + h 2 z = E k .
The winding number is defined in terms of this unit vector. The states of the system with periodic boundary conditions form a ring T 1 in the momentum space, while the unit vector ĥ(k) is defined on a unit circle S 1 in the yz-plane.

Thus the angle θ k is a mapping θ k : S 1 → T 1 . The winding number is the fundamental group of this mapping

w = dθ k 2π , (3.28)
where θ k = arctan -t sin(k) t cos(k) + µ (we perform the rotation so that the angle is defined in the xy-plane) and the integral is taken over the first Brillouin zone. The winding number shows how many times the unit vector ĥ(k) rotates in the yz-plane around the Brillouin zone (see Fig. 3.4). The absolute value of the winding number is equal to the number of Majorana fermions at each end of the chain. The winding number can be any integer and remains constant as long as the system is gaped. The winding number changes only when the gap closes at the topological phase transition and is not defined at the phase transition point since E k = 0 when the gap closes.

For the Kitaev model, the winding number w = 0 for t < |µ| implies the absence of Majorana fermions in the chain (or that the system is in non-topological phase) and w = -1 for t > |µ| imples the existence of one Majorana fermion at each of the chain (or that the system is in the topological phase) [START_REF] Trif | Resonantly tunable majorana polariton in a microwave cavity[END_REF].

Having determined the conditions for the topological phase transitions, we consider a finite chain with Majorana fermions emerging at its ends [START_REF] Alicea | New directions in the pursuit of Majorana fermions in solid state systems[END_REF][START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. Each site of the chain in the Kitaev model is described by a pair of creation c † j and annihilation c j fermionic operators. Let us formally define the Majorana operators by decomposing c j , c † j operators in the Kitaev Hamiltonian Eq.5.4 in terms of two Majorana operators

c j = 1 2 γ B,j + iγ A,j , c † j = 1 2 γ B,j -iγ A,j , (3.29)
where γ A,j and γ B,j are the Majorana operators that fulfill the relations

γ α,j = γ † α,j , {γ α,j , γ α ,j } = 2δ αα δ jj , (3.30)
with α = A, B and j = 1, ..., N.

In terms of the Majorana operators γ A,j and γ B,j , the initial Hamiltonian of the Kitaev chain Eq.5.4 becomes Let us first consider two special cases.

H K = - µ 2 N j=1 iγ B,j γ A,j + 1 - i 4 N-1 j=1 (∆ + t) γ B,j γ A,j+1 + (∆ -t) γ A,j γ B,j+1 . (3.31) γ A ,1 γ B, 1 γ A ,2 γ B, 2 γ B ,N γ A ,N γ A ,N -1 γ B ,N -1
γ A ,1 γ B, 1 γ A ,2 γ B, 2 γ B ,N γ A ,N γ A ,N -1 γ B ,N -1
1) Trivial case: |∆| = t = 0 and µ < 0. Then Eq.3.31 is simplified and it reads

H K = - µ 2 N j=1 iγ B,j γ A,j + 1 . (3.32)
In this case, the Hamiltonian H K contains pairing only between the Majorana operators γ A,j , γ B,j from the same site j (see Fig. 3.5). Therefore, all the Majorana operators are always paired. For |∆| = t = 0 and µ < 0, the Kitaev chain is in the topologically trivial (non-topological) phase.

2)Non-trivial case: |∆| = t > 0 and µ = 0. For this choice of parameters, the Hamiltonian of the Kitaev chain Eq.3.31 reads

H K = -i t 2 N-1 j=1 γ B,j γ A,j+1 . (3.33)
In this case, only the Majorana operators γ B,j , γ A,j+1 with j = 1, ..., N -1 from different sites are paired, while the operators γ A,1 and γ B,N do not enter the Hamiltonian (see

f j = 1 2 γ A,j+1 + iγ B,j . (3.34)
The Kitaev Hamiltonian Eq.(3.33) can be written in terms of these new operators f j , f † j as

H K = t N-1 j=1 f † j f j - 1 2 . (3.35)
Let us also combine two Majorana operators γ A,1 and γ B,N that dropped out from Eq.3.33 and introduce an ordinary fermionic operator

cM = 1 2 (γ A,1 + iγ B,N ) , (3.36)
that is a sum of two operators localized at sites 1 and N, respectively, and thus is a highly non-local object.

The finite size of the chain with open boundary conditions results in an overlap between

γ A,1 and γ B,N operators. This overlap can be described by an effective Hamiltonian [49]

H eff K = -i t 2 γ B,N γ A,1 , (3.37)
where t ∝ exp (-L/ξ) with L being the length of the chain and ξ being the coherence length.

The Hamiltonian H eff K can be written in terms of the ordinary fermionic operators as

H eff = 2 M c † M cM - 1 2 , ( 3.38)
where M is called the Majorana energy splitting and M = t 4 ∝ exp (-L/ξ). Let us introduce a number of particles operator

n M = c † M cM . (3.39)
Depending on the Majorana state being empty (n M = 0) or occupied (n M = 1), the Majorana ground state is either even or odd with energy -M or M , respectively.

All the aforementioned analysis of the topological properties of the Kitaev chain remains valid for arbitrary values of µ, ∆ and t. However, p-wave superconductors are very rare in nature (the only known p-wave superconductor is Sr 2 RuO 4 [START_REF] Jang | Observation of half-height magnetization steps in Sr2RuO4[END_REF]). So how one can realize a chain of spinless electrons in a laboratory?

The first theoretical proposal to effectively implement the Kitaev chain was to use an edge of a 2D topological insulator proximitized to an s-wave superconductor [START_REF] Fu | Superconducting proximity effect and Majorana fermions at the surface of a topological insulator[END_REF][START_REF] Fu | Josephson current and noise at a superconductor/quantumspin-Hall-insulator/superconductor junction[END_REF].

At the edge of the 2D topological insulator, electrons with spin up propagate clockwise around the edge while electrons with spin down propagate counterclockwise, making the system appear spinless. p-wave superconductivity is generated in the system when interfacing the topological insulator with a s-wave superconductor.

practical realization of majorana fermions in 1d p-wave superconductors: spin-orbit coupled nanowire

In this section, we discuss in detail one of the theoretical proposals of experimental realization of a 1D spinless p-wave superconductor: a spin-orbit coupled nanowire subject to a magnetic field and in proximity to an s-wave superconductor. We use in chapter 5. This theoretical model has been first proposed in [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF] based on earlier semiconductor-based proposals [START_REF] Alicea | Majorana fermions in a tunable semiconductor device[END_REF][START_REF] Sau | Generic new platform for topological quantum computation using semiconductor heterostructures[END_REF] and implemented experimentally in [START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF]. Three ingredients are combined in this model to engineer p-wave superconductivity: a 1D wire with strong spin-orbit coupling, a conventional s-wave superconductor, and a magnetic field. The scheme of the setup is shown in Fig. 3.7.

The Hamiltonian for the nanowire with spin-orbit coupling in the presence of the magnetic field and in proximity to an s-wave superconductor reads [START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF] spin-orbit interaction u is along z axis and a magnetic field B is along x axis.

H = Ψ † (y) HΨ (y) dy, (3.40)
H = p 2 2m -µ τ z + upσ z τ z + Bσ x + ∆τ x , (3.41)
where

Ψ = ψ † ↑ , ψ † ↓ , ψ ↓ , -ψ ↑ , ψ ↑,(↓) (y)
annihilates spin-up (spin-down) electrons at position y. The Pauli matrices σ i , τ i , i = x, y, z act in spin and particle-hole space, respectively. µ is the chemical potential, u is the Rashba spin-orbit interaction that aligns the spins in the z-direction, B is the magnetic field and ∆ is the s-wave pairing potential induced by proximity. We chose the wire to lie along the y axis, the spin-orbit interaction to be along the z axis, and the magnetic field to be along the x axis (see Fig. 3.7). The directions of the fields could be chosen arbitrary provided that the spin-orbit field and the magnetic field are orthogonal.

The bulk energy spectrum for the spin-orbit coupled nanowire reads

E 2 ± = B 2 + ∆ 2 + p 2 2m -µ 2 + (up) 2 ± 2 B 2 ∆ 2 + B 2 p 2 2m -µ 2 + (up) 2 p 2 2m -µ 2 . (3.42)
Let us discuss the energy spectrum of spin-orbit coupled nanowire and how it can emulate a spinless p-wave superconductor. In the absence of the magnetic field B = 0 and induced superconductivity ∆ = 0, the energy spectrum consists of two shifted parabolas that cross at the momentum p = 0. Due to the spin-orbit coupling, these two parabolas have opposite spins. When the magnetic field is applied, the crossing at p = 0 is lifted and the gap at p = 0 opens. If we tune the chemical potential µ so that it lies inside the field-induced gap, the wire appears spinless. The superconducting pairing ∆ opens gaps at the Fermi momentum p = ±p F = ± √ 2mµ and ensures that they always remain finite.

For large magnetic field B ∆ we can project away the upper energy band and the wire becomes effectively a spinless p-wave superconductor.

- The bulk energy spectrum E ± has two gaps:

3 -2 -1 0 1 2 3 -2 -1 0 1 2 p Energy -3 -2 -1 0 1 2 3 -2 -1 0 1 2 p (b) (a)
1) the gap at p = 0, which could be created by the magnetic field and is given by

E 0 = E(p = 0) = B -∆ 2 + µ 2 ; (3.43)
2) the gap at p = ±p F , which is created by the strong-spin orbit interaction and is always finite due to the induced superconductivity. The gap is given by

E 1 = E(p = ±p F ) = ξ p=±p F ± up F . (3.44)
The quantum phase transition occurs when the gap at p = 0 closes, E 0 = 0, and

B 2 = ∆ 2 + µ 2 (see Fig.3.8). When B 2 > ∆ 2 + µ 2 , E 0 is dominated by the magnetic field B,
the wire is in the topological phase with Majorana fermions at the ends of the wire.

When B 2 < ∆ 2 + µ 2 , E 0 is dominated by the pairing potential ∆, the wire is in the nontopological phase.

Now we demonstrate that in the limit of strong spin-orbit coupling, the nanowire model Eq.3.41 reduces at low energies to a spinless p-wave-superconductor.

Let us diagonalize the Hamiltonian for the spin-orbit coupled nanowire H = H 0 + ∆τ x in the absence of the induced pairing ∆ and then treat the latter perturbatively [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF][START_REF] Pientka | Magneto-Josephson effects and Majorana bound states in quantum wires[END_REF]]

H 0 = p 2 2m -µ τ z + upσ z τ z + Bσ x . (3.45)
In order to do so, we perform the unitary transformation

H0 = UH 0 U † with U = exp (iασ y τ z /2) . (3.46) If we choose α so that tan (α) = B up , (3.47)
H0 takes the form

H0 = p 2 2m -µ + u 2 p 2 + B 2 σ z τ z . (3.48)
Now let us reintroduce the pairing term ∆τ x and apply the transformation U to it.

Then HNW reads

HNW = p 2 2m -µ + u 2 p 2 + B 2 σ z τ z + ∆up u 2 p 2 + B 2 τ x - ∆B u 2 p 2 + B 2 σ y τ y . (3.49)
Neglecting the last term in Eq.C.50 for B ∆, µ the effective Hamiltonian reads

Heff = p 2 2m -µ + u 2 p 2 + B 2 σ z τ z + ∆up u 2 p 2 + B 2 τ x . (3.50)
The low-energy subspace at p = ±p F is formed by the bands with σ z = -1. The projection of Heff C.51 to the lower bands yields

Heff = p 2 2m -µ -u 2 p 2 + B 2 τ z + ∆up u 2 p 2 + B 2 τ x . (3.51)
The large spin-orbit coupling SO ∆ guarantees that we can neglect the coupling to high-energy degrees of freedom near ±p F . Linearizing the Hamiltonian Heff 3.51 around the Fermi momenta and using |up| ≡ SO B the effective Hamiltonian takes the form

Heff = u (|p| -p F ) τ z + sign (p) ∆τ x . (3.52)
This Hamiltonian describes a spinless p-wave superconductor with superconducting pairing ∆ p ∝ ∆p similar to the Kitaev model described in the previous section.

experimental signatures of majorana fermions: observation of zerobias peak

There are several experimental manifestations associated with the presence of Majorana fermions, such as the zero-bias peak in the topological region [START_REF] Churchill | Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover[END_REF][START_REF] Das | Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions[END_REF][START_REF] Deng | Anomalous zero-bias conductance peak in a Nb-InSb nanowire-Nb hybrid device[END_REF][START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF], the depen-dence of the Majorana energy splitting on the Zeeman splitting, or the size of the system

[1], enhanced conductance at zero energy in chain of magnetic atoms [START_REF] Nadj-Perge | Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor[END_REF], ac fractional

Josephson effect [START_REF] Rokhinson | The fractional ac Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles[END_REF][START_REF] Wiedenmann | 4 [pi]-periodic Josephson supercurrent in HgTe-based topological Josephson junctions[END_REF]. Let us discuss several of them in more detail.

First experimental observation of signatures of the Majorana fermions has been reported in [START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF]. They studied a nanowire by electrical measurements and observed a peak at zero bias, which is consistent with the presence of zero-energy (midgap) bound states at the ends of the wire.

The experimental setup consists of InSb nanowire connected to one normal and one superconducting electrode in the presence of magnetic field. This physical system has all three components required for the emergence of Majorana fermions [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF]: a nanowire with strong spin-orbit interaction, proximity-induced superconductivity and magnetic field. However, the two latter quantities should be related as The differential conductance dI/dV at voltage V for different values of magnetic field was measured in [START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF]. The peak at V = 0 appears for magnetic field B ≈ 100 mT. This peak remains stuck to zero over a substantial range of magnetic fields (up to B ≈ 400 mT). The zero-bias peak signals a presence of midgap states in the wire and could be interpreted as a transport signature of Majorana fermions. Thus, this experiment provides first evidence for the presence of the Majorana fermions in nanowires with proximityinduced superconductivity.

E z > µ 2 + ∆ 2 .
Recently, observation of exponential suppression of energy splitting with increasing wire length in InAs nanowire segment with epitaxial aluminium, which forms a proximityinduced superconducting Coulomb island, has been reported in [1]. For short devices of a few hundred nanometres, they found that energy of the sub-gap state oscillate as a function of magnetic field, as is expected for the Majorana fermions. 3.5 fractional josephson effect

Theoretical description of DC Josephson effect

The Josephson effect describes the appearance of a supercurrent in a weak link between two superconductors. For usual s-wave superconductors, the supercurrent appears due to the tunneling of the Cooper pairs with charge 2e across a weak link. In this section, we discuss DC Josephson effect, when there is no voltage applied to the junction and the phases of the superconductor are created by magnetic flux. The Josephson current depends on the phase difference of the superconductors and is given by given by

I J ∝ sin (φ R -φ L ), where φ L (φ R ) is
I J ∝ sin φ R -φ L 2
. This effect that arises because of the degeneracy of the zero energy Majorana modes is called the fractional Josephson effect.

The fractional Josephson effect is one of the hallmarks of the Majorana fermions [START_REF] Fu | Josephson current and noise at a superconductor/quantumspin-Hall-insulator/superconductor junction[END_REF][START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF][START_REF] Kwon | Fractional ac Josephson effect in p-and d-wave superconductors[END_REF][START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Pientka | Signatures of topological phase transitions in mesoscopic superconducting rings[END_REF]. It corresponds to 4π periodicity of the supercurrent in the superconductor phase difference across a weak link, as opposed to 2π periodicity for the usual Josephson effect in conventional s-wave superconductors.

Topo SC Topo SC γ L γ R φ L φ R Figure 3
.10: Scheme of a Josephson junction consisting of two topological wires (black lines) connected via a topologically trivial weak link (dashed line). p-wave superconductivity is created by a proximity to s-wave superconductors (blue rectangles). A weak link could be created by an insulating barrier (grey rectangle). The left (right) s-wave superconductor has a phase φ L (φ R ). The presence of the Majorana fermions γ L and γ R (brown circles) give rise to the 4π-periodic Josephson current across a weak link.

Let us consider a simplest setup required to observe the fractional Josephson effect:

the superconductor-insulator-superconductor junction (see Fig. 3.10). We assume that the topological superconductors are long enough, so that only a finite overlap between two Majorana fermions γ L and γ R at both sides of a weak link could be taken into account.

Let us define a usual fermionic operator that consists of two Majorana operators

cM = 1 2 (γ L + iγ R ) (3.53)
and a number of particles operator

n M = c † M cM . (3.54)
In the topological phase, the Josephson current is mainly transmitted by the Majorana states. Effective low-energy Hamiltonian for a setup shown in Fig. 3.10 reads

H eff = Γ 2 cos ∆φ 2 iγ L γ R = Γ cos ∆φ 2 n M - 1 2 , ( 3.55)
where ∆φ = φ R -φ L . Note that [H eff , n M ] = 0, so that the number of Majorana fermions is conserved.

The Majorana energy splitting is given by

E = ±Γ cos ∆φ 2 , ( 3.56)
wheresign corresponds to even parity (n M = 0) and + corresponds to odd parity (n M = 1). The energy spectrum is presented in Fig. 3.11.

The Josephson current could be defined as [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF] I J = -∂ H eff ∂∆φ and is equal to

I J = ± Γ 2 sin ∆φ 2 . ( 3

Observation of 4π-periodic AC Josephson current

In this section, we discuss other experimental manifestation of Majorana fermions, namely the fractional Josephson effect. In [START_REF] Rokhinson | The fractional ac Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles[END_REF], the first observation of the fractional AC Josephson effect in a hybrid superconductor-semiconductor nanowire junction has been reported.

When the junction is irradiated with a frequency ω 0 in the absence of an external magnetic field, quantized voltage steps (Shapiro steps) with a height hω 0 /2e are observed, as is expected for conventional superconductor junctions, where the supercurrent is carried by charge 2e Cooper pairs. At high magnetic fields the height of the first Shapiro step is doubled to hω 0 /e, suggesting that the supercurrent is carried by charge e quasiparticles.

This experimental result is a signature of the Majorana fermions.

In [START_REF] Rokhinson | The fractional ac Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles[END_REF], the experimental device consists of Nb/InSb/Nb Josephson junctions. Superconductivity in InSb is induced by the proximity effect from a Nb film placed on top of the InSb nanowire. The supercurrent is flowing only in the nanowire. This experimental realization is based on the theoretical proposal [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF].

(a) (b) In the presence of microwave excitation with frequency ω 0 , phase locking between the rf field and the Josephson supercurrent gives rise to constant-voltage Shapiro steps in the I -V characteristics at V n = n hω 0 q , where q is the charge of quasiparticles and n = 0, ±1, ±2, ... numbers of the Shapiro steps. At zero magnetic field B = 0 and ω 0 = 3

GHz the Shapiro steps with ∆V = 6µV were observed (up to n = 10). This value of ∆V is consistent with the Cooper pair (with charge 2e) tunneling. When a low magnetic field B I, B < 2 T is applied, first three Shapiro steps at V = 6, 12 and 18µ V are clearly visible [START_REF] Rokhinson | The fractional ac Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles[END_REF]. The first step with n = 1 at 6µV disappears above B ≈ 2 T. This observation could be explained if the current is carried by the charge e quasiparticles, e.g. the Majorana fermions. Thus, the doubling of the Shapiro step is a signature of the topological quantum phase transition and presence of the Majorana fermions.

Another experimental observation of the doubling of the Shapiro step has been reported in [START_REF] Wiedenmann | 4 [pi]-periodic Josephson supercurrent in HgTe-based topological Josephson junctions[END_REF]. In this paper, they study a Josephson junction based on HgT e, a 3D topological insulator and observe a missing n = 1 Shapiro step in I -V characteristic. This could be explained by the presence of a 4π-periodic supercurrent, e.g. the fractional Josephson effect.

discussion

The experiments Refs. [START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF][START_REF] Rokhinson | The fractional ac Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles[END_REF][START_REF] Wiedenmann | 4 [pi]-periodic Josephson supercurrent in HgTe-based topological Josephson junctions[END_REF] are in qualitative agreement with theoretical proposals for the emergence of Majorana fermions, yet not fully consistent with the predicted values of the conductance, the required values of the magnetic field for topological phase transition and the absence of all odd Shapiro steps. Therefore, zero-bias peaks obtained in transport experiments can be caused by other physical processes, such as disorder [START_REF] Bagrets | Class D spectral peak in Majorana quantum wires[END_REF][START_REF] Liu | Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and without Majorana end-states[END_REF], weak antilocalization [START_REF] Pikulin | A zero-voltage conductance peak from weak antilocalization in a Majorana nanowire[END_REF] or Kondo effect [START_REF] Lee | Zero-bias anomaly in a nanowire quantum dot coupled to superconductors[END_REF].

Cavity QED, discussed in chapter 2, offers an alternative approach to probe the signatures of Majorana fermions in topological wires by using the photonic field in the cavity.

Several theoretical works study the interplay of the Majorana fermions physics and microwaves in a superconducting cavity QED setup [START_REF] Cottet | Squeezing light with Majorana fermions[END_REF][START_REF] Müller | Detection and manipulation of Majorana fermions in circuit QED[END_REF][START_REF] Ohm | Majorana fermions coupled to electromagnetic radiation[END_REF][START_REF] Ohm | Microwave readout of Majorana qubits[END_REF][START_REF] Schmidt | Majorana qubit rotations in microwave cavities[END_REF][START_REF] Schmidt | Microwave-controlled coupling of Majorana bound states[END_REF][START_REF] Trif | Resonantly tunable majorana polariton in a microwave cavity[END_REF][START_REF] Yavilberg | Fermion parity measurement and control in Majorana circuit quantum electrodynamics[END_REF]. In contrast to the conventional electronic methods, this approach is unique in that it can be totally non-invasive, i.e. it does not alter the electronic system, a crucial feature in view of using these excitations for quantum computing. However, most studies concentrate on effective models that consider only the Majorana fermions interacting with the cavity field, which cannot account for the evolution of the Majorana fermions through the topological transition from a topological to a trivial superconductor. More specifically, the Majorana fermions emerge as edge modes in a topological superconductor: below the phase transition, the system is topological and shows edge modes, while above the phase transition, the system is trivial and has no edge modes. Majorana fermions are thus closely related to the bulk physics. We consider the whole system in chapter 5.

O U T -O F -E Q U I L I B R I U M Q U A N T U M D O T C O U P L E D T O A M I C R O WAV E C AV I T Y
In this chapter, we apply the cavity QED approach described in chapter 2 to study two mesoscopic conductors: a tunnel junction and a quantum dot tunnel coupled to metallic leads. We demonstrate that the measurement of the optical transmission coefficient which is related to the charge susceptibility can be used to extract information on the transport properties of the quantum dot. We show that the asymmetry of the capacitive couplings between the electronic reservoirs and the cavity plays a crucial role in relating optical observables to transport ones. In particular, we found that when the quantum dot is far from resonance, the charge susceptibility is directly proportional to the admittance.

However, when the quantum dot is near a resonance, such a relation generally holds only for the symmetric couplings between the leads and the cavity, at low frequency and for equal tunnel coupling or low voltage. Beyond this low-energy near equilibrium regime, the charge susceptibility offers new insights on the quantum conductors since the optical observables are not directly connected to transport quantities. For symmetric lead capacitive couplings, we show that the optical measurements can be used to reveal the Korringa-Shiba relation, connecting the reactive to the dissipative part of the susceptibility, at low frequency and low bias.

model hamiltonian

Our system, depicted in Fig. 4.1, consists of a quantum dot that is tunnel coupled to metallic leads. Both the quantum dot and the leads are capacitively coupled to a (superconducting) microwave cavity. We assume that the wavelength of the electromagnetic field (∼mm) is much larger than the mesoscopic system (∼µm), and thus the coupling is con-

53 Source Drain eV E(x)
A in exp (iφ in )

A out exp (iφ out )

β L β R λ Figure 4
.1: Sketch of the system. The quantum dot with discrete energy levels (in black) is coupled to the source and drain electrodes (blue), which have a voltage bias eV. Differential conductance of the system may be found by transport measurements. The entire setup is coupled to the microwave cavity (depicted as light between two mirrors; mirrors in gray). The photons are coupled to the left lead with coupling strength β L , to the right lead with β R and to the dot with λ. The electromagnetic field inside the cavity is probed by sending input field with amplitude A in and phase φ in and measuring the output field with A out and φ out . The difference between the outgoing and ingoing fields gives direct access to the charge susceptibility of the electronic system (see text).

stant over its length. The cavity supports many (in principle, infinitely many) modes, all of which should couple to the system. However, only cavity modes resonant with some electronic modes are relevant, the others giving negligible effect. From here on, we assume that only one mode, e.g. the lowest, couples to the system. Nevertheless, our analysis can be straightforwardly extended to a multimode cavity. We analyze the optical transmission and its relation to various observables of the mesoscopic system. We consider both the equilibrium and out-of-equilibrium situations where the voltage bias is large.

The Hamiltonian of the combined system reads:

H sys = H el + H el-c + H ph , (4.1)
H el = d d † d + kα t α c † kα d + h.c. + kα kα c † kα c kα , (4.2)
H el-c = λ(a + a † )d † d + (a + a † ) kα β α c † kα c kα , (4.3)
and H ph = ω c a † a. In these equations, c kα (c † kα ) is the annihilation (creation) operator of an electron in the left (right) lead α = L(R) with momentum k; d (d † ) is the annihilation (creation) operator of an electron in the quantum dot; a (a † ) is the annihilation (creation) operator of the photon field. We denote the energy level of the quantum dot as d , the tunneling amplitude between the quantum dot and the lead α as t α , and the cavity mode frequency as ω c . Finally, λ and β α are the electron-photon coupling constants to the quantum dot, and the lead α = L, R, respectively. Note that we include the capacitive coupling between the quantum dot and the cavity as well as between the cavity and the electronic reservoirs. We have neglected direct photon-induced tunneling terms in the Hamiltonian which are subleading compared to capacitive couplings terms [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF]. A general expression for the coupling constants λ and β α is given in Ref. [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF], involving an integral over the spatial variation of the vector potential in the corresponding part of the conductor, lead or quantum dot. The result depends on the geometry of the device and there is no general argument to favor λ over β α (or the opposite) in the general case.

electronic charge susceptibility

General relations obeyed by transport quantities

According to Eq. 2.40, the weak coupling between the cavity and the electronic system results in the modification of the transmission coefficient τ in the linear response regime [START_REF] Dmytruk | Out-of-equilibrium quantum dot coupled to a microwave cavity[END_REF] (see chapter 2 for a derivation of τ). In the time domain, the correlation function Π(ω) reads

Π(t -t ) = -iθ(t -t ) (λn d + β L n L + β R n R ) (t) , (λn d + β L n L + β R n R ) t , ( 4.4)
which corresponds to the total charge susceptibility of the electronic system capacitively coupled to the cavity, where n d = d † d is the number operator for electrons in the dot and

n α = k c † kα c
kα is the number operator of electrons in the lead α. Using the fact that total charge is conserved,

n d + n L + n R = N, with N proportional
to the total charge operator commuting with H sys , the susceptibility in Eq.4.4 can be rewritten as

Π t -t = (β L -λ) 2 Π LL t -t + (β R -λ) 2 Π RR t -t + (β L -λ) (β R -λ) Π LR t -t + Π RL t -t , (4.5)
where we introduced

Π αβ t -t = -iθ(t -t ) n α (t) , n β t . (4.6)
Using the equation of motion techniques, we can write

i∂ t Π αβ t -t = δ(t -t ) n α (t) , n β t + θ(t -t ) I α (t) , n β t /e, (4.7)
where

I α = e dn α (t) dt (4.8)
is the current operator. Let us introduce the correlation function

Y αβ (t, t ) = -iθ(t -t ) [I α (t), n β (t )] , (4.9)
which may be interpreted as an admittance or as a generalized non-equilibrium differential conductance (see e.g. [START_REF] Safi | Time-dependent theory of nonlinear response and current fluctuations[END_REF] with an opposite sign in the definition). Note that Y depends on the voltage bias V although it is not immediately visible from Eq.4.9. Using Eq.4.7 and Eq.4.9 we find the following relations between the susceptibility and the generalized differential conductance in Fourier space :

Π αα (ω) = i eω Y αα (ω), (4.10)
Π LR (ω) + Π RL (ω) = i eω [Y LR (ω) + Y RL (ω)] . (4.11)
From Eq.4.10 and Eq.4.11, we can relate Π(ω) defined in Eq.4.4 to a linear combination of the admittances. Such linear combination may be simplified in some limiting case as follows. For example, let us assume that there is no charge fluctuation in the quantum dot so that n d (t) ≈ const in the cavity frequency range of interest. In this limit,

Π(ω) ≈ i eω (β L -β R ) 2 Y(ω), (4.12)
where

Y = Y LL = Y RR = -Y LR = -Y RL . Π(ω)
is non-zero only for asymmetric couplings to the leads, β L = β R . The prediction of Eq. 4.12 is quite intuitive: the cavity electric field shakes the electrons in the reservoirs asymmetrically and therefore probes the admittance of the whole system. Eq. 4.12 also implies that the imaginary part of Π(ω) measured via the cavity resonance broadening is proportional to the real part of the admittance, which, at low frequency, reduces to the differential conductance. Thus the charge freez-ing limit n d (t) ≈ const is a particular case where the optical transmission coefficient directly measures the non-equilibrium admittance of the system. Charge freezing can be obtained by replacing the quantum dot by an insulating barrier, by tuning the dot level far off-resonance, or by considering an interacting quantum dot in the Kondo regime as in Ref. [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF]. We discuss this situation at the end of section 4.4.

Introducing the non-symmetrized noise at finite frequency at

C αβ (ω) = ∞ -∞ dt exp(iωt) I α (t)I β (0) , (4.13)
the admittance Y αα (ω) can be related to the current-current correlator [START_REF] Safi | Time-dependent theory of nonlinear response and current fluctuations[END_REF] through

C αα (ω) -C αα (-ω) = -2ωeY αα (ω). (4.14)
The first term on the left hand side of Eq. 4.14 corresponds to the emission noise and the second term corresponds to the absorption noise. Hence Eq. 4.14 directly relates the increase in the cavity resonance broadening to the balance between the emission and absorption noise resulting from photon exchange between the quantum conductor and the cavity [START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF].

Calculation of the electronic susceptibility

In order to calculate the components of the total charge susceptibility introduced in Eq. 4.6, we use the scattering formalism that is well suited for such non-interacting problems. In this formalism, the current operator I α (t) [START_REF] Ya | Shot noise in mesoscopic conductors[END_REF][START_REF] Rothstein | Noise spectra of a biased quantum dot[END_REF] can be written as

I α (t) = e 2π ∞ -∞ dE ∞ -∞ dE exp i(E -E )t × γγ A γγ (α, E, E )a † γ (E)a γ (E ), (4.15)
where

A γγ (α, E, E ) = δ αγ δ αγ -S * αγ (E)S αγ (E ). (4.16)
In Eq. 4.16, S αγ (E) are the elements of the scattering matrix characterizing the system (tunnel junction, quantum dot, etc.). The current operator I α is expressed in terms of annihilation (creation) a α (E) (a † α (E)) operators of the electrons in the reservoir connected to terminal α. These operators are normalized so that

a † α (E)a α (E ) = δ αα δ(E -E )f α (E), (4.17)
where

f α (E) = [exp(E -µ α )/k B T + 1] -1
denotes the Fermi function in the lead α and µ α is its chemical potential.

The Fourier transform of I α (t) reads:

I α (ω) = e ∞ -∞ dE γγ A γγ (α, E, E + ω)a † γ (E)a γ (E + ω). (4.18)
Using

n α (t) = i 2πe ∞ -∞ dω exp(-iωt) I α (ω) ω (4.19)
and Eq. 4.18, we can express the number operator as

n α (t) = i 2π ∞ -∞ dω exp(-iωt) 1 ω (4.20) × ∞ -∞ dE γγ A γγ (α, E, E + ω)a † γ (E)a γ (E + ω).
Performing manipulations detailed in the appendix B.1, we can rewrite Π αβ as

Π αβ (ω) = 1 2π 2 ∞ -∞ ∞ -∞ dω 2 dE 2 1 ω 2 2 1 ω + ω 2 + iη × γ 1 ,γ 1 F αβ γ 1 γ 1 (E 2 , ω 2 ) f γ 1 (E 2 + ω 2 ) -f γ 1 (E 2 ) , (4.21)
where we introduced a new function

F αβ γγ = A γγ (α, E + ω, E)A γ γ (β, E, E + ω). (4.22)
At zero temperature (T = 0), this expression can be simplified further. Introducing the integrals

K αβ (ω 2 ) = eV/2-ω 2 eV/2 dE 2 F αβ LL (E 2 , ω 2 ) + -eV/2-ω 2 -eV/2 dE 2 F αβ RR (E 2 , ω 2 ) + eV/2-ω 2 -eV/2 dE 2 F αβ LR (E 2 , ω 2 ) + -eV/2-ω 2 eV/2 dE 2 F αβ RL (E 2 , ω 2 ), (4.23)
the real and imaginary part of the susceptibility can be written as

Π αβ (ω) = 1 2π 2 P ∞ -∞ dω 2 1 ω 2 2 1 ω + ω 2 K αβ (ω 2 ), (4.24)
Π αβ (ω) = - π (2πω) 2 K αβ (-ω). (4.25)
In order to compute the susceptibility Π αβ for a non-interacting scatterer in question, we first need to compute the functions F αβ γγ in Eq. 4.22, and then the functions K αβ in Eq. (4.23). In the next section, we present these computations for a simpler system -the tunnel junction, -and we analyze a more complex quantum dot system in section .

tunnel junction

Cavity pull and dissipation

The Hamiltonian that combines the resonating cavity with the tunnel junction has the form of Eq. 4.1 with

H el = kα kα c † kα c kα + kq tc † kR c qL + h.c. (4.26)
and

H el-c = (a + a † ) kα β α c † kα c kα , (4.27
Π LL (ω) = Π RR (ω) = -Π LR (ω) = -Π RL (ω) (4.28)
so that the total electronic susceptibility in Eq. 2.40 reads

Π(ω) = α,γ β α β γ Π αγ (ω) = (β L -β R ) 2 Π LL (ω), (4.29)
as in Eq. 4.12. The dependence on the coupling difference β L -β R may be also viewed as a consequence of gauge invariance as discussed in section. 4.3.2.

Using the scattering formalism of section 4.2, we can compute the electronic susceptibility for the tunnel junction. The scattering matrix is energy independent and takes the form

S =   ir t t ir   , (4.30)
where unitarity implies r 2 + t 2 = 1 (we choose real parameters for simplicity). The functions introduced in Eq. 4.22 are then expressed as

F αβ LL = F αβ RR = ε αβ T 2 and F αβ LR = F αβ RL = ε αβ T (1 -T ), where T = t 2 is the tunnel junction transmission, ε αβ = 1 if α = β and -1
otherwise. Inserting these expressions in Eq. 4.23, we obtain

K αβ (ω 2 ) = -2T ω 2 ε αβ . (4.31)
Even at this level one may observe that the photonic transmission is not affected by the bias voltage V. This is due to the linearity in the I-V characteristic of the tunnel junction.

Integrating Eq. 4.24

P ∞ -∞ dω 2 ω 2 (ω + ω 2 ) = 0, (4.32)
we find that the real part of the susceptibility becomes zero, Π αβ = 0, hence the absence of the cavity pull. This result is in agreement with Ref. [START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF] where a different gauge was used, corresponding to the Eq. 4.35 derived in the next subsection. Therefore, any shift in the cavity resonant frequency is due to either a weak energy dependence in the electronic transmission or to higher order effects (backaction) in the electron-photon coupling. The imaginary part is obtained from Eq. 4.25 and reads Π αβ (ω) = -(T/2πω) ε αβ recovering the current conservation in Eq. 4.28. Finally, the total susceptibility of the tunnel junction

Π(ω) = -i(β L -β R ) 2 T 2πω (4.33)
corresponds to a purely dissipative effect of the electrons on the photonic transmission.

This expression can be also obtained by computing the admittance Y(ω) and using Eq. 4.12.

For the tunnel junction, the admittance Y(ω) does not depend on neither the frequency ω nor the voltage V. It is equal to the linear conductance G = ∂I ∂V .

Gauge invariance

Different choices of gauge are possible to describe the same system. They are related to each other via unitary transformations. In order to relate to the previous work, we discuss below two other gauges.

The first unitary transformation is given by the displacement operator

U 1 = e (β R /ω c )n(a † -a) ,
shifting the photon field a → a -(β R /ω c )n. The operator U 1 does not affect the dynamics of electrons since the total number of electrons n = n R + n L commutes with the Hamiltonian, hence U 1 H el U † 1 = H el . The operator U 1 substracts β R from the capacitive couplings thereby canceling the coupling to the right lead. More specifically, the transformed Hamiltonian is given by

U 1 (H ph + H el-c )U † 1 = ω c a † a + (a + a † )(β L -β R )n L - β R β L ω c n 2 - β R (β L -β R ) ω c n(n L -n R ). (4.34)
The first two terms describe the single mode resonator and its coupling to the left lead.

The second term particularly emphasizes the sensitivity of the photon transmission to the capacitive inhomogeneity β L -β R . Since n commutes with the total Hamiltonian, the third term is a constant and the last term describes an additional small bias voltage. The result of Eq. 4.33 may be obtained from Eq. 4.34 straightforwardly.

The second unitary transformation removes the linear coupling between the cavity and the electrons, i.e. Eq. 4.27, by dressing the tunneling amplitude between the two leads.

This transformation with the unitary operator U 2 = e A(a † -a) , with

ω c A = β L n L + β R n R ,
transforms the Hamiltonian into

H = U 2 HU † 2 = kq te (β R -β L )(a † -a)/ω c c † kR c qL + h.c. + kα kα c † kα c kα + ω c a † a - A 2 ω c . (4.35)
The last term in this expression A 2 /ω c describes capacitive electron-electron interaction across the tunnel junction and is usually neglected.

We remark that the different gauges correspond to shifted positions of the resonator vacuum. A vacuum in one gauge is different from the vacuum in another gauge.

quantum dot

Let us determine the charge susceptibility probed by the microwave cavity of the QD tunnel coupled to metallic leads and capacitively coupled to the microwave cavity. The scattering matrix for the single-level quantum dot reads

S(E) = -1 + ig(E)   Γ L √ Γ L Γ R √ Γ L Γ R Γ R   , (4.36)
where

g(E) = 1 E -d + iΓ/2 , ( 4.37)
and where simple expressions and conclusions can be obtained before addressing the general case.

Γ = Γ L + Γ R is

The Quantum RC-circuit limit

For the two following limits (or sets of parameters) our system reduces to the so-called quantum RC circuit, which has been largely studied theoretically [START_REF] Büttiker | Mesoscopic capacitors[END_REF][START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF][START_REF] Filippone | Giant charge relaxation resistance in the Anderson model[END_REF][START_REF] Lee | Effect of many-body correlations on mesoscopic charge relaxation[END_REF][START_REF] Mora | Universal resistances of the quantum resistance-capacitance circuit[END_REF][START_REF] Nigg | Mesoscopic charge relaxation[END_REF] as well as implemented and verified experimentally [START_REF] Bruhat | Cavity photons as a probe for charge relaxation resistance and photon emission in a quantum dot coupled to normal and superconducting continua[END_REF][START_REF] Fève | An on-demand coherent single-electron source[END_REF][START_REF] Gabelli | Violation of Kirchhoff's laws for a coherent RC circuit[END_REF].

Case Γ R = 0
This case corresponds to a quantum dot connected to a single reservoir by a junction. The quantum RC circuit is particularly interesting in the small frequency limit when some universal behavior is expected [START_REF] Mora | Universal resistances of the quantum resistance-capacitance circuit[END_REF][START_REF] Nigg | Mesoscopic charge relaxation[END_REF] and was observed [START_REF] Bruhat | Cavity photons as a probe for charge relaxation resistance and photon emission in a quantum dot coupled to normal and superconducting continua[END_REF].

When ω is smaller than all other energy scales at play, we can further simplify the expression of the susceptibility Π(ω) to obtain

Π (ω) = - 2Γ L (β L -λ) 2 π Γ 2 L + 4 2 d + O(ω 2 ) , (4.38)
for the real part, and

Π (ω) = - 4Γ 2 L ω(β L -λ) 2 π Γ 2 L + 4 2 d 2 + O(ω 3 ) , (4.39)
for the imaginary part. Note that these expressions correspond exactly to ∆ω a 0 and ∆Λ 0 in Ref. [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF] with Γ = Γ L /2 [up to a minus sign which can be traced back to our different definition of Π(t) in Eq. 4.4].

The real and imaginary contributions, Π and Π , respectively, are not independent and fulfill at the lowest order in ω the following relation

Π (ω) = -πω Π (0) 2 /(λ -β L ) 2 . (4.40)
Up to the electron-photon couplings, this is nothing but the Korringa-Shiba relation connecting the dissipative part of the susceptibility to its reactive part. Note that we defined the correlation function Π(t) in Eq. 4.4 with a different sign compared to, e.g., Ref. [START_REF] Mora | Universal resistances of the quantum resistance-capacitance circuit[END_REF],

and thus introduced the opposite sign in the Korringa-Shiba relation. Note that the use of a microwave cavity for verifying the Korringa-Shiba relation has first been proposed in Ref. [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF].

Case β L = β R
For equal capacitive coupling to the leads (β L = β R ), the cavity electric field effectively couples to a single larger lead as well as to the quantum dot. We thus expect to observe the same physics as in the previous case. Again, we assume ω to be smaller than the other energy scales in the system.

In this low-frequency limit, the real part of the total susceptibility Π reads

Π (ω) = - 2 (λ -β L ) 2 π Γ L (eV -2 d ) 2 + Γ 2 + Γ R (eV + 2 d ) 2 + Γ 2 + O(ω 2 ) , (4.41)
while the imaginary part reads

Π (ω) = - 4Γ ω (λ -β L ) 2 π    Γ L (eV -2 d ) 2 + Γ 2 2 + Γ R (eV + 2 d ) 2 + Γ 2 2    + O(ω 3 ). (4.42)
For zero bias voltage eV = 0, one recovers the Korringa-Shiba relation Eq. 4.40. Indeed, the two leads combine and the quantum dot effectively couples to a single lead.

In the latter case (β L = β R ), given that eV = 0, as well as in the former case (Γ R = 0, eV = 0), the cavity measures only local charge fluctuations on the dot -leading to the dissipation channel fulfilling the Korringa-Shiba relation, -but it is not sensitive to the charge transfer across the quantum dot.

4.4.2 The tunneling limit, d /Γ → -∞
The transmission coefficient for the quantum dot, T d , in the tunneling limit ( d → -∞)

reads

T d = Γ L Γ R 2 d . ( 4.43)
Using this equation, we can immediately evaluate the real and imaginary parts of Π:

Π (ω) = - T d 2π (β L -λ) 2 Γ R + (β R -λ) 2 Γ L + (β L -λ)(β R -λ) 1 Γ L + 1 Γ R T d , (4.44)
Π (ω) = - T d 2πω (β L -β R ) 2 . (4.45)
The imaginary part of the susceptibility of the QD, related to δA/A, Eq. 4.45, is equal to the susceptibility of the tunnel junction in Eq. 4.33. The real part of the susceptibility of the QD, related to the phase shift δφ, is not zero as opposed to the tunnel junction result.

The non-zero phase shift for the QD is explained by the existence of a localized level inside the barrier which is coupled virtually to the continuum of levels in the leads by the cavity field. On the other hand, such level is absent in a pure tunnel junction resulting in the zero real part of the susceptibility.

The low-frequency limit

Let us explore in detail the low-frequency regime when the cavity frequency is typically smaller than the electronic quantum dot resonance width, ω Γ . This regime is studied experimentally in [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF].

In this limit, one can take a series in ω/Γ of both the real and imaginary parts of the charge susceptibility:

Π αβ (ω) = Π αβ (0) + O(ω 2 ), (4.46)
Π αβ (ω) = 1 πω Λ (-1)
αβ + ωΛ

(1)

αβ + O(ω 3 ). (4.47)
We obtain the following expressions for the lowest order terms.

Π LL,RR (0) = - 2Γ L,R (eV ∓ 2 d ) 2 ± (Γ 2 L -Γ 2 R ) π (eV ∓ 2 d ) 2 + Γ 2 2 , ( 4.48)
Π LR (0) + Π RL (0) = - 4Γ Γ L Γ R π ×    1 (eV -2 d ) 2 + Γ 2 2 + 1 (eV + 2 d ) 2 + Γ 2 2    . (4.49)
And similarly

Λ (-1)
LL,RR = -

2Γ L Γ R (eV ∓ 2 d ) 2 + Γ 2 , ( 4.50)
Λ (-1)

LR + Λ (-1) RL = 2Γ L Γ R × × 1 (eV -2 d ) 2 + Γ 2 + 1 (eV + 2 d ) 2 + Γ 2 . (4.51)
The expressions for Λ

(1)

αβ read:

Λ (1) LL,RR = - 4Γ L,R 3π (eV ∓ 2 d ) 2 + Γ 2 3 × (3Γ L,R -2Γ R,L )Γ 2 + 3(Γ + Γ R,L )(eV ∓ 2 d ) 2 , (4.52) and Λ
(1)

LR + Λ (1) RL = - 4Γ L Γ R 3π    5Γ 2 -3(eV -2 d ) 2 (eV -2 d ) 2 + Γ 2 3 + 5Γ 2 -3(eV + 2 d ) 2 (eV + 2 d ) 2 + Γ 2 3    . ( 4.53)
Analyzing these equations, one may observe that both the real and imaginary parts of the susceptibility Π αβ (ω) have peaks at d = ±eV/2. Noticing that Λ (-1)

LL + Λ (-1) RR + Λ (-1) LR + Λ (-1)
RL = 0, we can further simplify the expression for Π (ω) as

Π (ω) = (β L -β R ) πω (β L -λ)Λ (-1) LL -(β R -λ)Λ (-1) RR + ω (β L -λ) 2 Λ (1) LL + (β R -λ) 2 Λ (1) RR + (β L -λ)(β R -λ)(Λ (1)
LR + Λ

(1)

RL ) . (4.54) Let us discuss both β L = β R and β L = β R cases. Case β L = β R
For β L = β R , we find again the effective quantum RC circuit case discussed above. In particular, it implies that at V = 0,

δφ(V = 0) ≈ - 2Γ (β L -λ) 2 πκ Γ 2 + 4 2 d , (4.55)
which corresponds to a Lorenztian shape as a function of the dot gate voltage ∝ d .

In order to analyze the general non-equilibrium V = 0 case, we plot in Fig. 4.2 both Π = Π /(β L -λ) 2 and Π = Π /(β L -λ) 2 as functions of d and eV. Both the real and imaginary parts of Π have peaks around eV = 0. For eV larger than Γ , the peaks split, and resonances are found for eV/2 = ± d . For β L = β R , the Korringa-Shiba relations are satisfied at low energy for V = 0. This implies that Π ∝ (Π ) 2 and therefore the resonances in Π are more narrow at eV = 0 than the resonances of Π .

Let us compare these results to the non-equilibrium differential conductance G(V) = dI/dV across the quantum dot that reads The differential conductance has a Lorentzian shape double peaked at eV/2 = ± d . Noticing that, for Γ L = Γ R , the expression of Π (ω, V) given in Eq. 4.41 is directly proportional to G(V). We thus expect that

G(V) = e 2 2π 2Γ L Γ R (eV -2 d ) 2 + Γ 2 + 2Γ L Γ R (eV + 2 d ) 2 + Γ 2 . ( 4
δφ(V) = - 4(β L -λ) 2 κe 2 Γ G(V), (4.57)
for

Γ L = Γ R = Γ/2.
Therefore, for both symmetric tunneling amplitudes and lead capacitive couplings, Eq. 4.57 directly relates the optical phase shift to the non-equilibrium conductance.

For small deviations from the equalities Γ L = Γ R and β L = β R , we expect the relation Eq. 4.57 to hold approximately. For strong asymmetric tunneling, even though the expressions for Π and G(V) are no longer proportional within each other, these two quantities still have a similar shape characterized by Lorentzian peaks around eV/2 = ± d .

Such quantitative comparison cannot be made between Π (and therefore δA/A) and G(V) for β L = β R because the term proportional to 1/ω in the series for Π (ω) is exactly zero. However, small deviations of the stringent condition β L = β R may allow comparing these quantities as we show next.

Case

β L = β R
In the more general case β L = β R , the expression for Π simplifies for eV = 0 as

Π (ω, V = 0) ≈ - (β L -β R ) 2 2πω 4Γ L Γ R 4 2 d + Γ 2 = - (β L -β R ) 2 2πω T d . (4.58)
In this limit, Π (ω) has the same structure as for the tunnel junction [see Eq. 4.33]. Hence the frequency broadening

δA A (V = 0) ≈ - (β L -β R ) 2 2πωκ T d . (4.59)
Therefore, the frequency broadening follows the conductance as a function of the dot gate voltage. The 1/ω term in the frequency series for Π dominates even for a very small asymmetry of the capacitive couplings. Note that, for β L ≈ β R , the phase shift is still given approximately by Eq. 4.55 and still has a Lorentzian shape. These equilibrium results (derived for V = 0) agree with experimental results obtained in Ref. [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF], where the cavity frequency is the smallest energy scale, and a small asymmetry between the capacitive couplings is present.

In order to explore how the susceptibility depends on the bias and dot gate voltage, we

introduce γ = (β L -λ) / (β R -λ) such that Π(ω) (β L -λ) (β R -λ) =γΠ LL (ω) + Π RR (ω) γ + Π LR (ω) + Π RL (ω).
We remind that for |γ -1| 1, and Γ L ∼ Γ R , the out-of-equilibrium relation in Eq. 4.57

holds. However, no such relation is present between δA/A and G(V) at finite bias.

We plotted both the real and imaginary parts of

Π/(β L -λ)(β R -λ) in

A quantum dot in the deep Kondo regime

As mentioned in section 4.2, when the quantum conductor is far from any resonance, the susceptibility and the admittance are related by Eq. 4.12. This relation holds for a non-equilibrium case and does incorporate the bias V. Since ω is usually fixed by the cavity frequency ω c , Π (V) is proportional to the non-equilibrium differential conductance Y (V) which is characterized by a Kondo peak around zero bias. Therefore, we expect the dissipative part of the transmission phase shift to be sensitive to this zero bias anomaly.

However, in the experimental data of Ref. [START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF], the phase shift δφ(V) ∝ Π (V) was surprisingly shown to exhibit a behavior similar to the differential conductance.

summary of the results

We studied a hybrid system consisting of either a tunnel junction or a quantum dot coupled to metallic leads while the whole system itself is coupled to a single mode of a microwave cavity field. We took into account the capacitive coupling between the cavity and the whole electronic system, including the leads. Even for non-interacting conductors, many energy scales characterize the system, for example the cavity frequency, the voltage bias, the position and width of a resonance and finally the asymmetry of the capacitive couplings.

1. For a tunnel junction with asymmetric capacitive coupling β L = β R , the imaginary part of the electronic susceptibility Π(ω) is proportional to the admittance Y(ω)

which is equal to the linear conductance dI/dV. Similarly, for a quantum dot far from resonance, therefore characterized by the absence of charge fluctuations, we found that the electronic susceptibility is directly proportional to the quantum dot admittance for all frequency and voltage. The proportionality constant is related to the asymmetry between the lead and cavity capacitive couplings.

2. The case of symmetric capacitive couplings to the leads, β L = β R , can be absorbed into a redefinition of the capacitive coupling to the quantum dot.

3.

At zero voltage and small frequency, we recover the Korringa-Shiba relation that relates the real and imaginary part of the electronic susceptibility, similarly to the case of a quantum dot connected to a single reservoir (the so-called quantum RCcircuit).

4.

In the case of symmetric tunneling amplitudes, Γ L = Γ R , we found that the optical phase shift is directly proportional to the differential conductance of the quantum dot for arbitrary bias. This strongly symmetric case thus provides a weakly invasive measurement of the non-equilibrium differential conductance, extracted from the cavity field response. Nevertheless, the lead capacitive couplings have no reason to be equal in general since they depend on the particular geometry of the conductor.

They are also hardly tunable.

5.

Finally, at zero voltage, we find that the imaginary part of the electronic susceptibility is proportional to the equilibrium zero-frequency admittance.

6.

Beyond the above mentioned limits however, the charge susceptibility and the admittance are generally not in correspondance.

conclusions

In this chapter, we studied an electronic system consisting of either a tunnel junction or a quantum dot with metallic leads capacitively coupled to a single mode of electromagnetic resonator. We considered and detailed different cases and our results are summarized in section 4.5. We demonstrated that the asymmetry of the capacitive couplings between the electronic reservoirs and the cavity plays actually a crucial role by rendering the cavity electric field sensitive to charge transfers between two different leads. We found that the cavity field can be used to probe the low finite frequency admittance Y αβ (where α, β are lead indices) of the electronic system. Beyond the low frequency regime, the charge susceptibility and therefore the optical transmission in general offers new insights on the quantum conductor. Since the optical observables are not in direct correspondence with standard transport quantities, they can be used as a non-invasive probe to better characterize the quantum conductor.

M A J O R A N A F E R M I O N S I N T O P O L O G I C A L S U P E R C O N D U C T O R S C O U P L E D T O A M I C R O WAV E C AV I T Y
In this chapter, we apply the methods of cavity QED to study Majorana fermions in topological superconductors. While most of the experimental efforts are targeted at probing these quasiparticles using electronic transport measurements, we propose instead to detect the Majorana fermions by probing the light exiting from the cavity. As shown in the chapter , optical transmission coefficient that is measured in the cavity QED experiments is related to the electronic susceptibility of the topological superconductor. We analyze two superconducting systems: the prototypical Kitaev chain, and a topological semiconducting wire. For both systems, we show that the electronic susceptibility allows us to determine the topological phase transition point, the emergence of the Majorana fermions, and the parity of their ground state.

model hamiltonian

We study a topological superconductor capacitively coupled to a microwave cavity. The sketch of the system is presented in Fig. 7.1.

In the following, we consider various models of topological superconductors coupled to a superconducting microwave cavity, such as the Kitaev model [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF], and the spin-orbit coupled nanowire subject to a magnetic field and in the proximity of an s-wave superconductor [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF].

The general Hamiltonian for the one-dimensional systems we consider here is of the form:

H sys = H el + H el-ph + H ph , (5.1)
A in e iφ in A out e iφ out Ê(x) where H el is the electronic Hamiltonian that describes p-wave superconductor, H el-ph is electron-photon coupling Hamiltonian that describes capacitive coupling of the p-wave superconductor to the cavity, and H ph is the free photon field Hamiltonian. While the electronic term is model specific, and it will be discussed below, the electron-photon coupling

Hamiltonian reads:

H el-ph = α N i=1 ni (a + a † ), (5.2)
where a † (a) is the photon creation (annihilation) operator, respectively. α is the electronphoton coupling constant that couples to the charge density n. This merely acts as to shift the chemical potential. We have a considered a global capacitive coupling between the electronic nanosystem and the cavity electric field. Such a coupling can be justified by a full microscopic approach (see Appendix C.1 for details and also [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF] that provides a microscopic description of the electric coupling between electrons in a nanocircuit and cavity photons).

The charge susceptibility of the electronic system (which can be here a 1D p-wave superconductor or a topological 1D wire) reads

Π(t -t ) = -iα 2 θ(t -t ) [ nI (t), nI (t )] , (5.3)
where nI (t) = U † (t) nU(t), with n being the total number of electrons operator and U(t) = exp (-iH el t) the evolution operator for the electronic system. We assume zero temperature limit (T = 0) so that the average . . . is taken over the superconducting ground state. Note that

Π(ω) = ∞ -∞ dt exp (iωt)Π(t)
and that Π(ω) ≡ 0 in the absence of superconductivity (∆ = 0), i.e. there are no effects from such a coupling for a wire in the normal state. We detail below the models for both topological 1D systems we consider in this chapter.

topological phase transition detection

Next we show that the topological phase transition can be inferred from the cavity response from the transmission τ(ω) via the susceptibility Π(ω). This function can be calculated straightforwardly in the case of a closed ring, i.e. for periodic boundary conditions (PBCs), so that c N+1 ≡ c 1 for the Kitaev chain (c N+1σ ≡ c 1σ for the SO nanowire).

Kitaev chain model

The simplest model of a p-wave superconductor that hosts Majorana fermions is the Kitaev chain model [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF], which we have discussed in chapter 3. Therefore, we first consider for the electronic part in Eq. (5.1), the Kitaev Hamiltonian H K el . As a reminder, it is expressed as

H K el = -µ N j=1 c † j c j - 1 2 N-1 j=1 (t c † j c j+1 + ∆ c j c j+1 + h.c.) , (5.4)
where t is the hopping parameter, ∆ is the p-wave superconducting pairing potential, µ is the chemical potential, and N is the total number of sites. Also, c † j (c j ) is the creation (annihilation) electronic operator at the site j. Note that the electronic operators are spinless, and the electronic density is given by n = N j=1 c † j c j . In the present setup, which is based on a microwave superconducting stripline cavity, the s-wave superconductor that induces superconducting correlations in the wire could be a part of the underlaying cavity. For example, the nanowire could be tunnel-coupled to the central (super-)conductor showed in Fig. 7.1. The fact that microwave photons effectively couple only to electrons of the Kitaev chain is accounted for in Appendix C.1.

For PBCs, we can utilize the Fourier description for the electronic operators:

c j = 1/ √ N k e ikj c
K el = k H K BdG (k), with H K BdG (k) = (-t cos k -µ) τ k z -∆ sin k τ k y , (5.5)
where

τ k = (τ k x , τ k y , τ k z )
are Pauli matrices that act in the Nambu (particle-hole) space, i.e. on the vectors c k = (c k , c † -k ). The coupling to the cavity, on the other hand, simply reads

H el-c = α k τ k z (a † + a) , (5.6)
so that the susceptibility in the time domain can be written as:

Π(t) = -iα 2 k 0|[τ k z (t), τ k z (0)]|0 , (5.7)
with

τ k z (t) = e iH K BdG (k)t τ k z e -iH K BdG (k)t .
Utilizing this description, after some lengthy but straightforward calculations, we obtain for the susceptibility (in the ω space):

Π(ω) = -α 2 k>0;p=± (∆ sin k) 2 E 2 k p ω + 2pE k + iη , ( 5.8)
where

E k = (-t cos k -µ) 2 + (∆ sin k) 2
is the Bogoliubov spectrum of the 1D p-wave superconductor [from diagonalizing the BdG Hamiltonian in Eq. (5.5)] [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF] and η is a small positive number that accounts for causality. For t = ∆, the imaginary part Π (ω)

acquires a simple analytical form, and it is given by

Π (ω) = α 2 tN 2µω 1 - (ω/2) 2 -t 2 -µ 2 2 4t 2 µ 2 , ( 5.9)
for |t + µ| < ω/2 < |t -µ| and is zero otherwise. The topological phase transition takes place at |µ| = t, with the system being in the topological (trivial) phase for |µ| < t (|µ| > t).

In Fig. 5.2, we plot Π (ω) (main plot) and Π (ω) (inset) as a function of the chemical potential µ for various values of the cavity frequency ω. We see that this function shows a large peak at the transition point (|µ| = t), which becomes narrower and more pronounced for smaller ω (compared to the gap ∆). Physically, this is due to the fact that the electronic levels close to the zero energy have larger curvatures, i.e. they are more susceptible close to the phase transition point. The real part also serves for detecting the phase transition, although not as directly as the imaginary part, as shown in Fig. 5.2, where the phase transitions are inferred from the kinks in this function. We have checked that the same peak structure holds for the cases when ∆ = t, too, the only modification being a shift in the scale for ω, which should be of the order of ω ∼ ∆.

Spin-orbit coupled nanowire model

A realistic system that can emulate, in some limits, the Kitaev chain consists of a nanowire with a spin-orbit interaction, subjected to an external magnetic field, and coupled by proximity effect to an s-wave superconductor [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF]. The entire system is then assumed for the nanowire with spin-orbit (SO) interaction in the presence of the magnetic field reads [START_REF] Rainis | Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions[END_REF][START_REF] Stoudenmire | Interaction effects in topological superconducting wires supporting Majorana fermions[END_REF]]

H W el = -t N-1 j=1 c † j+1α δ αβ c jβ -µ N j=1 c † jα δ αβ c jβ + ∆ N j=1 c † j↑ c † j↓ -iγ N-1 j=1 c † j+1α σ y αβ c jβ -V Z N j=1 c † jα σ x αβ c jβ + h.c. , (5.10)
where t and µ are the hopping amplitude and the chemical potential, respectively, γ is the spin-flip hopping amplitude (or the spin-orbit coupling), ∆ is the s-wave pairing potential induced by proximity, V Z is the Zeeman splitting energy (V Z = -gµ B B/2, with g and B being the g-factor and external magnetic fields, respectively). Also, c jσ (c † jσ ) are the annihilation (creation) operators for electrons at site j and spin σ =↑, ↓, and σ i , with i = x, y, z are the Pauli matrices that act in the spin space. This model accounts thus for spinful electrons. Note that we assumed the spin-orbit field and the magnetic field to be orthogonal. The coupling to the cavity is again capacitive, and the density reads in this case ni = σ c † iσ c iσ . The case of a realistic SO coupled nanowire is more complicated that the Kitaev model showed above, and so is the evaluation of susceptibility. This is so because the SO coupled wire has four bands (because of the spin), instead of two, and a more complicated spectrum. Nevertheless, writing the electronic operators in the Fourier space as c jσ = 1/ √ N k e ikj c kσ , we can write again the electronic Hamiltonian as

H W el = k H W BdG (k), with H W BdG = [(-t cos k -µ) + γ sin k σ z ]τ k z + V Z σ x + ∆τ k x , (5.11)
and the coupling to the cavity the same as in Eq. (5.6). However, the expression for Π(t)

becomes rather cumbersome for the general case and to get some analytical insights we need to resort to approximations. For that, the Hamiltonian can be put in a different form by the use of a unitary transformation (see Appendix C.3):

H W BdG (k) = -t cos k -µ + (γ sin k) 2 + V 2 Z σ z τ k z + ∆γ sin k (γ sin k) 2 + V 2 Z τ k x - ∆V Z (γ sin k) 2 + V 2 Z σ y τ k y , (5.12)
while the H el-ph stays unchanged. Progress can be made if we assume the limit of large magnetic field, V Z ∆, µ, in which case we can neglect the last term in the above Hamiltonian. By doing so, we recover two copies of the Kitaev chain, for σ z =↑, ↓. The susceptibility becomes:

Π(ω) = -α 2 k>0;p,σ=± (∆ eff sin k) 2 E 2 kσ p ω + 2pE kσ + iη , ( 5.13)
where E kσ are given by the Kitaev spectrum with:

µ eff kσ = µ -(γ sin k) 2 + V 2 Z σ z , (5.14)
∆ eff = ∆γ (γ sin k) 2 + V 2 Z .
(5.15)

All the results from the previous section apply to this case but with the k-dependent parameters showed above. The system is in the topological nontrivial (trivial) regime

for V Z > µ 2 + ∆ 2 (V Z < µ 2 + ∆ 2
). In Fig. 5.3 we plot the imaginary part of the susceptibility as a function of the Zeeman splitting V Z for two different values of the cavity frequency ω c . We see a similar behavior as in the case of the Kitaev chain: a peak emerges in Π (ω) at the topological phase transition point, which becomes narrower as ω becomes smaller. However, an extra peak emerges at a larger V Z , and it is due to the resonance condition with the gaps around the k ∼ k F (external gaps in the SO coupled nanowire spectrum).

majorana fermions detection

In this section, we consider a finite 1D topological system coupled to the cavity (therefore with open boundary conditions, or OBCs), so that there are two Majorana fermions emerging in the topological region, each localized at one of the two ends of the chain.

Taken together, they give rise to a zero-energy fermionic state in the infinite wire limit, which can be either empty or occupied, thus labeling the parity of a 1D p-wave superconductor [START_REF] Alicea | Non-Abelian statistics and topological quantum information processing in 1D wire networks[END_REF]. The Majorana wavefunctions decay exponentially in the wire on the scale of the superconducting correlation length ξ, and for a finite wire it can lead to a finite energy splitting M ∝ exp (-L/ξ) of the initially zero energy fermionic state [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. In the following, we show that both the presence of the Majorana fermions and the parity of the Majorana fermionic state can be inferred from the susceptibility Π(ω).

In the finite chain case, we can no longer obtain exact results for the electronic susceptibility Π(ω), and therefore we proceed to calculate this quantity numerically (see Appendix C.4). We treat the two models, the Kitaev chain and the SO coupled wire, on equal footing, showing that they give similar results.

For starters, the electronic Hamiltonian can be cast in the following form:

H el = 1 2 c † M c , (5.16
)

with c = ({c 1s } . . . {c Ns }{c † 1s } . . . {c † Ns }) , (5.17)
where s counts internal degrees of freedom, such as spin, band index, etc. For the Kitaev chain s = 1 (and thus we can disregard it), while for the SO coupled nanowire s =↑, ↓.

Here, M is a 2Ns × 2Ns matrix [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF], and we can write M = PWP T , with W 2p-s,2k-s = (-1) s+1 δ p,k k ; s = 0, 1.

(5.18)

P is a unitary matrix (PP T = P T P = 1) whose columns are the eigenvectors of M [START_REF] Yu | Unpaired Majorana fermions in quantum wires[END_REF]. Also, p , with p = 1, . . . , sN are the eigenenergies of the electronic Hamiltonian, including the Majoranas (if present). Thus, the electronic Hamiltonian can be rewritten as

H el = (1/2) C † W C, and C = P † C, where C = ({ c1s } . . . { cNs }{ c † 1s } . . . { c † Ns }) , (5.19)
with c † p ( cp ) are the creation (annihilation) operators for the Bogoliubov quasiparticles in the finite wire, with p = 1 . . . N labeling the energy levels. Finally, we can write

H el = p,s ps c † ps cps - 1 2 , ( 5.20)
and also define the spinorial wavefunction for the state of energy ± ps at position j as ψ ps (j) = (u j ps , v j ps ) T , where u j ps (v j ps ) = P 2j-1,p (P 2j,p ) are the electron (hole) components of the wavefunction at position j in the wire.

The electron-cavity coupling Hamiltonian can then be written in the new basis as follows:

H el-ph = p,p C (1) ps,p s c † ps cp s -iC (2) ps,p s c † ps c † p s + h.c. × (a † + a) , (5.21)
where C

(1,2) ps,p s are coefficients that depend on the transformation from the electronic basis C to the Bogoliubov basis C and read [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF][START_REF] Trif | Resonantly tunable majorana polariton in a microwave cavity[END_REF]:

C (1,2) ps,p s = α N j=1 ψ †
ps (j)τ z,y ψ p s (j) .

(5.22)

Here the pseudo-spin τ = (τ x , τ y , τ z ) acts in the Nambu (or particle-hole) subspace. In general, all C

(1,2) ps,p s = 0 for p = p and s = s , thus there are couplings between all the levels (and bands) via the cavity field, and that includes transitions between the Majorana and the bulk (or gaped) modes. This in turn affects the correlation function in Eq. ( 5.3), which can be written as Π(ω) = Π BB (ω) + Π BM (ω) + Π MM (ω), being the sum of the terms that contain only bulk states (bulk-bulk, or BB), cross terms between Majorana and the bulk (bulk-Majorana or BM), and Majorana contributions only (Majorana-Majorana or MM), respectively. However, Π MM (ω) ≡ 0 [START_REF] Cottet | Squeezing light with Majorana fermions[END_REF] due to the fact that the cavity cannot mix different parities, and consequently the only contribution from the Majorana modes comes through the cross terms Π BM (ω). We have found that for N 1, the Π BB (ω) contribution is given by the one obtained from the PBCs, i.e., Π BB ∝ N, while Π BM ∝ const, up to exponentially small terms in L/ξ.

In the following, we analyze the cross-terms contribution Π BM (ω). For M p ± ω, with p = M, we obtain:

Π BM (ω) = p,s =M 1 ps + ω + iη + 1 ps -ω -iη × |C (1) M,ps | 2 (n M -n ps ) -|C (2)
M,ps | 2 (n M -1 + n ps ) , (5.23)
where n ps and n M are the occupations of the bulk and Majorana states, respectively.

Inspecting the above expression, we see that it is strongly dependent on the Majorana state parity n M . Assuming that ps > 0 for p, s = M and n ps = 0 for n = M in the ground state, we obtain that Π + BM ∝ |C

(1)

M,ps | 2 (Π - BM ∝ |C (2)
M,ps | 2 ) for n M = 1 (n M = 0). To

= 2 (Π + BM -Π - BM )/(Π + BM + Π - BM )
as a function of µ in logarithmic scale, for ∆ = 0.1 (red-full), ∆ = 0.2 (green-dashed), and ∆ = 0.3 (blue-dotdashed). The size of ∆Π is exponentially reduced as a function of ∆. We used N = 50, ω = 0.2, ∆ = 0.1, t = 1, and all energies are expressed in terms of t.

get more physical insight into the resulting susceptibility, we write the coefficients C

(1,2) M,ps in the following way:

C (r) M,ps = j [(u j M δ r,1 + v j M δ r,2 )u j ps -(u j M δ r,2 + v j M δ r,1 )v j ps ] . (5.24)
Let us analyze the implication of the above result. When M = 0, we also have u j M = v j M , and thus C

(1)

Mp = C (2)
Mp , since electron and hole contributions are the same in the Majorana state. However, for a finite energy splitting M = 0, and thus we have that u j M = v j M , which in turn results in C

(1)

Mp = C (2)
Mp . All these suggest that the susceptibility Π(ω), via Π BM (ω) should allow us to infer both the parity of the ground state and the zeros in the Majorana energy M , assuming their spatial overlap is large enough. V z @meVD P'Hw,V z Lêa 2

= 2 (Π + BM -Π - BM )/(Π + BM + Π - BM )
, for different values of ∆. We see that the oscillations have the same periodicity as the Majorana energy splitting M ∼ exp (-L/ξ) cos (k F L). Notice that the oscillations of the Majorana splitting with the magnetic field (chemical potential) has been studied in detail [START_REF] Prada | Transport spectroscopy of N S nanowire junctions with Majorana fermions[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] together with the fact that the magnitude of the oscillations becomes exponentially suppressed in L/ξ [START_REF] Pientka | Signatures of topological phase transitions in mesoscopic superconducting rings[END_REF][START_REF] Zyuzin | Correlations between Majorana fermions through a superconductor[END_REF].

In the main figure in Fig. 5.5, we plot the real part of the susceptibility for a 1D topological wire as a function of the Zeeman splitting V Z for the two parities n M = 0, 1. The susceptibility Π for that figure was computed using realistic parameters that might be appropriate for an InSb wire such as in the experiments in Ref. [START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF]. We find similar features as for the Kitaev toy model, namely oscillations as a function of the Zeeman splitting above the topological transition. These oscillations around the ground state have opposite sign and different amplitudes for each parity. Like for the Kitaev model, they have the same periodicity as the Majorana energy M (see the inset of Fig. 5.5) and cross at points where M = 0. Notice that if the parity is not conserved in the system (for example, due to the quasi-particle poisoning), Π will follow the ground state and exhibit therefore sharp cusps as a function ∆ Z at the crossing points where M = 0 (see also Ref. [START_REF] Väyrynen | Microwave signatures of Majorana states in a topological Josephson junction[END_REF] for similar features in a topological Josephson junction). As for the Kitaev chain, we thus find that the cavity phase shift is thus able to detect the Majorana fermions and the parity of the ground state of a realistic topological wire.

The imaginary part of Π(ω) gives us also information on the presence of Majorana fermions. In Fig. 5.6, we show the dependence of Π (ω) on ω for the Kitaev chain, both in the topological and non-topological regimes, for t = ∆. We see that the Majorana fermions, through Π BM (ω), give rise to an extra peak in the susceptibility at half the effective superconducting gap ∆ eff = ||µ| -t| in the topological regime, while such a peak is absent for the same effective gap ∆ eff , but in the non-topological case. For completeness, we also show the result for PBCs, in which case there are no Majorana fermions. Finally, let us give some estimates for Π(ω), and in particular for Π BM (ω) and the resulting phase shift in the exiting photonic signal. We assume typical experimental values for the cavity frequency, ω c ≈ 2 × 10 -5 eV, and with a quality factor Q ≈ 10 5 , which results in photon escape rate κ = 2 × 10 -10 eV. For an estimate of the capacitive coupling α we refer, for example, to the case of carbon nanotubes, which have been under experimental scrutiny in the context of cavity QED [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF][START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF]. There, it was found that α ≈ 5.6 × 10 -7

eV, and we believe similar values should be relevant for semiconductor nanowires too.

The phase shift of the radiation exiting the cavity satisfies δφ ∝ (α 2 /t κ) so that we obtain δφ ≈ 0.3 which is a sizeable value.

conclusions

In this chapter, we studied two paradigmatic examples of 1D topological superconducting systems capacitively coupled to a microwave superconducting stripline cavity: the Kitaev chain and a 1D nanowire with strong SO interaction in the presence of a magnetic field and in proximity of a superconductor. We analyzed the electronic charge susceptibility of these systems that is revealed in the photonic transport through the microwave cavity via its transmission τ(ω). We demonstrated that this electronic susceptibility can actually be used to detect the topological phase transition, the occurrence of Majorana fermions and the parity of the Majorana fermionic state in a non-invasive fashion. Such effects are due to the interplay between the bulk and Majorana states, either via virtual or real transitions taking place between the two, and which are mediated by the photonic field.

D E T E C T I O N O F M U LT I P L E M A J O R A N A F E R M I O N S C O U P L E D T O A M I C R O WAV E C AV I T Y
In this chapter, we discuss the case of multiple Majorana fermions emerging at each edge of the wire. This situation becomes possible if the system has not only the nearestneighbor hopping, but also the next-nearest-neighbor hopping. We discuss what possible signatures of multiple Majorana fermions could be visible in cavity QED experiments. We start with establishing the phase diagram of the extended Kitaev model. We discuss for which set of parameters it is possible to obtain the oscillations of the Majorana energy splitting. Then we demonstrate that the electronic susceptibility can be used to probe the phase transitions from 0 to 1 to 2 Majorana fermions present in the system. Moreover, the frequency response of the cavity is sensitive to the transitions between two lowest-energy Majorana states. Also we show that there are oscillations of the electronic susceptibility as a function of chemical potential similar to the ones of the Majorana energy splitting.

These oscillations have different periodicity depending whether there is one or two Majorana fermions at each end of the chain.

model hamiltonian

Let us consider a model of the topological wire that allows for multiple Majorana fermions emerging in the system. The Kitaev model with the next-nearest-neighbor hopping that can show up to two Majorana fermions emerging at each end of the chain has been introduced in [START_REF] Niu | Majorana zero modes in a quantum Ising chain with longer-ranged interactions[END_REF]. Later the fractional Josephson effect with multiple Majorana fermions has been studied [START_REF] Sticlet | Josephson effect in superconducting wires supporting multiple Majorana edge states[END_REF] and it was shown that the experimentally accessible quantity differential conductance dI/dV should be doubled when there are two Majorana fermions at 95 each end of the chain. However, the doubling of the differential conductance has not been observed experimentally.

We consider the extended Kitaev chain model that includes also the hopping and superconducting pairing between next-nearest-neighbors (see Fig. 6.1).

γ A ,1 γ B, 1 γ A ,2 γ B, 2 γ B ,N γ A ,N γ A ,3 γ B ,
H el = -2µ N j=1 c † j c j - N-1 j=1 t 1 c † j c j+1 + ∆ 1 c j c j+1 + h.c. - N-1 j=2 t 2 c † j-1 c j+1 + ∆ 2 c j-1 c j+1 + h.c. , (6.1)
where µ is the chemical potential, t 1 is the nearest-neighbor hopping amplitude, ∆ 1 is the nearest-neighbor superconducting pairing potential, t 2 is the next-nearest-neighbor hopping amplitude and ∆ 2 is the next-nearest-neighbor superconducting pairing potential.

N is the total number of sites. c † j (c j ) are the electron at the site j creation (annihilation) operators, respectively. When t 2 = ∆ 2 = 0, this Hamiltonian in Eq.( 6.1) corresponds to the Kitaev model.

We consider capacitive coupling between the topological wire and the cavity,

H el-ph = α N j=1 c † j c j (a + a † ), (6.2)
where α is the electron-photon coupling constant and a (a † ) is the photon annihilation (creation) operator. In the previous chapter, we found the relation between the susceptibility of the Kitaev chain and the transmission coefficient of the cavity in the presence of the topological wire. As shown in [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF], the susceptibility for the topological wire coupled to the cavity reads

Π(t) = -iθ(t)α 2 c † j (t)c j (t), c † j (0)c j (0) . (6.3)
Here we compute the electronic susceptibility of the Kitaev chain with the next-nearestneighbor hopping.

phase diagram and energy spectrum

Let us start our study of the extended Kitaev chain model by establishing a phase diagram of the system. To do so, we close the wire into a ring by applying periodic boundary conditions, c N+1 ≡ c 1 , and perform Fourier transformation. The electronic Hamiltonian of the system in the k-space, H = k>0 H k (k), reads

H k (k) = [-2µ -2t 1 cos(k) -2t 2 cos(2k)]τ z + [-2∆ 1 sin(k) -2∆ 2 sin(2k)]τ y . (6.4)
In order to compute the winding number of the system, let us diagonalize the Hamiltonian H k (k) by performing a unitary transformation Hk

(k) = U † H k (k)U with U = exp(-iθ k τ x /2). tan(θ k ) = - ∆ 1 sin(k) + ∆ 2 sin(2k) µ + t 1 cos(k) + t 2 cos(2k) . (6.5)
The bulk energy spectrum of the extended Kitaev model reads

E k,± = ±2 (µ + t 1 cos(k) + t 2 cos(2k)) 2 + (∆ 1 sin(k) + ∆ 2 sin(2k)) 2 . (6.6)
The condition for the topological phase transition is closing of the gap, E k = 0. In general case, when t 1 = ∆ 1 and t 2 = ∆ 2 , the gap at k = 0 closes when t 2 = -µ -t 1 and the gap at k = π closes when t 2 = -µ + t 1 , while the third condition can be found only

electronic susceptibility

In this section, we calculate the electronic susceptibility of the extended Kitaev model for periodic and open boundary conditions for different chemical potential and frequency.

-1.0 -0.

Topological phase transition detection

As was demonstrated in chapter 3, the topological phase transition points can be inferred from the response of the cavity when the topological wire is closed in a ring. Electronic susceptibility for the extended Kitaev chain reads

Π(t) = -iθ(t) k,q>0
τ z (t), τ z (0) . (6.7)

After performing Fourier transformation and substituting sum with the integral (see appendix D for more details on the derivation), the susceptibility of the closed chain reads We can compute the imaginary part of the susceptibility Eq.6.8 analytically. We obtain that Π (ω) as a function of the chemical potential peaks at the values of the chemical potential that correspond to the topological phase transition (see Fig.

Π(ω) = N 2π π 0 dk 4(∆ 1 sin(k) + ∆ 2 sin(2k)) 2 E 2 k 1 ω -2E k + iη - 1 ω + 2E k + iη . (6.8) 1 0 2 0 -4 -3 -2 -1 0 1 2 3 4 -30
1. n M 1 = 0, n M 2 = 0 2. n M 1 = 1, n M 2 = 1 3. n M 1 = 0, n M 2 = 1 4. n M 1 = 1, n M 2 = 0,
where M 1 corresponds to the lowest-energy state, or first Majorana fermion, and M 2 corresponds to the second-lowest-energy state, or second Majorana fermion. If there are two Majorana fermions, the susceptibility of the wire can be decomposed as:

Π(ω) = Π BB (ω) + Π BM 1 (ω) + Π BM 2 (ω) + Π M 1 M 2 (ω) + Π M 1 M 1 (ω) + Π M 2 M 2 (ω), (6.9)
where Π BB corresponds to the transitions between different bulk levels, Π BM 1 (ω) (Π BM 2 (ω)) comes from the transition between bulk and M 1 (M 2 ) Majorana levels while

Π M 1 M 1 (ω) = Π M 2 M 2 (
ω) ≡ 0 and corresponds to the transitions between the same Majorana levels.

Electronic susceptibility of the chain for n M 1 = n M 2 = 0 presented in Fig. 6.6 shows the non-monotonic behavior as a function of frequency ω. Real part has a kink around

ω p = M 1 + M 2 ,
while imaginary part has a peak around the same value of frequency.

This peak (kink) at ω p comes from the low-energy response of the cavity, which is well separated from the higher-energy response when the frequency ω is in resonance with the effective p-wave gap. The Majorana part of the susceptibility is nonzero only when there are two Majorana fermions at each end of the chain [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF]. In spectroscopic response of the cavity there are four different peaks at ω = ± M 1 ± M 2 depending on the parity of the Majorana states.

Parity dependence and oscillations of the susceptibility

In this subsection, we discuss the dependence of the electronic susceptibility on chemical potential for different parities of the Majorana states. The real part of the susceptibility Π (µ) is presented in the main plot in Fig. 6.7. For -1 < µ < -0.5 there is one Majorana fermion at each end of the chain and we get oscillations of Π (µ) with the period equal to that for the Kitaev chain [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF] (see the inset in Fig.

M 1 = 0, n M 2 = 0, red line corresponds to n M 1 = 1, n M 2 = 0, blue line corresponds to n M 1 = 0, n M 2 = 1 and green line corresponds to n M 1 = 1, n M 2 = 1.
Topological phase transition point is at µ = -0.5. For µ < -0.5 there is one Majorana fermion at each end of the chain, while for µ > -0.5 there are two Majorana fermions at each end of the chain.

The change of the oscillations period should allow to distinguish between the number of the Majorana fermions.

conclusions

In this chapter, we studied the Kitaev model with the next-nearest-neighbor interaction that can show 0, 1 or 2 Majorana fermions emerging at each end of the finite chain. We found the phase diagram for the system for the regime when there are oscillations of the Majorana energy splitting (t 1 = ∆ 1 ). We demonstrated that the imaginary part of the electronic susceptibility for periodic boundary conditions is sensitive to the topological phase transitions, and the peak in Π (µ) have different shape depending on the type of the phase transition (0 → 1 or 0 → 2). We also found that the real part of the susceptibility for finite chain follows the structure of the Majorana energy splitting and shows the change of the period of the oscillations depending weather there is one or two Majorana fermions at each end of the chain. As for the spectroscopic features, the susceptibility can

show four peaks at ω = ± M 1 ± M 2 depending on the parity of the Majorana state. As an outlook, it would be interesting to apply the same approach to study the emergence of fractional fermions in the Su-Schrieffer-Heeger (SSH) model [START_REF] Wakatsuki | Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor[END_REF] and the transition from the latter to Majorana fermions.

J O S E P H S O N E F F E C T I N T O P O L O G I C A L S U P E R C O N D U C T I N G R I N G S C O U P L E D T O A M I C R O WAV E C AV I T Y
In this chapter, we study a one dimensional p-wave superconducting mesoscopic ring interrupted by a weak link and coupled inductively to a microwave cavity. We establish an input-output description for the cavity field in the presence of the ring, and identify the electronic contributions to the cavity response and their dependence on various parameters, such as the magnetic flux, chemical potential, and cavity frequency. We show that the cavity response is 4π periodic as a function of the magnetic flux in the topological region, stemming from the so called fractional Josephson current carried by the Majorana fermions, while it is 2π periodic in the non-topological phase, consistent with the normal Josephson effect. We find a strong dependence of the signal on the cavity frequency, as well as on the parity of the ground state. Our model takes into account fully the interplay between the low-energy Majorana modes and the gaped bulks states, which we show is crucial for visualizing the evolution of the Josephson effect during the transition from the topological to the trivial phase.

the system and model hamiltonian

In Fig. 7.1, we present a sketch of the system under consideration: a topological superconducting ring which contains a weak link at some position coupled to a superconducting microwave cavity [START_REF] Dmytruk | Josephson effect in topological superconducting rings coupled to a microwave cavity[END_REF]. The ring is subject to both a DC magnetic flux, and an ac (quantum) flux from the cavity, which gives rise to a coupling between the two systems, the ring (electrons) and cavity (photons). We consider the Kitaev model Hamiltonian as describing the electronic p-wave superconductor. Such a model could be implemented experimentally utilizing a semiconducting nanowire with spin-orbit coupling in the proximity of 107 Bcav bin bout This is probed in the input-output method, and the output signal bout is to be compared with the input one bin . Bottom: a) A zoom into the topological superconductor in the shape of a ring. The magnetic field of the cavity Bcav pierces the ring and gives rise to a fluctuating magnetic flux. A perpendicular, external magnetic field B ext is also applied, so that the total flux felt by the electrons in the ring is

L R t 0 ˆ cav + ext B ext t 0 B ext L R ˆ + ext b) a) I J I J
Φ tot = Φ ext + Φcav .
The by a weak link, that pertains to the hopping strength t t (t is the hopping strength within the ring, see the text). In the topological region, there are Majorana fermions on the left (γ L ) and on the right (γ R ) of the superconductor, and a Josephson current I J is flowing within the ring. b) A possible implementation of the geometry in a) based on a spin-orbit nanowire flux-biased in a SQUID geometry. Depicted in light red is a conventional superconducting loop that is interrupted so that there is the Josephson current flowing only through the wire.

s-wave superconductor and subject to a magnetic field [1, [START_REF] Lutchyn | Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures[END_REF][START_REF] Mourik | Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices[END_REF][START_REF] Oreg | Helical liquids and Majorana bound states in quantum wires[END_REF]. A flux bias within a nanowire hosting Majorana fermions can be implemented by attaching a SQUID to the wire, in the same spirit as in [START_REF] Pillet | Andreev bound states in supercurrent-carrying carbon nanotubes revealed[END_REF], as depicted in Fig. 7.1 b). We assume the ring is interrupted at some point by a weak link that emulates the Josephson effect in a SIS junction. The external magnetic flux allows for a continuous phase change across the weak link, which controls the Josephson current. On the other hand, the fluctuating cavity flux couples to the flux-biased system and monitors its dynamics.

The general form of the combined Hamiltonian reads:

H sys = H el + H ph + H T , ( 7.1)
H el = - N-1 j=1 (te iΦ jj+1 c † j c j+1 + ∆e iφ jj+1 c j c j+1 + h.c.) -µ N j=1 c † j c j , (7.2)
H T = -t e iΦ N1 c † N c 1 + h.c. , (7.3)
where H el is the Hamiltonian of the p-wave superconductor (Kitaev model), H T is the tunneling Hamiltonian between the ends in the presence of the magnetic flux, and H ph = ω c a † a is the Hamiltonian of a single mode cavity, where ω c is the cavity frequency and a † (a) is the photon creation (annihilation) operators, respectively. Also, t is the hopping parameter, ∆ is the p-wave superconducting pairing potential, µ is the chemical potential, and N is the total number of sites. c † j (c j ) is the electron creation (annihilation) operators, respectively, at the site j. Moreover, Φ jj+1 and φ jj+1 are the phases on the tunneling matrix elements t and the superconducting pairing ∆ at positions j caused by the external fluxes. They read:

Φ jj+1 = e h j+1 j dxA(x) ≡ 2πΦ tot NΦ 0 , ( 7.4
)

φ jj+1 = 4πΦ tot j NΦ 0 . (7.5)
The form of the second term (the phase on the pairing ∆) can be found from utilizing the following assumption: the p-wave pairing in the Kitaev chain is assumed to be induced by an s-wave superconductor underneath the wire that is interrupted at the link.

In such a case, we can consider that the instantaneous supercurrent flowing through the superconductor vanishes, namely:

J s = 2e m |ψ| 2 ( h∇φ -2eA) ≡ 0 , (7.6)
where m, |ψ| 2 , and φ are the electronic mass, the density of superconducting electrons in the s-wave superconductor, and its phase, respectively. The latter is directly imprinted into the wire by means of proximity effect. We can easily solve this differential equation, which then gives the phase on the p-wave superconductor [START_REF] Pientka | Signatures of topological phase transitions in mesoscopic superconducting rings[END_REF] φ(j) ≡ φ jj+1 = 4πΦ tot j/NΦ 0 .

We note that the system is in topological (non-topological) phase when |µ| < 2t (|µ| > 2t), and it supports Majorana zero energy end modes (no Majorana end modes). The magnetic flux contains both the DC flux Φ dc , as well as a time-dependent component Φac (t), or Φ = Φ dc + Φac (t). We assume that the AC flux is due to the cavity photons, and it is given by Φac = iλ(a -a † ), with λ being the coupling constant between the ring and the cavity. Following Ref. [START_REF] Dassonneville | Dissipation and Supercurrent Fluctuations in a Diffusive Normal-Metal-Superconductor Ring[END_REF], we could find that λ 2 = M 2 /L, where M is the coupling inductance between the ring and the resonator and L is the geometrical inductance of the resonator. The spectrum of the ring as a function of the external flux is shown in Fig. 7.2, both in the trivial (left) and topological regimes (right).

We can rewrite the full Hamiltonian in a simpler form by performing a unitary transformation U = exp [-2πi N j=1 (j -1/2)n j Φ dc /NΦ 0 ] so that the entire dependence on the DC flux Φ dc is transferred to the weak link tunneling Hamiltonian H T , or:

H T → U † H T U = -t e i2πΦ dc /Φ 0 e i2π Φcav /NΦ 0 c † N c 1 + h.c. (7.7)
We mention that such a transformation can only gauge away the DC flux (i.e. transfer it entirely to t ), while the ac (or cavity component) cannot be gauged away in a simple way because such a transformation would not commute with the photonic Hamiltonian H ph .

However, we assume that Φac Φ 0 , so that we can expand the exponentials in Φac up to second order in this quantity. With these approximations, the final Hamiltonian can be written as the sum of the electrons (in the presence of the DC flux), the photons, and their mutual interaction:

H sys ≈ H el + H T + H ph -Φac I + 1 2 ( Φac ) 2 D , (7.8)
I = 2πi NΦ 0 N-1 j=1 tc † j c j+1 + 2j∆c j c j+1 + t e i2πΦ dc /Φ 0 c † N c 1 -h.c. , (7.9)
D = - 2πi NΦ 0 2 N-1 j=1 tc † j c j+1 + (2j) 2 ∆c j c j+1 + t e i2πΦ dc /Φ 0 c † N c 1 + h.c. . (7.10)
This is the Hamiltonian that we will be utilizing in the following to describe the response of the cavity, up to second order in the electron-cavity coupling.

Such configuration (ring geometry) have been studied in the past in the context of persistent currents in normal rings, and it was shown to be physically equivalent to a SNS junction, where the low-energy system is described by the so called Andreev levels [START_REF] Cayssol | Isolated hybrid normal/superconducting ring in a magnetic flux: From persistent current to Josephson current[END_REF].

In both cases, the spectrum is quantized due to the finite size of the ring (N system), or the finite size of the N part (SNS junction). We mention that for the SNS junction, if the length of the normal part L ξ (ξ is the coherence length), the spectrum shows a gap that is modulated by the external flux, and can even vanish. This gap, or mini gap, is usually much smaller than the bulk superconducting gap, and these levels are responsible for all the low-energy transport properties of the junction. The Andreev levels in the presence of the flux give rise to Josephson current which, for the state n with energy n reads:

i n = - ∂ n ∂Φ dc . ( 7.11)
The Josephson current carried by these states is then given by I J = n f n i n , where f n are the occupations (Fermi-Dirac) of the level n.

7.2 electronic susceptibility and cavity response

Theoretical approach to susceptibility

In this section, we discuss the cavity response in the presence of the p-wave superconductor, and the dependence on various electronic parameters, such as the chemical potential and the applied magnetic flux Φ dc .

There are two ways to approach the problem: utilizing a quantum description, namely treating the photons as quantum objects, or assuming the ac component of the flux is a classical oscillating quantity, of the form Φ ac = Φ ac (0) sin(ωt). We employ the first method, but compare with the expected results utilizing a classical description. The reason for doing so is that the first method is suited to also enter the quantum regime, where the coupling between the photons and the electronic system is strong and one needs to employ a polaritonic-like description. While such a case is left for a future study, we believe that the formalism developed in this chapter will be of great usefulness.

In chapter 2, we derived explicitly the response of the cavity due to its coupling to an electronic system. Here we give a brief summary of the derivation, and depict the results for the particular type of couplings found here.

The second order contribution to the electron-photon coupling can be written as (a † ) 2 + a 2 + 2n ph + 1, with n ph ≡ a † a, and we see that the third and fourth terms renormalize the cavity frequency and the weak tunneling amplitude, respectively. The first two terms correspond to creation or annihilation of two photons, and we can disregard them in the weak coupling limit. With this approximation, we can insert this expression into the equation of motion for the photonic operator, switch to the frequency domain, and solve these equations iteratively up to second order in λ. By taking the average over the unperturbed (or uncoupled to the cavity) electronic Hamiltonian, we obtain:

-iωa = -i(ω c + λ 2 D )a - κ 2 a - √ κb in -iλ 2 Π(ω c )a + λ I , (7.12)
where Π(ω) = dt exp (iωt)Π(t) with

Π(t) = -iθ(t) [I(t), I(0)] . (7.13)
Note that all the expectation values are taken over the electronic system in the absence of the cavity. Let us discuss briefly the resulting equation of motion and further approximations. We assume the input field b in contains a large number of photons, and thus we can neglect the last term that acts as an extra input source (and which is entirely due to the current flowing through the weak link). The opposite limit, when the input field is only due to vacuum fluctuations, can in principle lead to emission of photons due to current fluctuations in the ring.

With this, we see that the coupling to the superconductor affects the cavity in two ways: by changing its frequency, and by changing its quality factor (or the escape rate κ). More precisely, we have:

δω = λ 2 [ D + ReΠ(ω c )] , (7.14)
κ = κ -λ 2 ImΠ(ω c ) . (7.15)
In Ref. [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF][START_REF] Dmytruk | Out-of-equilibrium quantum dot coupled to a microwave cavity[END_REF] we showed explicitly that the transmission τ of the cavity can be written as follows:

τ ≡ b out b in = κ -i(ω -ω c ) + κ + iλ 2 Π tot (ω c ) , (7.16)
where Π tot (ω c ) = Π(ω c ) + D . It is instructive to rewrite the two terms in more explicit forms:

D = n f n n|D|n , (7.17)
Π(ω) = n =m | n|I|m | 2 f n -f m -( n -m ) + ω + iη , (7.18)
where f n is the occupation of the electronic energy state n , and η is a small number that account for the dissipation. We can rewrite Π tot (ω) in a more transparent form by utilizing the following sum rule [START_REF] Ferrier | Phase-dependent Andreev spectrum in a diffusive SNS junction: Static and dynamic current response[END_REF][START_REF] Trivedi | Mesoscopic ring in a magnetic field: Reactive and dissipative response[END_REF]:

n|D|n + m =n | n|I|m | 2 n -m = 1 2 ∂ 2 n ∂Φ 2 , (7.19)
so that we obtain

Π tot (ω) = ∂I J ∂Φ + n ∂f n ∂Φ ∂ n ∂Φ -ω n =m f n -f m n -m | n|I|m | 2 ( n -m ) -ω -iη , (7.20)
where I J is the Josephson current flowing through the ring in the presence of a DC flux.

In this form, the expression for the susceptibility that is measured by the cavity (via its transmission) has the same form as the one derived previously for normal [START_REF] Trivedi | Mesoscopic ring in a magnetic field: Reactive and dissipative response[END_REF] and for superconducting rings [START_REF] Dassonneville | Dissipation and Supercurrent Fluctuations in a Diffusive Normal-Metal-Superconductor Ring[END_REF][START_REF] Ferrier | Phase-dependent Andreev spectrum in a diffusive SNS junction: Static and dynamic current response[END_REF]. However, in those previous works they utilized a classical description of the ac perturbation. Moreover, they assumed the electronic system is not closed, but coupled to an external bath which leads to a finite width of the electronic levels. To make correspondence between their results and ours in such a case, we can write:

Π tot (ω) = ∂I J ∂Φ + ω ω + iγ n ∂f n ∂Φ ∂ n ∂Φ -ω n =m f n -f m n -m | n|I|m | 2 ( n -m ) -ω -iγ , (7.21)
where γ is the relaxation rate of the levels (assumed, for simplicity, to be the same for all levels). We see that, in principle, there are three contributions to the cavity response: the Josephson current or non-dissipative part, the diagonal part corresponding to the decay of the levels, and finally the non-diagonal part stemming from the usual Kubo contribution.

In the following, we neglect the second term due to its smallness in the cavity frequency range. The first term is independent on the frequency ω, and thus any dependence on this parameter will be due to the last term.

For the classical derivation of the response we refer the reader to Refs. [START_REF] Ferrier | Phase-dependent Andreev spectrum in a diffusive SNS junction: Static and dynamic current response[END_REF][START_REF] Sticlet | Dynamical response of dissipative helical edge states[END_REF][START_REF] Trivedi | Mesoscopic ring in a magnetic field: Reactive and dissipative response[END_REF]. A few comments are in order. In the case of a SNS junction, the entire contribution to the cavity signal is coming from the sub gap Andreev states in the normal region, while any bulk effects can be neglected since the bulk gap is much larger than the minigap. That need not be the case for a p-wave superconductor, as this system can suffer a topological transition and close its bulk gap, which in turn should affect strongly the cavity response as a function of the magnetic flux. Thus, the energy levels in the expression for the susceptibility are the ones that correspond to the entire bulk spectrum, and not only the mini gap states.

Flux dependence of the cavity response

In Fig. 2.¥10 -4 FêF0 P J HF,0Lêl 2 0.0 0.2 0.4 0.6 0.8 1.0 topological regime, all susceptibilities are Φ 0 periodic, with large amplitudes variations.

-2.¥10 -4 -1.¥10 -4 0 1.¥10 -4 2.¥10 -4 FêF0 P J HF,0Lêl 2 0.0 0.2 0.4 0.6 0.8 1.0 -7 -6 -5 -4 -3 -2 -
The amplitude of the Φ 0 oscillations, however, diminishes as the system approaches the topological transition, and disappear after this transition, with only the Φ 0 /2 becoming manifest. Such Φ 0 /2 oscillations pertain to the normal Josephson effect, and are much smaller in amplitude, as can be seen from the top plot on the right column. We note that such oscillations exist also for the non-diagonal terms, but they are too small to be depicted (of the order 10 -6 ).

Let us now discuss the various terms contributing to the total susceptibility, as well as the effect of Majorana fermions on these quantities. As mentioned already, in the topological regime, there are Majorana fermions emerging at the edge of the ring, and which are coupled via the weak link. In a low-energy description, these Majoranas are responsible for the so called fractional Josephson effect, one of the hallmarks of the Majorana fermions physics. However, the response of the cavity is sensitive not only to the presence of Majoranas, but also to the interplay between these excitations and the bulk states of the superconductor, in particular close to the topological phase transition. The first term is insensitive to such transitions, as it is given by the derivative of the supercurrent with respect to the applied flux. The second term instead contains the matrix elements between the Majorana and the bulk states, it has both a real and imaginary parts, and depends on the cavity frequency ω c . We can thus decompose it as follows:

Π tot (ω) = Π BB (ω) + Π BM (ω) + Π MM (ω), (7.22)
where Π BB is the part of the susceptibility that comes from the transitions between (different) bulk levels, Π BM corresponds to the transitions between bulk and Majorana levels while Π MM corresponds to the transitions between Majorana levels and Π MM (ω) ≡ 0 in the present setup. Note that in the non-topological regime Π BM = 0, as there are no Majorana states present. To ascertain for the physical meaning to the oscillations of the susceptibility, it is instructive to cast this quantity in the following form:

Π tot (ω, φ) = ∞ j=0 a j cos (2πjΦ/Φ 0 + δφ j ) , (7.23)
where a j are (complex) Fourier coefficients of the expansion, and δφ j are phase shifts, all these quantities being functions of the chemical potential µ and the gap ∆. In the topological phase, the dominant terms are a 0 and a 1 , namely those corresponding to constant and Φ 0 oscillations, respectively. In the topologically trivial regime instead, a 1 = 0, and thus the usual Josephson effect dominates. Moreover, we find numerically that the coefficients with j = 2p + 1, p = 1, 2, . . . are zero above the topological transition (which does not host any Majoranas), while below such coefficients are in principle non-zero. We stress that the Josephson current has also a similar dependence on the applied flux: for a p-wave superconductor, the current I J (Φ) ∝ sin (4πΦ/Φ 0 ) in the non-topological region (normal Josephson effect), while this is I J (Φ) ∝ sin (2πΦ/Φ 0 ) in the topological region (fractional Josephson effect).

Parity dependence

In this section, we analyze the parity dependence of the electronic susceptibility, and implicitly of the cavity response. In Fig. 7.4, we plot the various contributions to the susceptibility in the topological regime for the two different parities of the ground state, f M = 0, 1. By inspecting the plots, we can conclude that there are two main features that separate the two parities: first, the oscillations (or maxima) of the susceptibilities are shifted by Φ 0 , and second, there is a strong asymmetry between the curves for the two parities, especially in the non-diagonal contribution. The shift by Φ 0 can be explained by means of the fractional Josephson effect, which is due to the presence of Majorana bound states that give rise to coherent transfer of charge e or -e, depending on the parity, between the two bound states. That in turn gives rise to dissipationless currents opposite in sign for the two parities, and which coincide for Φ = (2n + 1)Φ 0 /4, as can be seen also from the plots. The asymmetry of the Josephson contribution can be explained by inspecting the expression for the supercurrent flow:

I J = n=bulk i n + (-1) f M i M , i.e
. the bulk contribution is constant, as its parity is assumed unchanged, while the Majorana contribution depends on the parity of the ground state f M = 0, 1. The non-diagonal susceptibility shows a much stronger asymmetry, and it is due to an interplay between the occupancy of the Majorana and the matrix elements of the current operator I between the Majorana and the bulk states, or simply by Π BM given by:

Π BM (ω) = -ω n =M f n -f M n -M | n|I|M | 2 ( n -M ) -ω -iγ , (7.24)
where M stands for the Majorana.

Frequency dependence

Finally, we discuss briefly the frequency dependence of the susceptibility (thus of the cavity response) for different parity and flux realizations, respectively. We note that the Josephson contribution is frequency independent, and the only dependence on ω arises from the non-diagonal susceptibility. In Fig. 7.5, we plot the real part of the non-diagonal susceptibility as a function of the frequency in the topological regime, and for the two different parities. The non-monotonic behavior of the curves can be related to the resonance condition when the frequency ω matches the effective p-wave gap and the sign of the susceptibility changes. Interestingly, the dependence on ω is different for different parities f M = 0, 1 and for different values of the magnetic flux Φ. Such features can again be traced back to the dependence of Π BM on both the flux and the parity of the topological superconductor. We note, however, that the curves become identical (not shown) at flux values Φ = (2n + 1)Φ 0 /4, with n = 0, 1, . . . , at which instances the degeneracy of the Majorana energy levels is restored.

conclusions

In this chapter, we studied theoretically a Kitaev ring interrupted by a weak link and pierced by a magnetic flux, and coupled inductively to a microwave cavity. We established an input-output description for the cavity transmission, and show that the cavity response depends strongly on various electronic parameters, such as the chemical potential, the magnetic flux, and the parity of the superconducting ground state. We found a 4π (2π) variation of the cavity response with respect to the external magnetic flux in the topological (trivial) phase, and related such dependence to the fractional (normal) Joseph-son effect. As opposed to previous works, our theory takes into account, on equal footing, the low-energy Majorana modes, the gaped bulk states, and the interplay between these states in the presence of the cavity. Such a description allows one to describe not only the superconducting ring deep in the topological regime, but also the topological transition, and the crossover from the fractional to the normal Josephson effect in the presence of a magnetic flux.

While our theory treats the Kitaev toy model, we stress that it has been shown theoretically that such a model can be emulated by the spectrum of a spin-orbit semiconducting ring subject to an external magnetic field and coupled by the proximity effect to an swave superconductor. The main difference between such a model and the Kitaev chain is the number of states, which is doubled, as one needs to account for the spin degree of freedom. We believe, based on our previous findings in Ref. [START_REF] Dmytruk | Cavity quantum electrodynamics with mesoscopic topological superconductors[END_REF], that our main results on the visualization of the transition from the fractional to the normal Josephson effect by utilizing a microwave cavity are robust. In this thesis, we have studied how the measurement of the cavity microwave field can be used to extract information on the properties of the nanostructures coupled to the cavity. Current experiments allow to implement on-chip mesoscopic circuits coupled to high finesse resonators and to measure extremely small variations of the amplitude of the microwave optical signal. Cavity QED is an experimental technique that allows to study and manipulate atoms, superconducting qubits and quantum dots coupled to a microwave cavity. Using input-output theory [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF] in case of weak coupling between the electronic system and the photonic field in the cavity, we have related the transmission coefficient of the cavity to the electronic susceptibility of the nanostructure. The frequency shift is proportional to the real part of the electronic susceptibility while the frequency broadening is proportional to the imaginary part of the susceptibility. In this thesis, this general framework is applied to four specific mesoscopic conductors: a tunnel junction, a quantum dot coupled to the leads, a one-dimensional topological wire and a superconducting ring.

First, we studied a tunnel junction and a quantum dot tunnel coupled to metallic leads embedded in a microwave cavity. For the tunnel junction, the imaginary part of the susceptibility is directly proportional to the differential conductance dI/dV. We demonstrated that for the quantum dot tunnel coupled to the metallic leads, the phase shift is proportional to the imaginary part of the admittance far from quantum dot resonance. For symmetric lead-cavity coupling, equal tunneling and low-frequency, the phase shift is proportional to the real part of the admittance, or dI/dV. In the general case, there is no direct relation between optical observables, such as phase shift and frequency broadening, and transport quantities, such as differential conductance. Therefore, the transmission coefficient of the cavity gives access to finite frequency admittance of the quantum dot thus providing new information about the quantum conductor coupled to the cavity.

123

We have developed a formalism relating the electronic susceptibility to the optical transmission of the cavity. In addition to the mesoscopic conductors studied here in detail, our formalism is applicable to more complicated mesoscopic systems coupled to the cavity.

More specifically, further research should focus on a system with two out-of-equilibrium quantum dots coupled to the same microwave cavity. In such system, the cavity can be used to both induce and probe the correlated transport. Our formalism may allow to detail theoretically the behavior of the two quantum dots system, which is a universal gate ubiquitous for quantum computing. In a further perspective, our approach may shed more light on general quantum dot architecture.

Moreover, we propose to use the methods of cavity QED to study topological superconductors that can host Majorana fermions. We considered different theoretical models of topological superconductors with nearest-neighbor and next-nearest-neighbor hoppings that give rise to one or two Majorana fermions at each end of the wire, respectively. Using the photonic field of the cavity, within a single set of measurements, one can determine the topological transition point and the emergence of the Majorana fermions. One can also measure, in the same set, the parity of the ground state, the quantity of crucial interest for quantum computing with Majorana fermions, which is impossible to probe electronically since the transport inherently modifies the number of electrons in the system.

At the same time, cavity QED can be used to probe DC fractional Josephson effect. We continued by studying a superconducting ring with a weak link coupled to a microwave cavity. The ring is subject to both a DC magnetic flux, and an AC flux from the cavity, which gives rise to a coupling between the ring and cavity. We demonstrated that the electronic susceptibility has different periodicity as a function of magnetic flux in topological and non-topological regions. Different periodicity allows to probe the fractional Josephson effect, which is associated with the emergence of the Majorana fermions in the system. As opposed to previous works, our theory takes into account both the low-energy Majorana modes and the bulk states, thus allowing to visualize the transition from the fractional to the normal Josephson effect.

As we demonstrated, cavity QED methods can be used to extensively study 1D topological structures. We propose to generalize these methodology to theoretically investigate 2D systems, such as 2D superconductors or 2D topological insulators that have seen large interest in recent experimental studies [START_REF] Wiedenmann | 4 [pi]-periodic Josephson supercurrent in HgTe-based topological Josephson junctions[END_REF]. Furthermore, we suggest that a generalization of the input-output theory for a 2D system may allow to characterize theoretically the braiding of Majorana fermions.

In a broader perspective, we propose to manipulate electronic properties with light. The strong coupling between the electronic system and the electric field in the cavity enables such manipulation. Qubits -basic units for quantum computing -can be entangled, controled and measured in the strong-coupling regime. An experimental implementation of a superconducting qubit as a Cooper pair box has been already studied in the strongcoupling regime [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF]. A quantum dot can also serve as a physical implementation of the qubit [START_REF] Loss | Quantum computation with quantum dots[END_REF] as follows. The quantum dot with a single electron can be constructed by tuning the gate voltage. And the spin of this electron can encode a qubit. Several experiments have implemented the coupling between the spin qubit and photons inside a cavity [START_REF] Petersson | Circuit quantum electrodynamics with a spin qubit[END_REF][START_REF] Viennot | Coherent coupling of a single spin to microwave cavity photons[END_REF]. Cavity QED architecture allows to construct distant spin qubits that can be entangled thus allowing to implement two-qubit quantum algorithms. In order to obtain information about the spin dynamics and the parameters of the qubits, such as spin dephasing time, we suggest to define the spin susceptibility for spin qubit systems in the strong-coupling regime similarly to the electronic susceptibility.

Finally, cavity QED allows to study various mixed electron-photon states, including new topological states created by the interaction between light and matter. For instance, the Floquet topological insulator, which was constructed by applying a classical AC drive [START_REF] Lindner | Floquet topological insulator in semiconductor quantum wells[END_REF] to create topological electronic band structure. Coupling a single photon to an exciton leads to the emergence of a novel topological state, when a topological polariton (topolariton) [START_REF] Karzig | Topological polaritons[END_REF] is formed. The topolariton is a kind of polariton -a quasiparticle formed by photons and electrical dipoles that emerge in the strong light-matter coupling regime [START_REF] Carusotto | Quantum fluids of light[END_REF].

It is characterized by the formation of chiral edge states in the excitation spectrum. Generally, we expect the methods of cavity QED to generalize for probing any polaritons. As a direct extension of the current work, we plan to apply cavity QED to theoretically investigate "Majorana polaritons" which are highly entangled electron-photon degenerate states that may appear under a strong coupling between a single photon and Majorana fermions in a topological wire [START_REF] Trif | Resonantly tunable majorana polariton in a microwave cavity[END_REF]. This particular system gives us a pertinent example for A

A P P E N D I X A
In this appendix, we provide more details on the input-output theory. We consider a onesided cavity (κ 2 = 0 in Fig. 2.5) with a resonance frequency ω c [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF]. The Hamiltonian that describes the system reads

H = H sys + H bath + H int , (A.1)
where the bath Hamiltonian is given by

H bath = q ω q b † q b q , (A.2)
and the coupling Hamiltonian (withing the rotating wave approximation) reads

H int = -i q f q a † b q -f * q b † q a . (A.3)
The Heisenberg equation of motion (EOM) for the cavity mode is

ȧ = i [H, a] , (A.4)
where

127 κ(ω c ) = 2π q |f q | 2 δ(ω c -ω q ) (A.10) so that ∞ -∞ dν 2π κ(ω c + ν)e -iν(t-τ) = q |f q | 2 e -i(ω q -ω c )(t-τ) . (A.11)
Performing the Markov approximation, Eq.(A.11) reads q |f q | 2 e -i(ω q -ω c )(t-τ) = κδ(t -τ) (A.12) and q f q b q = q f q e -iω q (t-t 0 ) b q (t 0 ) + κ

t t 0 dτδ(t -τ)e iω c (τ-t) a(τ) (A.13) Using x 0 -∞ dxδ(x -x 0 ) = 1 2 (A.14)
we obtain q f q b q = q f q e -iω q (t-t 0 ) b q (t 0 ) + κ 2 a(t).

(A.15)

The cavity EOM reads b out (t) = 1 2πρ q e -iω q (t-t 1 ) b q (t 1 ), (A. In this section, we express the charge susceptibility of a non-interacting quantum conductor in terms of integral over the elements of its S matrix.

The electron number operator in terms of the current operator reads

n α (t) = i 2π ∞ -∞ dω exp(-iωt) 1 ω ∞ -∞ dE γγ A γγ (α, E, E + ω)a † γ (E)a γ (E + ω), (
F αβ (ω) = 1 2π 2 ∞ -∞ ∞ -∞ dω 2 dE 2 1 ω 2 2 1 ω + ω 2 + iη × γ 1 ,γ 1 A γ 1 γ 1 (α, E 2 + ω 2 , E 2 )A γ 1 γ 1 (β, E 2 , E 2 + ω 2 )× × f γ 1 (E 2 + ω 2 ) 1 -f γ 1 (E 2 ) . (B.4)
In the same way we can calculate X αβ (ω):

X αβ (ω) = - 1 2π 2 ∞ -∞ ∞ -∞ dω 2 dE 2 1 ω 2 2 1 ω + ω 2 + iη × γ 1 ,γ 1 A γ 1 γ 1 (α, E 2 + ω 2 , E 2 )A γ 1 γ 1 (β, E 2 , E 2 + ω 2 )× × f γ 1 (E 2 ) [1 -f γ 1 (E 2 + ω 2 )] . (B.5)
And finally Π αβ (ω) reads

Π αβ (ω) = 1 2π 2 ∞ -∞ ∞ -∞ dω 2 dE 2 1 ω 2 2 1 ω + ω 2 + iη × γ 1 ,γ 1 F αβ γ 1 γ 1 (E 2 , ω 2 ) f γ 1 (E 2 + ω 2 ) -f γ 1 (E 2 )    . (B.6)
where we introduced

F αβ γγ = A γγ (α, E + ω, E)A γ γ (β, E, E + ω). (B.7)
At zero temperature T = 0, we can further simplify Eq. (B.6) and using the identity In order to calculate the charge susceptibility for the quantum dot, we need the sixteen functions F γγ αβ which are given by [START_REF] Carusotto | Quantum fluids of light[END_REF]) The functions K αβ take the following form: Using the decomposition

1 ω + ω 2 + iη = P 1 ω + ω 2 -iπδ(ω + ω 2 ) , (B.
F LL LL (E, ω) = A LL (L, E + ω, E)A LL (L, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + ω 2 Γ 2 L , (B.9) F RR LL (E, ω) = A LL (R, E + ω, E)A LL (R, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R , (B.10) F LR LL (E, ω) = A LL (L, E + ω, E)A LL (R, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + iωΓ 2 L Γ R , (B.11) F RL LL (E, ω) = A LL (R, E + ω, E)A LL (L, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -iωΓ 2 L Γ R , (B.12) F LL RR (E, ω) = A RR (L, E + ω, E)A RR (L, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R , (B.13) F RR RR (E, ω) = A RR (R, E + ω, E)A RR (R, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + ω 2 Γ 2 R , (B.14) F LR RR (E, ω) = A RR (L, E + ω, E)A RR (R, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -iωΓ 2 R Γ L , (B.15) F RL RR (E, ω) = A RR (R, E + ω, E)A RR (L, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + iωΓ 2 R Γ L , (B.
F LL LR (E, ω) = A LR (L, E + ω, E)A RL (L, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + |g(E)| 2 Γ L Γ R , (B.17) F RR LR (E, ω) = A LR (R, E + ω, E)A RL (R, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + |g(E + ω)| 2 Γ L Γ R , (B.18) F LR LR (E, ω) = A LR (L, E + ω, E)A RL (R, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -g(E)g(E + ω)Γ L Γ R -iΓ L Γ R |g(E)g(E + ω)| 2 Γ R g * (E + ω) + Γ L g * (E) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R E 2 + Eω -2E d -d ω + 2 d + Γ 2 /4 -iΓ L ω/2 + iΓ R ω/2 = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R (E -d ) × (E + ω -d ) + Γ 2 /4 -iΓ L ω/2 + iΓ R ω/2 , (B.19) F RL LR (E, ω) = A LR (R, E + ω, E)A RL (L, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -g * (E)g * (E + ω)Γ L Γ R + iΓ L Γ R |g(E)g(E + ω)| 2 Γ R g(E + ω) + Γ L g(E) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R E 2 + Eω -2E d -d ω + 2 d + Γ 2 /4 + iΓ L ω/2 -iΓ R ω/2 = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R (E -d ) × (E + ω -d ) + Γ 2 /4 + iΓ L ω/2 -iΓ R ω/2 , (B.20) F LL RL (E, ω) = A RL (L, E + ω, E)A LR (L, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + |g(E + ω)| 2 Γ L Γ R , (B.21) F RR RL (E, ω) = A RL (R, E + ω, E)A LR (R, E, E + ω) = -|g(E)g(E + ω)| 2 Γ 2 L Γ 2 R + |g(E)| 2 Γ L Γ R , (B.22) F LR RL (E, ω) = A RL (L, E + ω, E)A LR (R, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -g * (E)g * (E + ω)Γ L Γ R + iΓ L Γ R |g(E)g(E + ω)| 2 Γ R g(E) + Γ L g(E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R E 2 + Eω -2E d -d ω + 2 d + Γ 2 /4 -iΓ L ω/2 + iΓ R ω/2 = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R (E -d ) × (E + ω -d ) + Γ 2 /4 -iΓ L ω/2 + iΓ R ω/2 , (B.23) F RL RL (E, ω) = A RL (R, E + ω, E)A LR (L, E, E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -g(E)g(E + ω)Γ L Γ R -iΓ L Γ R |g(E)g(E + ω)| 2 Γ R g * (E) + Γ L g * (E + ω) = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R E 2 + Eω -2E d -d ω + 2 d + Γ 2 /4 + iΓ L ω/2 -iΓ R ω/2 = |g(E)g(E + ω)| 2 Γ 2 L Γ 2 R -Γ L Γ R (E -d ) × (E + ω -d ) + Γ 2 /4 + iΓ L ω/2 -iΓ R ω/2 . (B.
K LL (x) = - 2Γ L Γ x 2 + 4 x 2 Γ + 4Γ R [arctan (a + x) -arctan (a -x)] + Γ L x -2 log 1 + a 2 + log 1 + (a + x) 2 + log 1 + (a -x) 2 , (B.26) K RR (x) = - 2Γ R Γ x 2 + 4 x 2 Γ + 4Γ L [arctan (b + x) -arctan (b -x)] + Γ R x -2 log 1 + b 2 + log 1 + (b + x) 2 + log 1 + (b -x) 2 , (B.27) K LR (x) = - Γ L Γ R Γ x 2 + 4 -2i(x -2i) [arctan (a + x) -arctan (a -x)] + 2i(x + 2i) [arctan (b + x) -arctan (b -x)] + (x -2i) -2 log 1 + a 2 + log 1 + (a + x) 2 + log 1 + (a -x) 2 + (x + 2i) -2 log 1 + b 2 + log 1 + (b + x) 2 + log 1 + (b -x) 2 , (B.28) K RL (x) = - Γ L Γ R Γ x 2 + 4 -2i(x -2i) [arctan (b + x) -arctan (b -x)] + 2i(x + 2i) [arctan (a + x) -arctan (a -x)] + (x -2i) -2 log 1 + b 2 + log 1 + (b + x) 2 + log 1 + (b -x) 2 + (x + 2i) -2 log 1 + a 2 + log 1 + (a + x) 2 + log 1 + (a -x) 2 , (B.
arctan(α + x) = 1 2i (log[1 + i(α + x)] -log[1 -i(α + x)]) , (B.32)
the function f(x) can be given the following alternative form

f(x) = 1 2i log 1 + ix 1 + ia + log 1 + ix 1 -ia -log 1 - ix 1 + ia -log 1 - ix 1 -ia . (B.33)
Therefore, L(y) can be written as a sum over four integrals. Let us detail the calculation of one of them. We consider the following principal part integral Using Cauchy's residue theorem, we obtain eventually:

L 1 (y) =P ∞ -∞ dx x 2 (x + y)(x + 2i)(x -2i) × log 1 + ix 1 + ia . (B.
-2πiRes[-2i] = C - = L 1 (y) + C -y + C 0 + C R (B.
+ Γ L πΓ 2 y 2 + 4 1 + 4Γ R y 2 Γ -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 , (B.41) Π LL (ω) = - 2Γ L πΓ 2 y 2 + 4 1 + 4Γ R y 2 Γ [arctan (a + y) -arctan (a -y)] - 2Γ 2 L πΓ 3 y y 2 + 4 -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 , (B.42) Π RR (ω) = - 4Γ 2 R πΓ 3 y y 2 + 4 [arctan (b + y) -arctan (b -y)] + Γ R πΓ 2 y 2 + 4 1 + 4Γ L y 2 Γ -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 , (B.43) Π RR (ω) = - 2Γ R πΓ 2 y 2 + 4 1 + 4Γ L y 2 Γ [arctan (b + y) -arctan (b -y)] - 2Γ 2 R πΓ 3 y y 2 + 4 -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 , (B.44) Π LR (ω) + Π RL (ω) = - 4Γ L Γ R πΓ 3 y y 2 + 4 [arctan (a + y) -arctan (a -y)] - 4Γ L Γ R πΓ 3 y 2 y 2 + 4 -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 - 4Γ L Γ R πΓ 3 y y 2 + 4 [arctan (b + y) -arctan (b -y)] - 4Γ L Γ R πΓ 3 y 2 y 2 + 4 -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 , (B.45) Π LR (ω) + Π RL (ω) = 8Γ L Γ R πΓ 3 y 2 y 2 + 4
[arctan (a + y)arctan (a -y)]

- 2Γ L Γ R πΓ 3 y y 2 + 4 -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 + 8Γ L Γ R πΓ 3 y 2 y 2 + 4 [arctan (b + y) -arctan (b -y)] - 2Γ L Γ R πΓ 3 y y 2 + 4 -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 . (B.46) Using Π(ω) = (β L -λ) 2 Π LL (ω) + (β R -λ) 2 Π RR (ω) + (β L -λ) (β R -λ) [Π LR (ω) + Π RL (ω)] , (B.47)
we obtain the final result for Π(ω):

Π (ω) = - 4Γ L πΓ 3 y y 2 + 4
[arctan (a + y)arctan (a -y)]

× (β L -λ) 2 Γ L + (β L -λ) (β R -λ) Γ R + Γ L πΓ 2 y 2 + 4 -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 × (β L -λ) 2 1 + 4Γ R y 2 Γ -(β L -λ) (β R -λ) 4Γ R y 2 Γ - 4Γ R πΓ 3 y y 2 + 4 [arctan (b + y) -arctan (b -y)] × (β L -λ) 2 Γ R + (β L -λ) (β R -λ) Γ L + Γ R πΓ 2 y 2 + 4 -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 × (β L -λ) 2 1 + 4Γ L y 2 Γ -(β L -λ) (β R -λ) 4Γ L y 2 Γ . (B.48)
and the imaginary part of the total charge susceptibility reads

Π (ω) = - 2Γ L πΓ 2 y 2 + 4
[arctan (a + y)arctan (a -y)] with A(r) being the electromagnetic field vector potential at position r, and the integration is performed between the sites i and i + 1. We will focus on the derivation of the effective Kitaev model in the tight-binding picture, as the microscopic, continuum model was described in great detail very recently in [START_REF] Cottet | Electron-photon coupling in mesoscopic quantum electrodynamics[END_REF]. We thus refer the reader to that paper for a detailed calculation of the cavity effects, as well as the derivation of the capacitive coupling starting from the minimal coupling.

× (β L -λ) 2 1 + 4Γ R y 2 Γ -(β L -λ) (β R -λ) 4Γ R y 2 Γ - 2Γ L πΓ 3 y y 2 + 4 -2 log 1 + a 2 + log 1 + (a + y) 2 + log 1 + (a -y) 2 × (β L -λ) 2 Γ L + (β L -λ) (β R -λ) Γ R - 2Γ R πΓ 2 y 2 + 4 [arctan (b + y) -arctan (b -y)] × (β L -λ) 2 1 + 4Γ L y 2 Γ -(β L -λ) (β R -λ) 4Γ L y 2 Γ - 2Γ R πΓ 3 y y 2 + 4 -2 log 1 + b 2 + log 1 + (b + y) 2 + log 1 + (b -y) 2 (β L -λ) 2 Γ R + (β L -λ) (β R -λ) Γ L . (B.
Here we give some details on the derivation of Eq. (5.4) starting from a non-superconducting nanowire coupled to a bulk p-wave superconductor with such a coupling being assisted by the cavity field. For simplicity, we assume the bulk to be not s, but p-wave paired, thus the presence of spin-orbit coupling in the wire is not a necessary ingredient. However, the present calculations can be straightforwardly generalized to more realistic system, such as nanowires with SOI. The total Hamiltonian of the system reads:

H sys = H b + H w + H T + H c , (C.2)
where

H p = -µ p j c † j,p c j,p + j t p c † j,p c j+1,p + ∆ p c † j,p c † j+1,p + h.c. (C.3)
with p = b(bulk), w(wire), and ∆ w = 0 (no intrinsic superconductivity in the wire), and ∆ b ≡ ∆ the p-wave pairing in the bulk superconductor. Here, c j,p (c † j,p ) and t p are the electronic annihilation (creation) operator at position j and the hopping parameter in system p = b, w, respectively. The tunneling Hamiltonian in the presence of the cavity reads:

H T = j t int e -i φj c † j,w c j,b + h.c. , (C.4)
where φj = Âj d j , with Âj = i(α j /ω c )(a † -a), d j , α j , ω c , and a (a † ), being the cavity vector potential, the coupling strength, the cavity frequency, and the cavity photon annihilation (creation) operators, respectively. Note that we assumed that the cavity field points perpendicularly to the wire, and it has no component along it. If instead such components would exists, we should have modified the wire Hamiltonian too in order to account for the cavity induced phase factors. In the following, we will assume that α j d j ≡ α j = α, namely it is constant along the entire wire. Finally, the Hamiltonian of the cavity reads:

H c = ω c a † a , (C.5)
with ω c being the (fundamental) frequency of the cavity. Before deriving an effective wire Hamiltonian, it is instructive to switch to the Fourier space, for both the bulk and wire

Hamiltonians. We get:

H b = k ξ k,b c † k,b c k,b - k>0 i∆ sin k c -k,b c k,b -c † k,b c † -k,b , (C.6) H w = k ξ k,w c † k,w c k,w , (C.7) H T = t int k e i φc † k,w c k,b + e -i φc † k,b c k,w , (C.8)
where ξ k,p = t p cos k -µ p , with µ α the chemical potential in the p = w, b system.

Next we perform the so called Lang-Firsov transformation on the system Hamiltonian, which means H sys = exp(S)H sys exp(-S) with S chosen as follows:

S = α ω c (a -a † ) q c † q,w c q,w . (C.9)
After some lengthy, but straightforward calculation we obtain the system Hamiltonian as follows:

H sys = H w + H b + α q c † q,w c q,w (a + a † ) H c-w + α 2 ω c q c † q,w c q,w N2 2 + t int k (c † q,w c k,b + h.c.) H T +H c , (C.10)
which implies we excluded the photonic field from the tunneling term at the expense of adding photon-dependent chemical potential shift in the wire (third term) as well as an interaction term (fourth term). Note that for t int = 0, the transformation does not affect the spectrum, as it can be simply undone. However, as will see in the following, in the presence of the tunneling term the photonic field in the form of the capacitive coupling can lead to real effects.

In the following, we aim at finding an effective Hamiltonian describing the wire only by integrating the bulk superconductor degrees of freedom up to second order in the tunneling t int . We choose to do so by employing the Schrieffer-Wolff transformation formalism, which means, as before, that we unitary rotate the system Hamiltonian as

H eff sys = e S SW H sys e -S SW = H w + H b + H w-c + H T + H c + [S SW , H w + H b + H w-c + H T + H c ] + . . . , (C.11)
and choose

[S SW , H w + H b ] = -H T , (C.12)
or S SW = (L w + L b ) -1 H T , with L α being a superoperator whose action is defined as

L α A = [H α , A],
∀A. This is equivalent to the following identity:

S SW = i lim η→0 +∞ 0 dte -ηt e i(H w +H b )t H T e -i(H w +H b )t . (C.13)
This term excludes the tunneling Hamiltonian H T in leading order (assuming there is no diagonal contribution caused by such a term). Then, we neglect the contributions of the higher order terms on the wire spectrum by averaging over the bulk ground state |0 b in order to derive a purely (renormalized) wire Hamiltonian:

H eff w ≈ 0 b |H b + H w + H c + H w-c + 1 2 [S SW , H T ] + [S SW , H w-c ] + . . . |0 b , (C.14)
In order to find S SW from Eq. (C.13) explicitly, let us perform Bogoliubov transformation for the bulk p-wave superconductor defined as

c k,b = u * k γ k,b + v k γ † -k,b , (C.15) c † -k,b = -v * k γ k,b + u k γ † -k,b , (C.16)
where k > 0 and

u k = 1/2 (1 + ξ k /E kb ) v k = 1/2 (1 -ξ k /E kb )e -iφ b (C.17)
with φ b the phase of the superconducting condensate (that we choose = 0 from now on)

and E kb = ξ 2 k + ∆ 2 sin 2 k the spectrum. We can then express the bulk Hamiltonian in terms of the γ k and γ -k operators:

H b = k>0 E kb γ † k,b γ k,b + γ † -k,b γ -k,b . (C.18)
Utilizing the fact that:

c k,w (t) = c k,w (0) exp(-iξ k,w t) , (C.19) and γ k,b (t) = γ k,b (0) exp(-iE k,b t) , (C.20)
we can readily find the transformation matrix S SW as follows (assuming also that ξ k,w E k,b , since we are interested in the energies well inside the band gap of the bulk superconductor):

S SW = k t int E kb |u k | 2 -|v k | 2 (c † k,w c k,b -c † k,b c k,w ) -2u k v k (c † k,w c † -k,b -c -k,b c k,w ) . (C.21)
Utilizing this expression for S SW , we can calculate the expectation values for the different commutators in Eq. (C.14). We obtain:

H ind,w ≡ 1 2 0 b | [S SW , H T + 2H c-w ] |0 b ≈ - k t 2 int E kb × |u k | 2 -|v k | 2 c † k,w c k,w -2u k v k c † -k,w c † k,w + h.c. , (C.22)
which can be interpreted as follows: the first term renormalizes the single particle spectrum in the wire, while the second term is responsible for the induced superconductivity in the wire. The full wire Hamiltonian thus becomes:

H eff w = k (ξ k,w + δξ k,w ξ eff k,w )c † k,w c k,w + 2 k ∆ ind c † k,w c † -k,w + h.c + α k c † k,w c k,w a + a † + α 2 ω c N2 w (C.23) with δξ k,w = t 2 int E kb |u k | 2 -|v k | 2 = t 2 int ξ k,b E 2 kb , (C.24) ∆ ind = t 2 int E kb u k v k = t 2 int 2E 2 kb ∆ sin k (C.25)
being the renormalization of the single-particle energies and the p-wave induced gap (∝ sin k). Note that the last term in Eq. (C.23) can be seen as a normalization of the single-particle spectrum in the mean-field, and thus finally we recover the same wire Hamiltonian defined in Eq. (5.4).

c.2 the susceptibility for the kitaev model in case of periodic boundary conditions

In this section, we give more details on the derivation of the susceptibility Π(ω) for the case of a ring geometry for which we can apply PBCs. The Kitaev chain Hamiltonian in real space was defined in Eq.(5.4) as:

H tot = H el + H el-c + H ph , (C.26) H el = -µ N i=1 c † i c i - 1 2 N-1 i=1 (tc † i c i+1 + ∆c i c i+1 + h.c.) , H int = α N i=1 c † i c i (a + a † ) , (C.27)
and H ph = ω c a † a. In this case, we can switch to the Fourier space, which implies we can write the fermionic operators as follows:

c j = 1 √ N k c k e ikj . (C.28)
where c k is the fermionic annihilation operator with momentum k = 2πn/N. We can then rewrite the electronic Hamiltonian in momentum space

H el = k>0 H K BdG (k) , (C.29) with H k BdG (k) = ξ k (c † k c k -c -k c † -k ) -i∆ sin k(c -k c k -c † k c † -k ), (C.30)
is the Bogoliubov de Gennes Hamiltonian and ξ k = -t cos k -µ. The interaction Hamiltonian between the electronic system and the cavity can as well be written in the k-space as

H el-c = k>0 α(c † k c k -c -k c † -k )(a + a † ). (C.31)
One can simply diagonalize the H el in the k-space and write:

H el = k>0 E k γ † k γ k + γ † -k γ -k , (C.32) with E k = ± (-t cos k -µ) 2 + (∆ sin k) 2 , (C.33)
being the eigenenergies, and we used

c k = u * k γ k + v k γ † -k , (C.34) c † -k = -v * k γ k + u k γ † -k . (C.35)
Here, the functions u k and v k are given by

|u k | 2 = 1 2 1 + ξ k E k , (C.36) |v k | 2 = 1 2 1 - ξ k E k . (C.37)
We can now calculate the susceptibility Π(ω) defined in the previous section, which quantifies the change in the photonic transmission τ due to the interaction with the electronic system. In the Fourier space, we obtain:

Π(t) = -iθ(t)α 2 k,q>0 c † k c k -c -k c † -k (t), c † q c q -c -q c † -q (0) . (C.
Π(ω) = α 2 k>0 (∆ sin k) 2 E 2 k 1 ω -2E k + iη - 1 ω + 2E k + iη , (C.39)
with the small η > 0 assuring the convergence of the time-integrals. For large N 1, we can transform the sum into integral, and also write Π(ω) = Π (ω) + iΠ (ω), with:

Π (ω) = 2Nα 2 π P π 0 dk (∆ sin k) 2 E k 1 ω 2 -4E 2 k , (C.40) Π (ω) = Nα 2 2 π 0 dk (∆ sin k) 2 E 2 k [δ(ω -2E k ) -δ(ω + 2E k )] , (C .41)
where P . . . means the principal value of the function and we used the fact that:

1 x -a + i = P 1 x -a -iπδ(x -a) . (C.42)
We can perform the integral over k for the imaginary part Π (ω) to obtain the expression Eq.(5.9):

Π (ω) = α 2 tN 2µω 1 - (ω/2) 2 -t 2 -µ 2 2 4t 2 µ 2 , (C.43)
for |t + µ| < ω/2 < |t -µ|, and being zero otherwise. For the real part Π (ω) we found no simple solution, and so we chose not to depict it. Note that the susceptibility Π(ω) ∝ N, i.

e. it scales linearly with the number of sites.

The projection onto σ z = +1

In this case the effective Hamiltonian linearized around p = 0 reads

Heff = p 2 2m * -µ + B τ z + ∆up B τ x , (C.68)
where an effective mass m * was introduced as

1 m * = 1 m + u 2 B . (C.69)
The bulk energy spectrum reads

p = ± B -µ + p 2 2m * 2 + ∆ 2 u 2 p 2 B 2 ≈ ± (B -µ) 2 + cp 2 , (C .70)
where

c = B -µ m * + ∆ 2 u 2 B 2 . (C.71)
The density of state reads

ρ( ) = c [ 2 -(B -µ) 2 ] . (C.72) And sin(γ) 2 ( ) = ∆ 2 u 2 2 -(B -µ) 2 B 2 2 c . (C.73)
where M is a 2N × 2N matrix. Moreover, assuming all entries in the matrix are real, M is also symmetric, we can write it as follows:

M = PWP † , (C.79)
where W is a diagonal matrix with eigenvalues on its diagonal and P is a unitary matrix (PP † = P † P = I) whose columns are eigenvectors of M. The matrix W is ordered so that

W =            1 0 • • • 0 0 0 -1 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • N 0 0 0 • • • 0 -N            , (C.80)
with ± n being the eigenenergies of the BdG Hamiltonian and n = 1, . . . N. That pertains to the following diagonal Hamiltonian:

H 1D = N m=1 m c † m cm - 1 2 , (C.81)
where cm ( c † m ) are the annihilation (creation) operators for the Bogoliubov quasiparticles, which are defined as follows:

c = P † c (C.82) and c = ( c1 , c † 1 , c2 , c † 2 , . . . , cN , c † N ) T . (C.83) It is instructive to introduce the wavefunctions ψ k (i) = (u k,i , v k,i ) T , where u k,i (v k,i ) = p 2i-1,k (p 2i,k
), and which are describing the state k = 0, . . . , 2N at position i = 1, . . . , N in the lattice and accounts for the electron (u) and hole (v) components, respectively. That allows us to write:

c i = N k=1 u 2k-1,i ck + u 2k,i c † k , (C.84) c † i = N k=1 v 2k-1,i ck + v 2k,i c † k . (C.85)
so that we can rewrite Eq. (C.75) in terms of u k,i and v k,i as follows:

Π(ω) = N i,j,k,m=1 (1 -n k )(1 -n m )v 2k-1,i u 2m-1,i × u 2k,j v 2m,j -v 2k,j u 2m,j × 1 ω + iη -2k-1 -2m-1 - 1 ω + iη + 2k-1 + 2m-1 + (1 -n k )n m v 2k-1,i u 2m,i u 2k,j v 2m-1,j -v 2k,j u 2m-1,j × 1 ω + iη -2k-1 + 2m-1 - 1 ω + iη + 2k-1 -2m-1 + n k (1 -n m )v 2k,i u 2m-1,i u 2k-1,j v 2m,j -v 2k-1,j u 2m,j × 1 ω + iη + 2k-1 -2m-1 - 1 ω + iη -2k-1 + 2m-1 + n k n m v 2k,i u 2m,i u 2k-1,j v 2m-1,j -v 2k-1,j u 2m-1,j × 1 ω + iη + 2k-1 + 2m-1 - 1 ω + iη -2k-1 -2m-1 . (C.86)
Let us now introduce the coefficients C (s) , s = 1, 2 defined in Eq.(5.24):

C (s) = N i=1 (u 2M,i δ s,1 + v 2M,i δ s,2 )u 2k,i -(u 2M,i δ s,2 + v 2M,i δ s,1 )v 2k,i , (C.89)
which we can utilize to rewrite Π BM (ω) as follows:

Π BM (ω) = N-1 k=1 1 2k + ω + iη + 1 2k -ω -iη × (n M -n k ) C (1) 2 -(n M -1 + n k ) C (2) 2 , (C.90)
and which correspond to the expression Eq. (5.23).

D

A P P E N D I X D

In this appendix, we will derive the electronic susceptibility for the chain that can host multiple Majorana fermions in case of periodic boundary conditions, c N+1 ≡ c 1 .

c j = 1 √ N k c k e ikj , (D.1) c † j = 1 √ N k c † k e -ikj . (D.2)
Then Eq.( 6.1) in the k-space reads

H = k>0 H k (k), (D.3)
where

H k (k) = c † k c -k   -2µ -2t 1 cos(k) -2t 2 cos(2k) 2i∆ 1 sin(k) + 2i∆ 2 sin(2k) -2i∆ 1 sin(k) -2i∆ 2 sin(2k) 2µ + 2t 1 cos(k) + 2t 2 cos(2k)     c k c † -k   (D.4) τ x = c -k c k + c † k c † -k , (D.5) τ y = -i(c † k c † -k + c k c -k ), (D.6) τ z = c † k c k -c -k c † -k . (D.7) H k (k) = [-2µ -2t 1 cos(k) -2t 2 cos(2k)]τ z + [-2∆ 1 sin(k) -2∆ 2 sin(2k)]τ y . (D.8)
Let us diagonalize the Hamiltonian H k (k) by performing a unitary transformation

Hk (k) = U † H k (k)U = E k τ z (D.9) with U = exp(-iθ k τ x /2) = cos(θ k /2) -iτ x sin(θ k /2). (D.10) tan(θ k ) = - ∆ 1 sin(k) + ∆ 2 sin(2k) µ + t 1 cos(k) + t 2 cos(2k) . (D.11)
The winding number that measures the number of Majorana fermions reads

w = 2π 0 dθ k 2π . (D.12)
Energy is

E k,± = ±2 (µ + t 1 cos(k) + t 2 cos(2k)) 2 + (∆ 1 sin(k) + ∆ 2 sin(2k)) 2 . (D.13)
The condition for the topological phase transition is closing of the gap, E k = 0.

E k (k = 0) = 2(t 1 + t 2 + µ) = 0 ⇒ t 2 = -µ -t 1 , (D.14) E k (k = π) = 2(-t 1 + t 2 + µ) = 0 ⇒ t 2 = -µ + t 1 , (D.15)
Electronic susceptibility reads

Π(t) = -iθ(t) k,q>0 c † k (t)c k (t) -c -k (t)c † -k (t), c † q (0)c q (0) -c -q (0)c † -q (0) , (D.16) or Π(t) = -iθ(t) k,q>0
τ z (t), τ z (0) . (D.17)

Applying unitary transformation Eq.(D.10) to τ z , electronic susceptibility becomes

Π(t) = -iθ(t) τ z (t) cos(θ k ) + τ y (t) sin(θ k ), τ z (0) cos(θ k ) + τ y (0) sin(θ k ) . (D.18)
Using Eq.(D.9),

τ z (t) = τ z (0), (D.19) τ y (t) = e iE k τ z t τ y (0)e -iE k τ z t = cos(2E k t)τ y (0) + sin(2E k t)τ x (0). (D.20)
Then assuming that τ z (0) = -1, the electronic susceptibility reads

Π(t) = -2θ(t) sin 2 (θ k ) sin(2E k t). (D.21)
Performing the Fourier transform, the susceptibility Π(ω) becomes L'Hamiltonien du réservoir :

Π(ω) = k>0 sin 2 (θ k ) 1 ω -2E k + iη - 1 ω + 2E k + iη . ( D
Π(ω) = N 2π π 0 dk 4(∆ 1 sin(k) + ∆ 2 sin(2k)) 2 E 2 k 1 ω -2E k + iη - 1 ω + 2E k + iη . (D.25) E S Y N T H È S E E N F R A N Ç A I S
H b = q ω q b † q b q , (E.2)
où b q (b † q ) sont les opérateurs d'annihilation (création) pour le mode de réservoir ayant énergie ω q , avec leurs numéros quantiques étiquetés q. L'Hamiltonien de la cavité est

H c = ω c a † a, (E.3)
où a (a † ) est l'opérateur d'annihilation (création) pour le mode de cavité ayant énergie ω c . L'Hamiltonien cavité-réservoir est H c-b = -i q f q a † b q -f * q b † q a , (E.4) où les coefficients complexes f q sont des paramètres de couplage entre la cavité et le réservoir extérieur. L'Hamiltonien électron-cavité est H el-c = α(a + a † )n, (E.5) où α est la puissance de couplage entre le champ de la cavité et l'opérateur de nombre total dans le système n ≡ N j=1 c † j c j , avec c j (c † j ) étant l'opérateur d'annihilation (création) pour les électrons (degrés de liberté fermioniques) sur le site j dans le système électronique. Finalement, H el est l'Hamiltonien uniquement électronique. L'idée principale de la théorie des entrées-sortie est de trouver les photons (ou le champ) de sortie en termes de ceux d'entrée, comme nous montrons schématiquement sur la V z @meVD P'Hw,V z Lêa 2

Figure 2 . 1 :

 21 Figure 2.1: Scheme and equivalent lumped circuit representation of implementation of cavity QED using superconducting circuits. The 1D transmission line resonator consists of a full-wave section of superconducting coplanar waveguide. A Cooper-pair box qubit is placed between the superconducting lines and is capacitively coupled to the center conductor at a maximum of the voltage standing wave, yielding a strong electric dipole interaction between the qubit and a single photon in the cavity. The Cooper pair box consists of two small Josephson junctions, configured in a loop to permit tuning of the effective Josephson energy by an external flux Φ ext . Input and output signals are coupled to the resonator, via the capacitive gaps in the center conductor, from transmission lines which allow measurements of the amplitude and phase of the cavity transmission.Reproduced from[START_REF] Blais | Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[END_REF].

  2.3, b) thus allowing to probe the phase and the amplitude of the microwave probe beam transmitted through the resonator. The transmission of the cavity is changed due to the presence of the Cooper pair box inside the cavity.

Figure 2 . 2 :

 22 Figure 2.2: Integrated circuit for cavity QED. a) The superconducting niobium coplanar waveguide resonator is fabricated on chip using optical lithography. The stripline resonator consists of the center conductor and two lateral ground planes. The meandering resonator is coupled by a capacitor at each end of the resonator (see b) to an input and output feed line, fanning out to the edge of the chip and keeping the impedance constant. b) The capacitive coupling to the input and output lines and hence the coupled quality factor Q is controlled by adjusting the length and separation of the finger capacitors formed in the center conductor. c) False color electron micrograph of a Cooper pair box (blue) fabricated onto the silicon substrate (green) into the gap between the center conductor (top) and the ground plane (bottom) of a resonator (beige). The Josephson tunnel junctions are formed at the overlap between the long thin island parallel to the center conductor and the fingers extending from the much larger reservoir coupled to the ground plane. Reproduced from [101].

Fig. 2 .

 2 4, a). Coulomb diamonds as well as the Kondo ridge at zero bias associated with the Kondo effect have been observed. The variation of the phase δφ of the microwave signal transmitted through the cavity near the cavity resonance frequency f = 4.976 GHz for the same parameters is presented in Fig.2.4, b). Comparing the differential conductance dI/dV and the phase shift δφ as a function of the source-drain bias V sd over the Kondo ridge (see Fig.2.4,c), we can see these two quantities look very similar. Thus, this experiment demonstrated that photonic observable, such as the phase shift δφ, gives access to the properties of the quantum dot obtained by measuring the transport quantity, such as dI/dV. However, the explanation why the phase shift and the differential conductance have a similar behaviour as a function of voltage bias was not given. Also, in this work[START_REF] Delbecq | Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip[END_REF] the value of the electron-photon coupling λ for carbon nanotube constant has been measured and is equal to 140 MHz. We use this estimate for the coupling constant when giving quantitative estimate of the phase shift in chapter 5.

Figure 2 . 3 :

 23 Figure 2.3: (a) Scanning electron microscope (SEM) picture in false colors of the coplanar waveguide resonator. Both the typical coupling capacitance geometry of one port of the resonator and the 3-terminals geometry are visible. (b) False colors SEM picture of a single wall carbon nanotube dot inside an on-chip cavity embedded in a schematics of the measurement setup. Reproduced from [29].

Figure 2 . 4 :

 24 Figure 2.4: (a) Color scale plot of the differential conductance in units of 2e 2 /h measured along three charge states exhibiting the conventional transport spectroscopy. A Kondo ridge is visible at zero bias. (b) Color scale plot of the phase of the microwave signal at resonance frequency f = 4.976 GHz, measured simultaneously with the differential conductance. (c) Differential conductance (black line) and phase of the transmitted microwave signal (red line) at the resonance frequencyf = 4.976 GHz as a function of source-drain bias V sd over a Kondo ridge in between two Coulomb peaks. Reproduced from [29].

Figure 2 . 5 :

 25 Figure 2.5: Sketch of a cavity system. The input fields b in and c in are sent from the right and left mirrors, respectively, towards the cavity, and an output fields b out and c out are collected on the same sides. The cavity field, which is quantified by the bosonic operators a and a † , interacts with the electronic system via capacitive coupling, affecting the cavity. The coupling between the cavity field and the external modes is quantified by the decay rate κ 1(2) = 2πρ|f 1(2) | 2 , with ρ the bath density of states and |f 1(2) | 2 ≡ f 1(2) f *

Figure 3 . 1 :

 31 Figure 3.1: Exchange of two Majorana fermions γ 1 and γ 2 counterclockwise.

Figure 3 . 2 :

 32 Figure 3.2: Scheme of the Kitaev chain.

Figure 3 . 3 :

 33 Figure 3.3: Bulk energy spectrum E k of the Kitaev chain for parameters: t = ∆ = 1. a) The gap at k = 0 is open at µ = -0.5 in the topological phase (|µ| < t), b) the gap at k = 0 closes at µ = -1.0 at the topological phase transition (|µ| = t) , c) the gap at k = 0 opens again at µ = -1.5 in the non-topological (trivial) phase (|µ| > t).

Figure 3 . 4 :

 34 Figure 3.4: The pseudospin vector ĥ(k) for parameters: t = ∆ = 1. a) µ = -0.5; the pseudospin winds once along the Brillouin zone, i.e. w = -1, b)µ = -1.5; the pseudospin does not wind along the Brillouin zone, i.e. w = 0.

Figure 3 . 5 :

 35 Figure 3.5: Schematic illustration of the Hamiltonian in Eq.3.32. Majorana operators γ A,j , γ B,j from the same site j are paired.

Figure 3 . 6 :

 36 Figure 3.6: Schematic illustration of the Hamiltonian in Eq.3.33. Majorana operators γ B,j , γ A,j+1 from adjacent sites are paired. Majorana operators γ A,1 and γ B,N drop out of the Hamiltonian.

Fig. 3 .

 3 Fig.3.6). Let us introduce a new fermionic operator that combines the Majorana operators from different sites

Figure 3 . 7 :

 37 Figure 3.7: Scheme of the superconductor-semiconductor wire setup. 1D wire lies along the y axis, spin-orbit interaction u is along z axis and a magnetic field B is along x axis.

Figure 3 . 8 : 2 .

 382 Figure 3.8: Bulk energy spectrum E ± (p) of the spin-orbit coupled nanowire for parameters µ = 0 and ∆ = 0.25. The momentum (energy) scale is set by mu mu 2 2 . The blue line corresponds to the upper energy branch E + and the red line corresponds to the lower energy brunch E -. a) The gap at p = 0 is open for B = 0.5 in the topological phase, B > ∆ 2 + µ 2 , b) the gap at p = 0 closes at B = 0.25 at the topological phase transition point, B = ∆ 2 + µ 2 .

  In order to satisfy this relation, one needs to create Zeeman energy E z = gµ B B/2 (µ B is the Bohr magneton) large enough to exceed the induced superconducting gap ∆ ≈ 250µeV. Choosing bulk InSb with g ≈ 50, one can enter a topological regime at µ = 0 (E z > ∆) given the magnetic filed B ≈ 0.15T. At the same time, InSb nanowires have a strong spin-orbit interaction α = 0.2 eV•Å.

Figure 3 . 9 :

 39 Figure 3.9: a) Experimental setup. Scanning electron microscope image of the device with normal (N) and superconducting (S) contacts. The S contact only covers the right part of InSb nanowire. The underlying gates, numbered 1 to 4, are covered with a dielectric. An external magnetic field B is aligned parallel to the wire in order not to destroy superconductivity. The Rashba spin-orbit interaction is perpendicular to the nanowire and to magnetic field as required. Parameters of the wire: α = 0.2 eV•Å, ∆ ≈ 250µeV. b) dI/dV as a function of V for different values of magnetic field B: from 0 to 490 mT from in 10 mT steps. Green arrows indicate the induced gap peaks. Reproduced from [64].

  a phase of the left (right) superconductor. In case of topological superconductors (see Fig.3.10) the supercurrent appears due to fusing the Majorana fermions across a weak link. The current is carried by charge e quasiparticles instead of the Cooper pairs and is

. 57 ) 2 . 2 .Figure 3 . 11 :

 5722311 Figure 3.11: Energy spectrum E as a function of the phase difference across the junction ∆φ. Energy comes back to its initial value at 4π.

Figure 3 .

 3 Figure 3.12: a) Experimental setup. AFM image of a Josephson junction. The light areas are Nb. A light brown halo around Nb is a thin 2-3 nm Nb layer which defines the width of the semiconductor wire. Green arrow indicated the direction of the spin-orbit field B SO . b) I -V characteristic of a Nb/InSb/Nb Josephson junction in the presence of magnetic field B I measured with voltage V rf between 3 and 6 mV with 0.6 mV steps. For B < 2 T, Shapiro steps with a height ∆V = hω 0 /2e = 6µV are clearly observed. For B > 2 T the plateau at 6µV disappears, as emphasized by dashed ovals, and the first step is observed at hω 0 /e = 12µV. This doubling of the first Shapiro step is a signature of the presence of the Majorana fermions in the system. Reproduced from [81].

  ) characterized by the capacitive coupling β L (β R ) of the left (right) lead to the cavity. The tunneling amplitude t being independent of energy is equivalent to instantaneous electron scattering. As a result, no charge can accumulate in the junction, the total charge in the leads N = n R + n L is constant, and the current is conserved I R = -I L . The different charge susceptibilities are related to each other

Figure 4 . 2 :

 42 Figure 4.2: Π (ω) ≡ Π (ω)/ (β L -λ) 2 (left) and Π (ω) ≡ Π /(β L -λ) 2 (right) as a function of eV and d for β L = β R , Γ L = Γ R = 0.5 and ω = 0.1. All energies are expressed in terms of Γ = Γ L + Γ R .

Fig. 4 . 3 .Figure 4 . 3 : 4 . 4 . 4 Figure 4 . 4 :Figure 4 . 5 :

 43434444445 Figure 4.3: Π (ω) ≡ Π (ω)/γ (β R -λ) 2 (left) and Π (ω) ≡ Π /γ(β R -λ) 2 (right) as a function of eV and d for Γ L = Γ R = 0.5, γ = 0.5, and ω = 0.1. All energies are expressed in terms of Γ = Γ L + Γ R .

Figure 5 .

 5 Figure 5.1: A sketch of the system: a one dimensional system (red rectangle) is placed at the maximum of the electrical field (green straight arrows) inside a superconducting microwave cavity (blue). The electromagnetic field inside the cavity is probed by sending input fields of amplitude and phase A in and φ in , respectively, and measuring the field at the end with A out and φ out . The difference between the two gives a direct access to the electronic correlation function in the wire (see text). The presence of Majorana end modes in the finite wire (black curves) is also signaled in the cavity response.

  to be capacitively coupled to the microwave cavity. The tight-binding Hamiltonian H

Figure 5 . 2 :

 52 Figure 5.2: The imaginary part of the electronic susceptibility [Π (ω)] as a function of µ for the Kitaev model. The topological phase transition takes place at µ = -1, where this function reaches its maximum, indicating the transition point. Inset: The real part of the electronic susceptibility [Π (ω)], which also shows features (kinks) around the topological phase transition point. The full, dashed, dot-dashed curves correspond to the ω = 0.2, 0.3, and 0.4, respectively. We assumed t = ∆ = 1, N = 50 and all energies are expressed in terms of t.

Figure 5 . 3 :

 53 Figure 5.3: The imaginary part of the electronic susceptibility [Π (ω)] as a function of the Zeeman splitting V Z for a SO coupled nanowire in case of PBCs for parameters L = 2 µm, t = 0.5 • 10 -2 eV, α = 0.4 meV, µ = -10 -2 eV, ∆ = 0.25 meV, N = 80, and ω = 0.1 meV (ω = 0.2 meV in the Inset). The topological transition takes place for V Z ≈ 0.25 meV, for which the susceptibility reaches its maximum. The emergence of a second peak is due to the resonance condition around the external gaps (k ∼ k F ).

2 µFigure 5 . 4 :

 254 Figure 5.4: Dependence of Π (ω, µ) on the chemical potential µ. The blue (dot-dashed), red (full),and the black (dashed) lines correspond to the susceptibility for PBCs, OBCs for parity n M = 0, and OBCs for parity n M = 1, respectively. Lower Inset: the relative strength of the susceptibility∆Π = 2 (Π + BM -Π - BM )/(Π + BM + Π - BM )as a function of µ in logarithmic scale, for ∆ = 0.1 (red-full), ∆ = 0.2 (green-dashed), and ∆ = 0.3 (blue-dotdashed). The size of ∆Π is exponentially reduced as a function of ∆. We used N = 50, ω = 0.2, ∆ = 0.1, t = 1, and all energies are expressed in terms of t.

Figure 5 . 5 :

 55 Figure 5.5: Dependence of Π (ω, V Z ) on the Zeeman splitting V Z . Above the topological transition (indicated in the figure), the red (full) and black (dashed) curves correspond to Π (ω, V Z ) for parity n M = 0 and n M = 1, respectively. Π (ω, V Z ) shows oscillations as a function of V Z , that are different in amplitude for the two parities n M = 0, 1 (but having the same period), crossing at points where M = 0 (see Inset). Below the topological transition the susceptibility reproduces well the one for PBCs. The parameters we took are [8]: L = 2µm, t = 0.5 • 10 -2 eV, α = 0.4meV, µ = -10 -2 eV, ∆ = 0.25meV, ω = 0.02meV, and N = 80.

2 ωFigure 5 . 6 :

 256 Figure 5.6: Dependence of Π (ω) on the cavity frequency ω for N = 50. The full-red (dashedblack) line corresponds to the susceptibility in the topological regime for µ = -0.2, while the dot-dashed-blue (dotted-green) corresponds to the non-topological regime with open (periodic) BCs with µ = -1.8, so that the effective gap is the same ∆ eff = ||µ| -t| = 0.8 in both regions. Inset: a zoom in the region where the Majorana peak emerges. For all the plots we used t = ∆ = 1, and all energies are expressed in terms of t.

Fig 2 Figure 6 . 2 :Figure 6 . 3 :

 26263 Fig.6.4).

Figure 6 . 4 :

 64 Figure 6.4: Energy spectrum as a function of chemical potential µ in case of open boundary conditions for parameters: N = 100, t 1 = 1, t 2 = 1.5,∆ 1 = ∆ 2 = 0.2. Red line corresponds to the lowest-energy state, first Majorana fermion (M 1 ) and blue line corresponds to the second lowest-energy state, second Majorana fermion (M 2 ). Phase transition point is µ = -0.5.

Figure 6 . 5 :

 65 Figure 6.5: a) Imaginary part Π (ω) as a function of µ in case of periodic boundary conditions for parameters: N = 100, t 1 = 1, t 2 = 1.5, ∆ 1 = ∆ 2 = 0.2, ω = 0.15. Numbers in the plot indicate the number of the Majorana fermions in the region between the phase transitions obtained by computing the corresponding winding number. b) Bulk energy spectrum E k,± as a function of k in case of periodic boundary conditions for parameters: t 1 = 1, t 2 = 1.5, ∆ 1 = ∆ 2 = 0.2. Red line corresponds to µ = -2.5 (gap closes at k = 0, which corresponds to the topological phase transition from 0 to 1 Majorana fermions), green line corresponds to µ = -0.5 (gap closes at k = π, which corresponds to the topological phase transition from 1 to 2 Majorana fermions) and blue line corresponds to µ ≈ 1.25 (which corresponds to the topological phase transition from 2 to 0 Majorana fermions).

6 . 3 . 2

 632 Fig.6.5, a)) that in the energy spectrum corresponds to the gap closing at k = 0 (see Fig.6.5, b)). This value of the chemical potential corresponds to the topological phase transition from 0 to 1 Majorana fermion (see Fig.6.3). Then the maximum of Π (µ) at µ = -0.5 corresponds to the energy gap closing at k = π. µ = -0.5 corresponds to the topological phase transition from 1 to 2 Majorana fermions. And finally the double peak in the susceptibility at µ ≈ 1.25 corresponds to the topological phase transition from 2 to 0 Majorana fermions.

Figure 6 . 6 :

 66 Figure 6.6: Electronic susceptibility as a function of frequency ω in case of open boundary conditions for parameters: N = 100, t 1 = 1, t 2 = 1.5, ∆ 1 = ∆ 2 = 0.2, η = 0.001 and µ = -0.1. First bulk energy level is at B1 = 0.17. a) Real part of the susceptibility Π (ω) has a kink at ω = M1 + M2 = 0.008 (n M 1 = n M 2 = 0). b) Imaginary part of the susceptibility Π (ω) has a peak at ω = M1 + M2 = 0.008 (n M 1 = n M 2 = 0).

6 - 4 . 8 - 4 . 7 - 4 . 6 - 4 . 5 -Figure 6 . 7 :

 64847464567 Figure 6.7: Real part of the total susceptibility Π (ω) as a function of chemical potential µ in case of open boundary conditions for parameters:N = 100, t 1 = 1, t 2 = 1.5, ∆ 1 = ∆ 2 = 0.2, η = 0.01 and ω = 0.05. Black line corresponds to n M 1 = 0, n M 2 = 0, red line corresponds to n M 1 = 1, n M 2 = 0, blue line corresponds to n M 1 = 0, n M 2 = 1 and green line corresponds to n M 1 = 1, n M 2 = 1.Topological phase transition point is at µ = -0.5. For µ < -0.5 there is one Majorana fermion at each end of the chain, while for µ > -0.5 there are two Majorana fermions at each end of the chain.

Figure 7 . 1 :

 71 Figure 7.1: Top: A sketch of the combined topological superconductor fluxed-biased by the cavity.This is probed in the input-output method, and the output signal bout is to be compared with the input one bin . Bottom: a) A zoom into the topological superconductor in the shape of a ring. The magnetic field of the cavity Bcav pierces the ring and gives rise to a fluctuating magnetic flux. A perpendicular, external magnetic field B ext is also applied, so that the total flux felt by the electrons in the ring isΦ tot = Φ ext + Φcav .The by a weak link, that pertains to the hopping strength t t (t is the hopping strength within the ring, see the text). In the topological region, there are Majorana fermions on the left (γ L ) and on the right (γ R ) of the superconductor, and a Josephson current I J is flowing within the ring. b) A possible implementation of the geometry in a) based on a spin-orbit nanowire flux-biased in a SQUID geometry. Depicted in light red is a conventional superconducting loop that is interrupted so that there is the Josephson current flowing only through the wire.

Figure 7 . 2 :

 72 Figure 7.2: The spectrum of the ring as a function of the flux Φ in the non-topological (left) and topological (right) regions, respectively. In red are depicted the Majorana energy levels that show oscillations with a period Φ 0 .

7 . 3 ,

 73 we plot separately the different contributions to the susceptibility measured by the cavity, as a function of the external flux: the Josephson current contribution (top), the real (middle) and the imaginary (bottom) non-diagonal contributions both in the topological (left column) and the non-topological phases (right column). We see that in the

Figure 7 . 3 :

 73 Figure 7.3: Left column (right column), from top to bottom: Dependence of the Josephson, nondiagonal real, non-diagonal imaginary susceptibilities respectively, in the topological (non-topological) region on the magnetic flux Φ/Φ 0 for f M = 0. The red, green, and blue lines correspond to µ = -1.5 (µ = -2.5), µ = -1 (µ = -3), and µ = -0.5 (µ = -3.5), respectively. The phase transition point is at µ = -2. The parameters for the plots are ∆ = 0.1, t = 0.2, N = 50, and all energies are expressed in terms of t = 1.

2 Figure 7 . 4 :

 274 Figure 7.4: Dependence of the Josephson, non-diagonal real, and non-diagonal imaginary susceptibilities, respectively, on the magnetic flux Φ and parity f M . The dashed-black (full-red) curves correspond to parity f M = 0 (f M = 1). The oscillations of the susceptibilities for the two parties are both shifted by Φ 0 , as well as asymmetric in amplitudes (once superimposed). The parameters for the plots are µ = -0.5, ∆ = 0.1, t = 0.2, N = 50, and all energies are expressed in terms of t = 1.

2 Figure 7 . 5 :

 275 Figure 7.5: Dependence of the real part of the non-diagonal susceptibility on the frequency ω for two values of the external flux, Φ = 0 (left) and Φ = Φ 0 /2 (right), and for the two parity states f M = 0 (dashed-black) and f M = 1 (full-red) lines, respectively. The parameters for the plots are: µ = -0.5, ∆ = 0.1, t = 0.5, N = 50, and all energies are expressed in terms of t = 1.

C

  O N C L U S I O N S A N D P E R S P E C T I V E S

  tackling the broader problem of theoretically describing the transition from the classical regime to the quantum regime: we can study the creation of Majorana polariton by applying a classical AC field as well as by coupling it to a cavity. Extensively studying both the classical and quantum description as well as the relation between them may help us shed some extra light on the fundamental relation between classical physics and quantum mechanics.

B

  A P P E N D I X B b.1 scattering matrix approach for a non-interacting quantum conductor

8 )

 8 we obtain Eq. (4.25) and Eq. (4.24). b.2 calculation of the quantum dot susceptibility b.2.1 Expressions of the intermediate F γγ αβ functions

29 )

 29 The functions K αβ directly provide Π αβ .b.2.3 Calculation of the real part of the susceptibilityWe give technical details on how to compute the real part of the charge susceptibility. Let us compute one of the integralsL(y) = 2iP ∞ -∞ dx x 2 (x + y)(x 2 + 4) f(x), (B.30) where f(x) = arctan(a + x)arctan(a -x). (B.31)

34 )

 34 L 1 (y) can be computed with standard complex plane integration techniques. The integrand has five singularities in the complex plane: poles at z = 0, z = -y, z = -2i, z = 2i and a branch point at z = i -a. A semi-circle is added around z = 0 and z = -y. The contour is closed in the lower half-plane, avoiding both the pole at z = 2i and the branch cut [Fig. B.1].

Figure B. 1 :

 1 Figure B.1: Complex plane z with poles of the integrand, Eq.(B.34).

2 L

 2 πΓ 3 y y 2 + 4[arctan (a + y)arctan (a -y)]

A

  of the effective kitaev hamiltonian in the presence of the cavity fieldIn this section, we provide theoretical arguments for the wire Hamiltonian utilized in Eq.(5.4), and the effective electron-cavity Hamiltonian used. In a continuum description, the natural way to account for the interaction between the electrons and the electromagnetic field is via the minimal coupling, i.e. p → p -(e/c)A in the electronic Hamiltonian, with A being the electromagnetic field vector potential and p being the momentum of the electrons in the material. In a tight-binding picture instead, one accounts for the coupling between light and matter by performing the Peierls substitution to the hopping parameters t ii+1 between neighboring sites i and i + 1, namely t ii+1 → t ii+1 e i i+1 i

. 22 )

 22 Using Eq.(D.11), sin 2 (θ k ) = 4(∆ 1 sin(k) + ∆ 2 sin(2k))

2 Figure E. 1 :

 21 Figure E.1: Schéma d'un système de cavité. Les champs d'entrée b in et c in sont envoyés vers la cavité depuis les miroirs gauche et droit respectivement, et les champs de sortie b out and c out sont mesurés des mêmes côtés. Le champ de la cavité, quantifié par les opérateurs bosoniques a et a † , interagit avec le système électronique à travers le couplage capacitif affectant la cavité. Le couplage entre le champs de la cavité et les modes externes est quantifié par le taux de décroissance κ 1(2) = 2πρ|f 1(2) | 2 où ρ est la densité des états dans le réservoir et |f 1(2)| 2 ≡ f 1(2) f * 1(2)est le couplage moyen entre la cavité et les modes de réservoir à droite (gauche). Afin de simplifier la première partie, nous supposons que κ 2 = 0, ce qui correspond à une cavité unilatérale.

Fig

  Fig. E.1. D'après Ref. [20], étant donné la cavité unilatérale (κ 2 = 0 in Fig. E.1), nous obtenons, dans l'équation de mouvement de cavité :ȧ = i [H sys , a] -κ 2 a -√ κb in , (E.6)

Figure

  Figure E.2: Π (ω) ≡ Π (β L -2 (à gauche) et Π (ω) ≡ Π /(β L -λ) 2 (à droite) en fonction de eV et d pour β L = β R , Γ L Γ R = 0.5 et ω = 0.1. Toutes les énergie sont exprimées en termes de Γ = Γ L + Γ R .

Figure E. 3 :

 3 Figure E.3: La partie imaginaire de la susceptibilité électronique [Π (ω)] en tant qu'une fonction de l'énergie Zeeman V Z pour un nanofil avec interaction spin-orbite dans le cas de conditions périodique aux limites ayant les paramètres L = 2 µm, t = 0.5 • 10 -2 α = 0.4 meV, µ = -10 -2 eV, ∆ = 0.25 meV, N = 80, et ω = 0.1 meV (ω = 0.2 meV dans l'encart). La transition topologique a lieu pour V Z ≈ 0.25 meV, pour lequel la susceptibilité atteint son maximum. L'émergence d'un deuxième pic est due à la condition de résonance autours de gaps extérieurs (k ∼ k F ).

Figure E. 4 :|Figure E. 5 :

 45 Figure E.4: Dépendance entre Π (ω, V Z ) l'énergie Zeeman V Z . Au dessus de la transition topologique (indiqué sur l'image), les courbes rouge (continue) et noire (pointillée) correspondent à Π (ω, V Z ) pour respectivement la parité n M = 0 et n M = 1. Π V Z ) comprenne des oscillations en tant que function de V Z . Ces oscillations ont des amplitudes différentes pour les parités n M = 0, 1 (mais les périodes égales) et se croise dans le points où M = 0 (voir l'encart). Au dessous de la transition topologique, la susceptibilité reproduit celle des conditions périodiques aux limites. Nos paramètres sont [8]: L = 2µm, t = 0.5 • 10 -2 eV, α = 0.4meV, µ = -10 -2 eV, ∆ = 0.25meV, ω = 0.02meV, and N = 80.

  Being lengthyand not essential for further discussion, the intermediate expressions for the functions F αβ γγ and K αβ (ω) are given in Appendix []. The functions K αβ directly provide Π αβ through Eq. 4.25. However, in order to obtain Π αβ , we need to perform a final tedious integration over energy in Eq. 4.24, the details of which are given in the Appendix B.2.The final expressions for the real and imaginary parts of the total susceptibility are given in Eq. B.48 and Eq. B.49, respectively, of the Appendix B.2. However, such expressions are intricate and difficult to interpret directly. Therefore, we consider some limiting cases

the width of the level. Using this expression for the S-matrix we can compute the functions A γγ (α, E, E ) defined in Eq. 4.16, then the sixteen functions F αβ γγ introduced in Eq. 4.22, and finally the functions K αβ (ω) defined in Eq. 4.23.

  Figure 6.1: Scheme of the extended Kitaev chain. t 1 is the nearest-neighbor hopping amplitude and t 2 is the next-nearest-neighbor hopping amplitude.

	t 1
	3
	t 2

The Hamiltonian of the Kitaev chain model with next-nearest-neighbor hopping and superconductivity reads

[START_REF] Niu | Majorana zero modes in a quantum Ising chain with longer-ranged interactions[END_REF]

  B.1)where A γγ have been defined in Eq. (4.16). We can now calculate Π αβ (ω) ≡ F αβ (ω) +

	X αβ (ω) with
	F αβ (t) = -iθ(t) n α (t)n β (0) ,	(B.2)
	X αβ (t) =iθ(t) n β (0)n α (t) .	(B.3)
	Let us first start with F αβ (ω) = -i	∞ 0 dte iωt n α (t)n β (0) . Using Eq. (B.1) and applying
	Wick's theorem, we obtain
		133

  Expressions of the intermediate K αβ functions

	b.2.2 Let us introduce the following dimensionless variables
	a =	eV -2 d Γ	, b =	2 d + eV Γ	, x =	2ω 2 Γ	and y =	2ω Γ	.	(B.25)
										24)

  Dans cette thèse, nous avons étudié la façon dont la mesure du champ micro-ondes de la cavité peut être utilisée pour extraire des informations sur les propriétés des nanostructures couplées à la cavité. Les expériences actuelles permettent de mettre en oeuvre des circuits mésoscopiques sur puce couplés à un résonateur à haut facteur de qualité et de mesurer de très petites variations de l'amplitude du signal optique micro-ondes. Nous démontrons que ces deux types de couplages entraînent une réponse de la cavité totalement différente et donc sondent différentes observables associées au conducteur quantique. Nous trouvons des relations entre les observables optiques, tels que la transmission de la cavité, et les quantités liées au transport, telles que la conductance différentielle dI/dV. Ces relations sont valables jusqu'à des valeurs élevées de la fréquence et de la tension pour les conducteurs quantiques loin de résonance (donc dans le régime hors équilibre). Et elles sont valables à basse fréquence et basse tension lorsque la boîte quantique est proche de la résonance. Au-delà de ces régimes, des mesures du champ photonique de la cavité fournissent de nouvelles informations sur le conducteur quantique.De plus, nous appliquons l'approche de la QED en cavité pour trouver des traces de fermions de Majorana dans les fils topologiques. Nous considérons un supraconducteur en onde p couplé capacitivement à une cavité micro-ondes. Le déphasage et l'élargissement en fréquence dû à la présence du supraconducteur en onde p dans la cavité est proportionnel à la susceptibilité électronique du fil. Les observables photoniques peuvent être utilisées pour obtenir des indices de la présence de fermions de Majorana, comme une transition de phase topologique ou des oscillations de la susceptibilité élec-

L'électrodynamique quantique (QED) en cavité est une technique expérimentale qui permet d'étudier et de manipuler des atomes qubits supraconducteurs et des boîtes quantiques couplées à une cavité micro-ondes. En utilisant la théorie des entrées-sorties dans le cas d'un couplage faible entre le système électronique et le champ photonique dans la cavité, nous avons relié le coefficient de transmission de la cavité à la susceptibilité électronique de la nanostructure. Le décalage de fréquence est proportionnel à la partie réelle de la susceptibilité électronique alors que l'élargissement de la fréquence est proportionnel à la partie imaginaire de la susceptibilité. Dans cette thèse, cette approche est appliquée à quatre conducteurs mésoscopiques spécifiques : une jonction tunnel, une boîte quantique couplée aux fils, un fil topologique à une dimension et un anneau supraconducteur.

Tout d'abord, nous utilisons la réponse photonique de la cavité pour étudier une jonction tunnel et une boîte quantique couplée par effet tunnel à des fils métalliques. Nous considérons à la fois le couplage capacitif de la boîte quantique à la cavité ainsi que le couplage des fils à la cavité. tronique dans la phase topologique. En outre, des expériences de QED en cavité devraient permettre de sonder la parité de l'état Majorana, qui n'est pas accessible dans les expériences de transport.

[H int , a] = i q f q b q .

(A.5)

Introducing Eq.A.5 into Eq.A. [START_REF] Alicea | Non-Abelian statistics and topological quantum information processing in 1D wire networks[END_REF] we obtain

The EOM for the bath variables reads

which is the EOM of a harmonic oscillator driven by the motion of the cavity degrees of freedom. For t 0 < t being a time in the distant past, the solution of Eq.A.7 can be written as b q (t) = e -iω q (t-t 0 ) b q (t 0 ) + t t 0 dτe -iω q (t-τ) f * q a(τ).

(A.8)

Introducing Eq.(A.8) into the last term in Eq.(A.6), we obtain q f q b q = q f q e -iω q (t-t 0 ) b q (t 0 ) + t t 0 dτe -iω q (t-τ) f * q a(τ) = q f q e-iω q (t -t 0 )b q (t 0 ) + q |f q | 2 t t 0 dτe -i(ω q -ω c )(t-τ) e iω c (τ-t) a(τ), (A.9)

where the last term in square brackets is a slowly varying function of τ.

where the last term is a linear damping term for the cavity mode. Assuming

being a constant and defining the density of states by ρ = q δ(ω c -ω q ), the decay rate becomes

We can now define the input mode

Finally, the cavity EOM reads

Let us define the output mode b out (t) that is radiated into the bath. For t 1 > t being a time in the distant future after the input field b in (t) has interacted with the cavity, the solution of Eq.(A.7) in terms of the final condition of the bath reads b q (t) = e -iω q (t-t 1 ) b q (t 1 ) -

Defining the output mode as where L 1 (y) = -πi 2Res[-2i] + Res[-y] + Res (2) [0] , (B.36)

and

Res (2) [0] = i 4y(1

and Res (2) [a] ≡ Res (2)

] are the residues of the function f(z) at the pole a of the first and second order, respectively. Summing everything, we finally obtain

The other terms appearing in L(y) can be calculated using a similar approach.

b.2.4 General expression for the quantum dot susceptibility

Gathering all terms, the real and imaginary parts of the susceptibility read

Let us diagonalize the Hamiltonian H NW = H 0 + ∆τ x in the absence of the induced pairing ∆ and then treat the latter perturbatively

In order to do so, let us perform the unitary transformation [START_REF] Pientka | Magneto-Josephson effects and Majorana bound states in quantum wires[END_REF]

where

H0 takes the from

Now let us reintroduce the pairing term ∆τ x and apply the transformation U to it.

Then HNW reads

Neglecting the last term in Eq.(C.50) the effective Hamiltonian reads

The bulk energy spectrum of Heff for

and for σ z = +1 is

The electronic susceptibility is defined as

In order to diagonalize the Hamiltonian in Eq. (C.51), let us perform a transformation

where

Then the electronic susceptibility reads

Sum can be transformed into integral as

Then the imaginary part of the susceptibility Π(ω) = Π (ω) + iΠ (ω), Eq. (C.60) reads

The projection onto the lower band σ z = -1

The low-energy subspace at p = ±p F is formed by the bands for which σ z = -1. p F = 2mu

when µ = 0.

The effective Hamiltonian for SO = mu 2 >> B, µ = 0 linearized around ±p F reads

The bulk energy spectrum reads

the correlation function for open boundary conditions

In this part, we present details on the calculation of the susceptibility for a finite wire with open boundary conditions. In this case, the excitation spectrum changes compared to the previous case, as in the topological region the zero energy Majorana fermions emerge.

The susceptibility reads:

and which can be written as

being the sum of a bulk susceptibility, that can be constructed from only the bulk (or gaped) states, and cross terms that involve both bulk and Majorana states, respectively. We use the discrete lattice model to numerically diagonalize the Hamiltonian for an electronic system with N fermionic sites. This can be written in a compact form as follows:

Next we extract from this expression only the Π BM (ω) component. This reads:

and, using that u 2k-1,i = v 2k,i , it can be simplified even further to give:

Pour un fil fini dans le régime topologique, la susceptibilité peut être écrite comme

M,ps | 2 ) pour n M = 1 (n M = 0). Afin d'obtenir la compréhension physique haut niveau de la susceptibilité résultante, nous écrivons le coefficients C

(1,2) M,ps d'une manière suivante :

Analysons l'implication de ce résultat. Quand M = 0, nous avons également u j M = v j M , et donc C

(1)

Mp puisque les contributions des électrons et des trous sont les mêmes dans l'état de Majorana. Néanmoins, M = 0 pour le fractionnement fini de l'énergie, et nous avons donc u j M = v j M , ce qui entraîne à son tour C

(1)

Mp . Tous ces résultats suggèrent que la susceptibilité Π(ω) devrait nous permettre de déduire, à travers Π BM (ω),