G Pecci

G Aupetit-Diallo

M Albert

P Vignolo

Keywords: ultracold quantum gases, Fermi gases, one-dimensional systems ultracold quantum gases, Fermi gases, one-dimensional systems

In this thesis, we study the coherence and the dynamics of ultracold atomic quantum gases, with a particular regard on Fermi gases. For what concerns the coherence, we study a Fermi gas confined on a ring threaded by an artificial gauge field. First, we show that the persistent current flowing in the ring can be used as probe for the different regimes of the BCS-BEC crossover. We propose a readout protocol for our results, based on the co-expansion of the gas trapped on the ring and a reference gas placed in the center of the ring itself. Next, we show that using the same expansion protocol we can extract information about the phase of the quantum gas. At intermediate times, the expanding cloud provides information about the first order coherence of the Fermi gas. In few-body systems, we can also access the number of particles in the system. At long times, we extract information about the many-body coherence coherence of the gas. Subsequently we focus on the dynamics of a strongly interacting two-component Fermi gas trapped in a harmonic potential. We consider a spin-separated state and then we follow the dynamics until long times. At short times, the dynamics is superdiffusive with dynamical exponents of the Kardar-Parisi-Zhang universality class. At intermediate times, the dynamics is dominated by damped spin oscillations unveleing a spin-drag effect. At long times, we observe thermalization. Ultimately, we study a multicomponent Bose gas in the strongly interacting regime. We first provide the exact solution for the wavefunction and we discuss its symmetry. Afterwards, we study the response of the gas to an artificial gauge field, with particular regard to the period of the persistent current as a function of the gauge flux.

Resumé

Dans cette thèse, nous étudions la cohérence et la dynamique des gaz quantiques atomiques ultra-froids, avec un regard particulier sur les gaz de Fermi. Pour ce qui concerne la cohérence, nous étudions un gaz de Fermi confiné sur un anneau en presence de un champ de jauge artificiel. Nous montrons tout d'abord que le courant persistant qui circule dans l'anneau peut être utilisé comme sonde pour les différents régimes du BCS-BEC crossover. Nous proposons un protocole de lecture de nos résultats, basé sur la coexpansion du gaz piégé sur l'anneau et d'un gaz de référence placé au centre de l'anneau lui-même. Ensuite, nous montrons qu'en utilisant le même protocole d'expansion, nous pouvons extraire des informations sur la phase du gaz quantique. Aux temps intermédiaires, le nuage en expansion fournit des informations sur la cohérence de premier ordre du gaz de Fermi. Dans les systèmes à plusieurs corps, nous pouvons également accéder au nombre de particules dans le système. Aux temps longs, nous extrayons des informations sur la cohérence à plusieurs corps du gaz. Par la suite, nous nous concentrons sur la dynamique d'un gaz de Fermi à deux composants en forte interaction, piégé dans un potentiel harmonique. Nous considérons un état séparé en spin, puis nous suivons la dynamique jusqu'aux temps longs. Aux temps courts, la dynamique est superdiffusive avec des exposants dynamiques de la classe d'universalité de Kardar-Parisi-Zhang. Aux temps intermédiaires, la dynamique est dominée par des oscillations de spin amorties dévoilant un effet de traînée de spin. Aux temps longs, nous observons une thermalisation. Finalement, nous étudions un gaz de Bose multicomposant dans le régime de forte interaction. Nous fournissons d'abord la solution exacte de la fonction d'onde et nous discutons de sa symétrie. Ensuite, nous étudions la réponse du gaz à un champ de jauge artificiel, avec une attention particulière à la période du courant persistant en fonction du flux de jauge.

Mots clés : Gaz quantiques ultra-froids, gaz de Fermi, systèmes unidimensionnels

Introduction

Introduction

Les gaz ultrafroids sont une plateforme très polyvalente qui permet d'étudier différentes branches de la physique. Ils peuvent être utilisés pour simuler des systèmes de matière condensée ou des modèles quantiques théoriques à mettre en oeuvre dans la simulation et le calcul quantiques. Le grand avantage de ce type de systèmes réside dans leur haut degré de contrôle. La grande capacité de réglage des gaz quantiques atomiques ultrafroids permet de simuler et de réaliser expérimentalement des systèmes de faible dimension. En particulier, il est possible de mettre en oeuvre des modèles unidimensionnels sur une plate-forme physique et d'exploiter les propriétés particulières des systèmes quantiques unidimensionnels pour étudier de nouveaux phénomènes et applications physiques. Dans une dimension, les corrélations entre les particules sont renforcées et les interactions jouent un rôle majeur puisque les particules ne peuvent pas se contourner les unes les autres. Dans ces systèmes, les fluctuations quantiques sont particulièrement importantes et affectent dans une large mesure le comportement du système, entraînant par exemple une réduction de la cohérence et d'autres caractéristiques particulières telles que la fermionisation. Du point de vue théorique, il existe une grande variété de méthodes permettant de décrire le système à des interactions arbitraires. Par exemple, on peut citer l'Ansatz de Bethe qui permet d'écrire exactement la fonction d'onde de systèmes homogènes unidimensionnels, où les particules interagissent via un potentiel de contact. De plus, des méthodes numériques spécifiques ont été développées permettant de décrire à la fois les propriétés d'équilibre et les propriétés dynamiques.

Dans cette thèse, nous étudions les propriétés des gaz quantiques atomiques ultrafroids confinés dans des géométries unidimensionnelles. Nous nous concentrons principalement sur les gaz de Fermi et nous étudions les propriétés de cohérence et le comportement hors-équilibre.

Dans le chapitre 1, nous donnons un aperçu des gaz quantiques ultrafroids. Nous partons du concept de condensation de Bose-Einstein et de la première réalisation expérimentale de ces systèmes. Ensuite, nous fournissons un résumé des techniques que nous pouvons utiliser pour régler et contrôler la température, la forme et les interactions dans les gaz quantiques ultrafroids. Dans la deuxième partie du chapitre, nous nous concentrons sur les gaz de Fermi. Nous décrivons les principaux phénomènes que nous étudierons dans les chapitres suivants et nous soulignons comment la nature fermionique du gaz affecte la physique, contrairement à ce qui est observé dans les gaz de Bose. Introduction que nous utilisons dans la thèse pour fournir des solutions exactes aux modèles unidimensionnels. Nous fournissons des exemples explicites de modèles dans différentes géométries. Enfin, nous présentons les méthodes numériques que nous mettons en oeuvre et qui sont fréquemment appliquées aux systèmes unidimensionnels pour lesquels la solution analytique n'est pas accessible ou ne permet pas de calculer les observables.

Dans le chapitre 3, nous exploitons les propriétés de cohérence des gaz de Fermi atomiques à deux composants pour étudier le passage de BCS à BEC. Nous considérons un gaz de Fermi confiné sur un anneau mésoscopique, enfilé avec un flux de jauge artificiel : ceci induit un courant de masse persistant circulant dans l'anneau. Nous montrons que cette quantité a des caractéristiques nettement différentes dans le régime BCS et dans le régime BEC et permet donc de discerner les différentes étapes du crossover. En particulier, nous nous concentrons sur la période du courant par rapport au flux et sur la dépendance du courant à la parité du nombre d'atomes confinés sur l'anneau.

Nous proposons un protocole de lecture pour détecter ces caractéristiques, basé sur l'analyse de la co-expansion du gaz initialement confiné sur l'anneau et d'un gaz de Fermi de référence placé au centre de l'anneau. Une fois les pièges désactivés, les deux sous-systèmes se dilatent et interfèrent : le motif résultant fournit des informations sur le courant circulant dans le système.

Dans le chapitre 4, nous proposons d'utiliser le même protocole d'expansion pour étudier la phase d'un gaz de Fermi quantique ultra-froid. En particulier, à des temps intermédiaires après le début de l'expansion, la figure d'interférence fournit des informations sur la cohérence de premier ordre du gaz de Fermi. De plus, pour les systèmes à quelques corps, ce protocole permet d'accéder au nombre de particules dans le gaz. Nous comparons notre approche avec le protocole analogue mis en oeuvre pour les gaz de Bose, qui a déjà été analysé dans la littérature, et nous remarquons des différences pertinentes entre les deux.

Aux temps longs après l'expansion, le protocole que nous proposons est équivalent aux expériences de temps de vol : le nuage en expansion fournit des informations sur les corrélateurs de momentum du gaz piégé. Dans ce régime temporel, nous comparons les interactions attractives et répulsives : nous constatons que l'appariement fermionique qui se produit dans ce dernier cas est mis en évidence par un renforcement des corrélations entre les particules de moment opposé. Par conséquent, l'image résultante est sensiblement différente pour les interactions attractives et répulsives. De plus, nous étudions la dépendance des fonctions de corrélation momentum-momentum par rapport à un champ de jauge artificiel. Nous montrons qu'à partir de l'imagerie du temps de vol, nous pouvons extraire des informations sur le courant persistant induit dans l'anneau.

Dans le chapitre 5, nous nous concentrons sur le gaz de Fermi à deux composants en forte interaction piégé dans un potentiel harmonique. Nous étudions la dynamique de mélange de spin de ce système, après avoir préparé l'état initial de telle sorte que toutes les particules de spin up se trouvent du côté gauche du piège et que toutes les particules de spin down se trouvent du côté droit du piège. Nous étudions la dynamique dans tous les régimes de temps.

Aux temps courts, nous montrons un comportement superdiffusif du courant de spin circulant depuis le centre du piège avec un exposant critique associé à la classe d'universalité de Kardar-Parisi-Zhang. Nous montrons également que l'aimantation satisfait à certaines propriétés d'échelle associées à l'universalité de Kardar-Parisi-Zhang.

Introduction

Aux temps intermédiaires, la dynamique est caractérisée par des oscillations de spin amorties, qui sont liées à l'effet de traînée de spin. De plus, à l'échelle mésoscopique, nous trouvons un changement d'échelle universel de la fréquence d'oscillation en fonction du nombre de particules à la puissance 1/4. Aux temps longs, nous étudions la thermalisation du système et nous dévoilons des comportements chaotiques quantiques en opposition à l'intégrabilité.

Dans le chapitre 6, nous nous intéressons à un gaz de Bose multicomposant confiné sur une géométrie en anneau. Nous nous concentrons en particulier sur la limite des interactions fortes, en discernant le régime fermionisé symétrique SU(2) du régime où nous considérons les interactions intraspécifiques comme infiniment répulsives et les interactions intraspécifiques comme grandes mais finies.

Nous fournissons une solution exacte pour la fonction d'onde dans ce dernier régime et nous caractérisons la symétrie de l'état fondamental en termes de tableaux de Young. Dans les deux régimes de forte interaction, nous étudions la réponse du gaz à un champ de jauge artificiel : nous montrons que la période du courant par rapport au flux de jauge est réduite d'un facteur égal au nombre total de particules, une caractéristique marquant clairement le régime de fermionisation.

Pour conclure le manuscrit, nous fournissons quelques perspectives et aperçus sur les futures directions de recherche possibles suite au travail présenté dans cette thèse. Une liste de publications basées sur cette thèse suit : Introduction Introduction U ltracold gases are a very versatile platform which allows studying different branches of physics. They can be used to simulate condensed matter systems or theoretical quantum models to be implemented in quantum simulation and quantum computation. The great advantage of these systems comes from their high degree of control. The great tunability of ultracold atomic quantum gases allows one to simulate and experimentally realize low-dimensional systems. In particular, one can implement one-dimensional models on a physical platform and exploit the peculiar properties of one-dimensional quantum systems to study novel physical applications and phenomena. In one dimension, correlations among the particles are enhanced and interactions play a major role since the particles cannot circumvent each other. In these systems, the quantum fluctuations are particularly relevant and affect to a large extent the behaviour of the system, yielding for instance reduced coherence and other special features such as fermionization. From the theoretical point of view, a large variety of methods are available and allow one to describe the system at arbitrary interactions. For instance, one may cite the Bethe Ansatz which allows to obtain the exact wavefunction of one-dimensional homogeneous systems, where particles interact via contact potential. Furthermore, specific numerical methods have been devised allowing to describe both equilibrium and dynamical properties.

In this thesis, we study the properties of ultracold atomic quantum gases confined in one-dimensional geometries. We focus primarily on Fermi gases and we study the coherence properties and the out-of-equilibrium behaviour.

In Chapter 1 we give an overview of the ultracold quantum gases. We start from the concept of Bose-Einstein Condensation and from the first experimental realization of these systems. Afterwards, we provide a summary of the techniques we can use to tune and control the temperature, the shape and the interactions in ultracold quantum gases. In the second part of the Chapter, we focus on Fermi gases. We describe the main phenomena we will study in the following Chapters and we stress how the fermionic nature of the gas affects the physics, as opposed to what is observed in Bose gases.

In Chapter 2 we introduce the physical phenomena and the methods typical of onedimensional systems. First, we summarize some of the properties which make onedimensional systems remarkable from the point of view of theoretical and experimental physics. Next, we present the analytical methods we use in the thesis to provide an exact solution to one-dimensional models. We provide explicit examples of models in different confinement geometries. Last, we present the numerical methods we implement and which are frequently applied to one-dimensional systems where the analytical solution is not accessible or is not suitable to calculate observables.

Introduction

In Chapter 3, we exploit the coherence properties of atomic two-component Fermi gases to study the BCS-BEC crossover. We consider a Fermi gas confined on a mesoscopic ring, threaded with an artificial gauge flux: this induces persistent mass-current flowing in the ring. We show that this quantity has markedly different features in the BCS and in the BEC regime and therefore enables one to discern the different stages of the crossover. In particular, we focus on the period of the current with respect to the flux and on the dependence of the current on the parity of the number of atoms confined on the ring.

We propose a readout protocol to detect these features, based on the analysis of the co-expansion of the gas initially confined on the ring and a reference Fermi gas placed in the center of the ring. Once the traps have been switched off, the two sub-systems expand and interfere: the resulting pattern yields information about the current flowing in the system.

In Chapter 4, we propose to use the same expansion protocol to study the phase of an ultracold quantum Fermi gas. In particular, at intermediate times after the beginning of the expansion, the interference pattern provides information about the first-order coherence of the Fermi gas. Moreover, for few-body systems, this protocol allows one to access the number of particles in the gas. We compare our approach with the analogous protocol implemented for Bose gases, which has already been analyzed in the literature, and we notice relevant differences between the two.

At long times after the expansion, the protocol we propose is equivalent to time-offlight experiments: the expanding cloud provides information about the momentum correlators of the trapped gas. In this time regime, we compare attractive and repulsive interactions: we see that the fermionic pairing occurring in the latter case is highlighted by an enhancement of the correlations between particles with opposite momenta. Therefore, the resulting imaging is substantially different for attractive and repulsive interactions. Moreover, we study the dependence of the momentum-momentum correlation functions on an artificial gauge field. We show that from the time-of-flight imaging we can extract information about the induced persistent current in the ring.

In Chapter 5, we focus on the strongly interacting two-component Fermi gas trapped in a harmonic potential. We study the spin-mixing dynamics of this system, after preparing the initial state such that all the spin up particles are on the left side of the trap and all the spin down particles are on the right side of the trap. We study the dynamics in all the time regimes.

At short times, we show a superdiffusive behaviour in the spin current flowing from the center of the trap with a critical exponent associated with the Kardar-Parisi-Zhang universality class. We also show that the magnetization satisfies some scaling properties associated with Kardar-Parisi-Zhang universality.

At intermediate times the dynamics is characterized by damped spin oscillations, which are related to the spin-drag effect. Moreover, on mesoscopic scales, we find a universal scaling of the oscillation frequency in terms of the number of particles to power 1/4. At long times, we study the thermalization of the system and we unveil quantum chaotic behaviours as opposed to integrability.

In Chapter 6, we focus on a multicomponent Bose gas confined on a ring geometry. We focus in particular on the strongly interacting limit, discerning the SU(2) symmetric Introduction fermionized regime from the regime where we consider the intraspecies interactions as infinitely repulsive and the intraspecies interactions as large but finite.

We provide an exact solution for the wavefunction in this latter regime and we characterize the symmetry of the ground state in terms of Young tableaux. In both strongly interacting regimes, we study the response of the gas to an artificial gauge field: we show that the period of the current with respect to the gauge flux is reduced by a factor equal to the total number of particles, a feature clearly marking the fermionization regime.

To conclude the manuscript, we provide some outlooks and insights on future possible research directions following the work presented in this thesis. A list of publications based on this thesis follows:

Publications and preprints • G. Pecci, P. Naldesi, L. Amico, and A. Minguzzi, Probing the BCS-BEC crossover with persistent currents, Phys. Rev. Research 3, L032064 (2021).

• G.Pecci, P. Vignolo and A. Minguzzi, Universal scaling of spin mixing dynamics in a strongly interacting one-dimensional Fermi gas, Phys. Rev. A 105, L051303

• G. Pecci, P. Naldesi, A. Minguzzi, and L. Amico, Single-particle versus many-body phase coherence in an interacting Fermi gas , Quantum Science and Technology 8 (1), 01LT03

• G.Aupetit-Diallo, G. Pecci, C. Pignol, F. Hébert, A. Minguzzi, M. Albert, P. Vignolo Exact solution for SU(2)-symmetry breaking bosonic mixtures at strong interactions, Phys. Rev. A 106, 033312 (2022)

• G. Pecci, G.Aupetit-Diallo, M. Albert, P. Vignolo, A. Minguzzi Persistent currents in Bose-Bose mixtures on a ring, arXiv:2211.16194 (2022)currently under peer-review 1

Ultracold gases of atoms

In this chapter, we provide a general overview of quantum gases of ultracold atoms. Starting from the first experimental realization of a Bose-Einstein condensate, we discuss the significant impact that this system had on the development of novel experimental and theoretical methods affecting different branches of physics, ranging from condensed matter to quantum computation and quantum simulation. In the second part of the Chapter, we focus on ultracold degenerate Fermi gases and on their specific physical features. We address in particular the BCS-BEC crossover and the behaviours of such systems in low dimensions and in mesoscopic geometries.

Bose-Einstein condensation

Ultracold atomic gases are a system which has had a steadily increasing impact on experimental and theoretical physics since the last two decades. A great breakthrough in this field occurred in 1995, when the first observations of Bose-Einstein Condensation (BEC) was performed in a dilute vapor of Rubidium-87 and in a gas of Sodium atoms [2,[START_REF] Davis | Bose-Einstein condensation in a gas of Sodium atoms[END_REF]. This phenomenon, predicted by Bose and Einstein several years before (see eg [4][5][START_REF] Ketterle | Making, probing and understanding Bose-Einstein condensates[END_REF][START_REF] Griffin | Bose-Einstein Condensation[END_REF] and references therein for further details), concerns the emergence of quantum features on a macroscopic scale in a non-interacting bosonic system at ultra-low temperatures. In this regime, uniquely because of the statistics of the particles, the ground state of the system, ie the condensate, is macroscopically occupied: at zero temperature all the particles occupy the same single-particle state. At small but finite temperature, some particles escape the condensate because of thermal excitations: the fraction of particles in the condensate decreases as the temperature increases and eventually goes to zero at a critical temperature, which for an ideal Bose gas is given by T c = 2π h2 1.897mk B n 2/3 [START_REF] Griffin | Bose-Einstein Condensation[END_REF], where n and m are respectively the density and the mass of the gas and k B is the Boltzmann constant. Notably, the decrease in the condensate fraction may occur at any temperature below T c by increasing the strength of the repulsive interactions among the particles: this phenomenon is called quantum depletion [START_REF] Dalfovo | Theory of Bose-Einstein condensation in trapped gases[END_REF][START_REF] Leggett | Bose-Einstein condensation in the alkali gases: Some fundamental concepts[END_REF].

Nowadays, this condensation mechanism is considered to underlie different phenomena related to macroscopic quantum coherent behaviours, including superfluidity and Bardeen-Cooper-Schrieffer (BCS) superconductivity, and has several implications on the coherence of a many-body quantum system [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF][START_REF] James | Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems[END_REF][START_REF] Andrews | Observation of interference between two Bose condensates[END_REF][START_REF] Castin | Relative phase of two Bose-Einstein condensates[END_REF].

The low temperatures these systems experience plays a key role in determining their physical features. Once they have been trapped, quantum gases can be cooled down to temperatures below microkelvin using laser cooling techniques [START_REF] Schreck | Laser cooling for quantum gases[END_REF][START_REF] William | Nobel lecture: Laser cooling and trapping of neutral atoms[END_REF]. These techniques are based on slowing down the atoms exploiting the recoil force deriving from the scattering with a counter-propagating laser beam. However, laser cooling does not allow one to cool the atoms to quantum degeneracy since it is limited by photon reabsorption and re-emission. To further cool down the particles, one can use other techniques such as evaporative cooling [START_REF] Bederson | Advances in atomic, molecular, and optical physics[END_REF]. This last procedure is based on decreasing in a controlled fashion the confinement potential of the gas such that the particles having kinetic energy above a fixed threshold evaporate, ie are ejected from the trap as a result of elastic collisions. After evaporation, just the atoms at lower kinetic energy remain in the bottom of the trap: Chapter 1. Ultracold gases of atoms the average kinetic energy of the atoms corresponds to the resulting temperature of the gas, which consequently cools down.

Atom trapping and manipulation

One of the most striking features of quantum gases at very low temperature is the possibility to manipulate these systems by means of external potentials and fields. Usually, quantum gases are trapped in magnetic or optical fields that couple with the magnetic dipole moments of the atoms and produce dipolar forces via the potential

V dip (r) = -d dip • E(r)
, where d dip is the dipole momentum of the atoms and E(r) is the electric field, whose amplitude square is proportional to the intensity of the employed light beam. This procedure makes possible to shape the gas in a desired spatial configuration (see Fig. 1.1 for an example of the great degree of control achievable on these systems). Using this technique, one can implement confining potentials acting on the gas and optical lattices. The latter can be realized using counterpropagating laser beams [START_REF] Bloch | Ultracold quantum gases in optical lattices[END_REF][START_REF] Schäfer | Tools for quantum simulation with ultracold atoms in optical lattices[END_REF] to generate standing waves in the form V con f = V 0 sin(k 0 x), where V 0 is the depth of the optical wells and k 0 is the wavevector depending on the wavelength of the laser. An example of this is presented in Fig. 1.2, where we see a three-and a two-dimensional optical lattice. In the latter, the atoms are tightly confined in an array of one-dimensional tubes. By tuning other parameters of the beams, such as the angle of incidence or the depth of the lattice wells, it is possible to generate different lattice geometries. A pseudo-disorder can be realized by using bichromatic lattices with incommensurate frequencies. Finally, it is possible to tune the interactions among the atoms by means of the so-called confinement induced resonances (CIR) [START_REF] Olshanii | Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons[END_REF][START_REF] Bergeman | Atom-atom scattering under cylindrical harmonic confinement: Numerical and analytic studies of the confinement induced resonance[END_REF] (see Sec. 1.2.1 below).

As we mentioned, one can control with a large degree of precision the shape and the energy scales of the magnetic trap. This feature can also be exploited to apply very strong 1.1. Cold atoms confined in an external potential confinement along one or two spatial directions, constraining the degrees of freedom of the gas and experimentally accessing low-dimensional physics. This paved the way for a novel branch of research: before the advent of ultracold gases, two-and one-dimensional physics were considered fascinating theoretical models whose relevancy for real-world physics was very limited due to the few possible experimental realizations. The design of these simulation platforms enables to access and study phenomena which don't occur in three dimensions, such as Berezinskii-Kosterlitz-Thouless (BKT) phase transitions [START_REF] Krüger | Critical point of an interacting two-dimensional atomic Bose gas[END_REF][START_REF] Schweikhard | Vortex proliferation in the Berezinskii-Kosterlitz-Thouless regime on a two-dimensional lattice of bose-einstein condensates[END_REF][START_REF] Cladé | Observation of a 2d Bose gas: From thermal to quasicondensate to superfluid[END_REF][START_REF] Hadzibabic | The trapped two-dimensional Bose gas: from Bose-Einstein condensation to Berezinskii-Kosterlitz-Thouless physics[END_REF][START_REF] Tung | Observation of the presuperfluid regime in a two-dimensional Bose gas[END_REF], and topological phases such as the Hofstader and Haldane models [START_REF] Goldman | Topological quantum matter with ultracold gases in optical lattices[END_REF][START_REF] Jotzu | Experimental realization of the topological Haldane model with ultracold fermions[END_REF][START_REF] Aidelsburger | Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms[END_REF] in two dimensions. It was also possible to confine the atoms in one spatial dimension [START_REF] Dettmer | Observation of phase fluctuations in elongated Bose-Einstein condensates[END_REF][START_REF] Görlitz | Realization of Bose-Einstein condensates in lower dimensions[END_REF][START_REF] Moritz | Exciting collective oscillations in a trapped 1d gas[END_REF][START_REF] Tolra | Observation of reduced three-body recombination in a correlated 1d degenerate Bose gas[END_REF]: examples of physical phenomena occurring in these systems will be presented in detail in Chapter 2.

Feshbach resonances

Ultracold atoms have a large degree of versatility because it is also possible to tune the sign and the strength of interaction. The essential tool used to tune interactions in quantum gases is the Feshbach resonance (see [START_REF] Chin | Feshbach resonances in ultracold gases[END_REF][START_REF] Inouye | Observation of Feshbach resonances in a Bose-Einstein condensate[END_REF] and references therein for an exhaustive treatment). The underlying physical mechanism is the following. At low temperatures, two-body scatterings solely depend on the scattering length associated to the elastic collisions in the system. It can be shown that this quantity can be tuned with high precision using an external magnetic field or some optical methods. The explicit dependence of the scattering length on an external magnetic field is displayed in the bottom panel of Chapter 1. Ultracold gases of atoms Fig. 1.3. Remarkably, this technique allows not only to tune the strength of the coupling among the particles, but also to switch from repulsive to attractive interactions and vice versa. However, as can be seen from the top panel of Fig. 1.3, this process might also increase the number of three-body or inelastic collisions inducing loss of particles in the system. Feshbach resonances allow one to resonantly couple a bound state and a scattering state. This can be done by tuning the scattering length such that the energies of the two states coincide. In this way, the bound state becomes energetically favourable and the formation of molecules in the system is enhanced. The explicit dependence of the scattering length a sc on the external magnetic field B is a sc (B) = a of f 1 -∆ FR B-B 0 [START_REF] Moerdijk | Resonances in ultracold collisions of 6 Li, 7 Li, and 23 Na[END_REF], where a of f is the scattering length far from the resonance, ∆ FR is the resonance width and B 0 is the resonant value of the magnetic field.

Atom imaging

One of the most popular measurement techniques in the framework of ultracold gases is the so-called time-of-flight (TOF) imaging [START_REF] Gerbier | Expansion of a quantum gas released from an optical lattice[END_REF][START_REF] Tenart | Two-body collisions in the time-of-flight dynamics of lattice Bose superfluids[END_REF][START_REF] Bergschneider | Spin-resolved single-atom imaging of 6 Li in free space[END_REF][START_REF] Carcy | Momentum-space atom correlations in a Mott insulator[END_REF]. This technique consists in a sudden ramp down of the trap confining the gas, which is let free to expand nearly ballistically in the real space. During the expansion, one can measure the spatial distribution of the cloud, which preserves the information about the state of the system right before the trap is switched off. We will focus at length on this property in Chapters 3 and 4. The spatial distribution can be acquired by means of absorption imaging. This technique is based on shining the expanding gas with a laser beam: by observing the absorption pattern of the atoms one can reconstruct the spatial density of the expanding cloud. To increase the resolution, one can record the absorption pattern with a charge-coupled-device camera. This latter procedure is called in situ absorption imaging [START_REF] Hung | In situ imaging of atomic quantum gases[END_REF] and can be applied also to gases inside the initial confining potential. Special care needs to be taken when one has to image very few atoms [START_REF] Serwane | Deterministic preparation of a tunable few-fermion system[END_REF]. In this case, an exquisite control on particle number is possible [START_REF] Wenz | From few to many: Observing the formation of a Fermi sea one atom at a time[END_REF]. The only parameter varying in the three panels is the temperature. Figure from [2].

Cold atoms confined in an external potential

As we will show in Chapters 3 and 4, the spatial density of the expanding gas can be used to extract different correlation functions. For long expansion times, the density distribution approaches a quantity which is proportional to the momentum distribution of the trapped gas [START_REF] Gerbier | Expansion of a quantum gas released from an optical lattice[END_REF][START_REF] Pedri | Expansion of a coherent array of Bose-Einstein condensates[END_REF][START_REF] Murthy | Matter-wave Fourier optics with a strongly interacting twodimensional Fermi gas[END_REF][START_REF] Cayla | Single-atom-resolved probing of lattice gases in momentum space[END_REF]. This makes possible to extract information about the velocity of the particles in the many-body quantum state before the trap is switched off.

An example of the velocity distribution of a trapped gas measured within this approach is reported in Fig. 1.4. Remarkably, one can also detect and therefore manipulate the gas on a single-atom level using a quantum microscope [START_REF] Waseem S Bakr | A quantum gas microscope for detecting single atoms in a hubbard-regime optical lattice[END_REF][START_REF] Ott | Single atom detection in ultracold quantum gases: a review of current progress[END_REF][START_REF] Gross | Quantum gas microscopy for single atom and spin detection[END_REF][START_REF] Urban | Observation of Rydberg blockade between two atoms[END_REF][START_REF] Kuhr | Quantum-gas microscopes: a new tool for cold-atom quantum simulators[END_REF]. This guarantees single-site resolution and permits to simulate few-body physics. An example of single-site level imaging in a Bose gas in an optical lattice is shown in Fig. 1.5.

Ultracold atoms as quantum simulators

All the properties we mentioned so far created an increasing interest in atomic gases at low temperatures as a tool to simulate quantum systems. Quantum gases found application in several different fields of research, ranging from many-body physics to condensed matter and out-of-equilibrium quantum physics [START_REF] Bloch | Many-body physics with ultracold gases[END_REF][START_REF] Gross | Quantum simulations with ultracold atoms in optical lattices[END_REF][START_REF] Langen | Ultracold atoms out of equilibrium[END_REF][START_REF] Ueda | Quantum equilibration, thermalization and prethermalization in ultracold atoms[END_REF][START_REF] Lewenstein | Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems[END_REF].

The field of quantum simulation, which includes ultracold gases as one of the suitable platforms (see [START_REF] Acín | The quantum technologies roadmap: a European community view[END_REF] and references therein), has been developed to overcome the intrinsic limitations of simulating a quantum system using classical resources [START_REF] Richard P Feynman | Simulating physics with computers[END_REF]. A quantum simulator, in particular, is a physical platform which can be externally manipulated and controlled in order to mimic a targeted quantum system. Using a quantum simulator, one would be able to study regimes typically difficult to access with classical computers, such as long-times dynamics [START_REF] Eisert | Quantum many-body systems out of equilibrium[END_REF] or large systems sizes. From the quantum simulation perspective, implementing artificial gauge fields in quantum gases plays a very important role [START_REF] Dalibard | Colloquium: Artificial gauge potentials for neutral atoms[END_REF][START_REF] Dalibard | Introduction to the physics of artificial gauge fields[END_REF][START_REF] Fetter | Rotating trapped Bose-Einstein condensates[END_REF]. This procedure consist in simulating the action of an electromagnetic field acting on the system regardless the neutral charge of Chapter 1. Ultracold gases of atoms the particles. This can be achieved in different ways. A technique which is particularly suitable for rotationally invariant confining potentials, is to stir the gas with a potential barrier, inducing the particles to rotate. Thanks to the analogy between the Lorentz and the Coriolis forces, this simulates a magnetic field of an intensity proportional to the angular velocity of the gas and directed along the axes with respect to which the confining potential is rotationally invariant [START_REF] Gross | Quantum simulations with ultracold atoms in optical lattices[END_REF][START_REF] Langen | Ultracold atoms out of equilibrium[END_REF]. Other methods, which don't rely on the symmetries of the confinement, involve optical couplings inducing geometric phases on the system (see [START_REF] Goldman | Light-induced gauge fields for ultracold atoms[END_REF][START_REF] Lin | Synthetic magnetic fields for ultracold neutral atoms[END_REF] and references therein for a more exhaustive treatment).

Atomtronics

Artificial gauge fields induce a current flow in the gas, which can be exploited to design matter-wave circuits using cold atoms. This is the main focus of atomtronics [START_REF] Amico | Roadmap on Atomtronics: State of the art and perspective[END_REF][START_REF] Amico | Atomtronic circuits: from many-body physics to quantum technologies[END_REF], that investigates how to provide atomic counterparts of electronic devices and furthermore study the applications of atomic circuits in different research fields, ranging from quantum simulation to quantum sensing. The properties of the current in ultracold gases can be accessed experimentally using the above-mentioned time-of-flight technique. As we stated before, the current is related to the presence of a geometrical phase in the many-body wavefunction, which can be accessed by measuring first-and second-order correlators. Typically, the presence of 1.1. Cold atoms confined in an external potential the current can be detected by looking at the momentum distribution [START_REF] Amico | Quantum many particle systems in ring-shaped optical lattices[END_REF][START_REF] Aghamalyan | Coherent superposition of current flows in an atomtronic quantum interference device[END_REF] whereas more quantitative details can be extracted by employing more sophisticated protocols. For instance, one could consider a system composed of a gas subjected to an artificial gauge field and a second static condensate, playing the role of reference phase. By looking at the co-expansion of the two systems in a TOF experiment and evaluating the corresponding interference pattern created by the two gas in the real space [START_REF] Corman | Quench-induced supercurrents in an annular Bose gas[END_REF][START_REF] Eckel | Interferometric measurement of the current-phase relationship of a superfluid weak link[END_REF][START_REF] Mathew | Self-heterodyne detection of the in situ phase of an atomic superconducting quantum interference device[END_REF][START_REF] Roscilde | From quantum to thermal topological-sector fluctuations of strongly interacting bosons in a ring lattice[END_REF][START_REF] Haug | Readout of the atomtronic quantum interference device[END_REF], one can provide a precise estimation of the intensity and of the direction of the current. Typically, a non-zero current is revealed by a spiral-shaped interference pattern and the number of branches of the spiral is proportional to the intensity of the current (See Fig. 1.6). The direction of the current coincides with the orientation of the spirals. This protocol will be better detailed in Chapters 3 and 4. However, the general spiral-like shape of the interference pattern can be summarized as follows for a non-interacting system. Consider initially the gas trapped in a ring-shaped optical lattice and an additional site in the center. Each site can be modeled as a harmonic trap. Once the traps are released, the expanding wavefunctions interfere and the resulting pattern is made of parallel fringes, since it originates from two point-like sources. In a ring lattice, the interference pattern originated by sources placed on the sites and a single source located in the center is a linear superposition of the fringes arising from the interference of each site with the central one. As we mentioned, because of the artificial gauge field, each wavefunction defined on the ring is subjected to a phase winding. Therefore, different sources along the ring are characterized by different phases. In particular, two nearest-neighbour sources have constant phase difference, which implies that the interference fringes of neighbouring sources with the central disk are shifted by a constant factor. Summing together all those contributions we obtain a spiral-like shape. We illustrate this idea in Fig. (1.7), where we show the separate contributions of three consecutive lattice sites, the shift of the interference fringes along the cuts connecting the two sources, and the total interference pattern. The spiral-like pattern is hence due to the relative shift of the maxima of the fringes of each single-site contribution.

Chapter 1. Ultracold gases of atoms

Ultracold gases of fermions

In the following, we focus in particular on ultracold Fermi gases [START_REF] Giorgini | Theory of ultracold atomic Fermi gases[END_REF][START_REF] Inguscio | Ultra-cold Fermi gases[END_REF], which is the main topic of this thesis work. In these systems, the statistics of the particles induces marked differences with respect to the bosonic case. First of all, scattering processes between particles with the same spin are forbidden by the Pauli exclusion principle. This affects both the theoretical treatment of such systems and the experimental realization. In particular, a constrained number of collisions reduces the particles losses, which allows better control on the number of particles, but also makes evaporative cooling processes more challenging. Experiments on ultracold Fermi gases started to proliferate just on the verge of this century [START_REF] Demarco | Onset of Fermi degeneracy in a trapped atomic gas[END_REF][START_REF] Km O'hara | Observation of a strongly interacting degenerate Fermi gas of atoms[END_REF], using Potassium and Lithium isotopes. In these systems, reaching the so-called quantum degenerate regime, i.e. the regime where the statistics of the particles becomes relevant, is particularly difficult due to the low efficacy of evaporative cooling methods. A major contribution in this sense was given by the sympathetic cooling techniques [START_REF] Roati | Fermi-Bose quantum degenerate 40 K-87 Rb mixture with attractive interaction[END_REF][START_REF] Schreck | Quasipure Bose-Einstein condensate immersed in a Fermi sea[END_REF][START_REF] Truscott | Observation of Fermi pressure in a gas of trapped atoms[END_REF] which consist in realizing a mixture of a Fermi gas and a Bose gas and then in cooling the whole system. In this way, the bosonic gas acts as a refrigerator for the Fermi gas and the number of collisions in the system increases because of the interactions between Bose and Fermi gases. A further difference between fermionic and bosonic gases concerns the condensation process. While in Bose gases the latter is energetically favourable for zero inter-1.2. Ultracold gases of fermions actions among the atoms, fermionic condensation necessarily requires attractive interactions. This implies that Feshbach resonances, as introduced in Sec. 1.1.3, play a key role in the manipulation of fermionic systems to produce macroscopically coherent states. We previously mentioned that the statistics of the particles restrains the particle losses depicted in the top panel of Fig. 1.3 at the resonance point of very large interaction. In this interacting regime, called unitary limit [START_REF] Randeria | Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein condensation and the unitary Fermi gas[END_REF][START_REF] Zwerger | The BCS-BEC crossover and the unitary Fermi gas[END_REF], the scattering length diverges: there is no length scale associated with the interaction. The different interacting regimes that one can explore by means of Feshbach resonance define one of the central phenomena occurring in ultracold Fermi gases: the so-called BCS-BEC crossover.

BCS-BEC crossover

The BCS-BEC crossover finds relevant applications in a very wide range of research fields, from condensed matter physics to nuclear matter and neutron stars [START_REF] Randeria | Pre-pairing for condensation[END_REF][START_REF] Strinati | The BCS-BEC crossover: From ultra-cold Fermi gases to nuclear systems[END_REF][START_REF] Baldo | Deuteron formation in expanding nuclear matter from a strong coupling BCS approach[END_REF]. Consider a spin 1 2 Fermi gas with attractive interactions. In the ground state, the atoms form two-body bound states, called the Cooper pairs in analogy with superconducting systems. The total spin of a Cooper pair is zero. Typically, in this regime interactions are weak and the correlation length among the particles is larger than the average interparticle distance. On the other hand, as attractive interactions increase, the correlation length among the particles in real space is smaller than the mean interparticle distance: each pair can be considered as a unique point-like particle of integer spin. Therefore, since the strong interparticle attractive interaction hinders the fermionic nature of the atoms, the pairs can condense as they effectively were a Bose gas with half of the density of the initial Fermi gas.

The first regime is called BCS regime, whereas the second one is called BEC regime. The BCS-BEC crossover represents a continuous evolution between these two regimes. Using a Feshbach resonance, one can tune continuously the value of the scattering length, and therefore of the interaction, to drive the system from weak to strong attractions and then, crossing the unitary limit, from attractive to repulsive interactions, in the deep BEC regime where the particles are tightly bound in molecules. A qualitative phase diagram of the gas undergoing the BCS-BEC crossover as a function of interaction is displayed in Fig. 1.8.

The BCS-BEC crossover highlights the similarities between BCS superconductivity and Bose-Einstein condensation, that in this context appear as different regimes of the same system. As a consequence of this, BCS-BEC crossover has assumed a significant relevance for the understanding of the physical mechanism underlying high-Tc superconductivity (for an extensive treatment see [START_REF] Chen | BCS-BEC crossover: From high temperature superconductors to ultracold superfluids[END_REF] and references therein, in particular [START_REF] Uemura | Bose-Einstein to BCS crossover picture for high-Tc cuprates[END_REF] for an experimental analysis).

As the interest in low-dimensional physics increases, the crossover has been studied also in one and two dimensions [START_REF] Fuchs | Exactly solvable model of the BCS-BEC crossover[END_REF][START_REF] Ren | Bethe-ansatz analysis of near-resonant twocomponent systems in one dimension[END_REF]. The former in particular will be detailed more in Chapter 3. In one-dimensional systems, the high degree of control on the coupling strength, which is required for the crossover, is ensured by the confinement-induced resonances. This phenomenon relies on the explicit dependence of the one-dimensional scattering length on its three-dimensional counterpart and on the frequency of the trapping potential applied to reduce the dimensionality. This allows us to establish a relation between the parameter that can be tuned and controlled experimentally, ie the threedimensional scattering length, and the parameter that effectively dominates the physics in the one-dimensional regime, ie the one-dimensional scattering length.

Chapter 1. Ultracold gases of atoms

Fermi gases on a ring

A further connection between superconducting materials and ultracold Fermi gases is related to the response to an external magnetic field. This is particularly evident by comparing two systems with the same geometries, such as Fermi gases trapped in a ring-shaped potential and mesoscopic metallic rings. When pierced by an external magnetic flux, these systems display very similar physical features [START_REF] Viefers | Quantum rings for beginners: energy spectra and persistent currents[END_REF][START_REF] Manninen | Quantum rings for beginners ii: Bosons versus fermions[END_REF]. The field induces a persistent current in the ring, in analogy to the one we introduced in Sec. 1.1.6. Moreover, in metallic rings, the current is quantized and is periodic as a function of the applied flux (see Fig. 1.9). The period of the current is called quantum of flux and depends only on fundamental constants and is inversely proportional to the electron charge. This result follows from a theorem proved by Leggett [START_REF] Leggett | Granular Nanoelectronics[END_REF]. The period of the current can be used to probe the phase of the metal: if the system is driven below the critical temperature and therefore enters in the superconducting phase, the flux quantum is reduced by a factor of two. This is due to the microscopic mechanism of superconductivity: in this phase the charge carrier in the metal are paired electrons and their charge is double with respect to the one of the single particle. Therefore, by measuring the period of the current as a function of the applied flux, one can discern whether the charge carriers in the metallic ring are single electrons or Cooper pairs. This allows one to determine whether the system is in the normal or in the superconducting phase.

Figure 1.9: Energy levels as a function of the magnetic flux flowing in a metallic ring. We see that the energy is a periodic function of the flux. The current also reflects the same behaviour, being the derivative of the energy with respect to the flux. In this plot, the system is in the diamagnetic phase. Figure from [START_REF] Leggett | Granular Nanoelectronics[END_REF].

The second part of the Leggett theorem regards the sign of the current in metallic rings. Due to the statistics of the particles, the sign of the current in a metallic ring threaded with magnetic flux depends on the number of particles. Assuming that all the charge carriers have the same spin indeed, according to the parity of the number of particles, the metal behaves as a paramagnet or as a diamagnet. In the first case, occurring when the number of electrons is even, the energy of the system decreases when the magnetic flux is switched on. On the other hand, when the number of charge carriers is odd, the metal behaves as a diamagnet: a non-zero external field drives the system to an energetically disadvantageous configuration. As the current reflects the slope of the energy as a function of the magnetic flux, in the two cases it has opposite signs. This phenomenon, called parity effect [START_REF] Schilling | Number-parity effect for confined fermions in one dimension[END_REF], can be generalized for particles with spin [START_REF] Waintal | Persistent currents in one dimension: The counterpart of Leggett's theorem[END_REF].

These typical features of metallic rings have their counterpart in ultracold Fermi gases confined in ring potentials: this equivalence will be studied in detail in Chapter 3.

These connections highlight the potentiality of quantum gases as a suitable tool for 1.2. Ultracold gases of fermions quantum simulation, with particular regard to Fermi gases when comes to study systems subjected to Fermi statistics.

Large-spin systems and SU(r) symmetry

Another remarkable property of Fermi gases is the possibility to simulate systems with more than two spin components [START_REF] Sonderhouse | Thermodynamics of a deeply degenerate SU(N)-symmetric fermi gas[END_REF][START_REF] Pagano | A one-dimensional liquid of fermions with tunable spin[END_REF]. Ultracold Ytterbium or alkaline-earth atoms can be used to study quantum magnetism in systems with large spin [START_REF] Miguel A Cazalilla | Ultracold gases of ytterbium: ferromagnetism and mott states in an SU (6) Fermi system[END_REF][START_REF] Vyacheslavovich Gorshkov | Two-orbital SU (N) magnetism with ultracold alkaline-earth atoms[END_REF] or to perform high-precision measurements [START_REF] Livi | Synthetic dimensions and spin-orbit coupling with an optical clock transition[END_REF][START_REF] Ludlow | Optical atomic clocks[END_REF]. Furthermore, they can be used also as quantum simulators [START_REF] Hofrichter | Direct probing of the Mott crossover in the SU(n) Fermi-Hubbard model[END_REF][START_REF] Scazza | Observation of two-orbital spin-exchange interactions with ultracold SU(N)-symmetric fermions[END_REF][START_REF] Chetcuti | Persistent Current of SU(N) Fermions[END_REF][START_REF] Decamp | High-momentum tails as magnetic-structure probes for strongly correlated SU(κ) fermionic mixtures in one-dimensional traps[END_REF]. A schematic comparison between spin with a large number of spin components as opposite to spinless fermions and spinless bosons is shown in Fig. 1.10. The states of these systems present an internal symmetry with respect to the transformation of the SU(r) group, r being the number of spin components: such symmetry can be classified in terms of standard Young tableaux (SYT). A Young tableau is a visual representation of a partition of a set of integers. Consider a set of N boxes, we can build different partitions of such set by gathering the boxes in different numbers of rows and columns. A Young tableau is defined as a partition of a fixed shape, ie a fixed number of rows and columns, where the number of boxes in each column is not increasing from left to right. For a fixed shape, we can fill each box with an integer: for each Young tableau of N boxes, there are N ! possible fillings, corresponding to all the possible permutations of N integers. Each filling where the integers are monotonically increasing from top to bottom and from left to right, defines a SYT. In Fig. 1.11 we show an example of Young tableaux for N = 4 on the left and the complete set of SYT for a fixed tableau shape on the right. The SYT have particular relevance in quantum physics since they can be used to characterize the symmetry of a state under particle exchange [START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF][START_REF] Nataf | Exact diagonalization of heisenberg SU(n) models[END_REF]. Given a system with N particles with r different spin-components, one can associate each element of the basis of the corresponding Hilbert space to a Young tableau of N boxes and at most r rows. This splits the Hilbert space into sub-spaces generated by the elements of the basis associated to Young tableaux with the same shape. The dimension of each sub-space coincides with the number of possible SYT one can build from the corresponding Young tableau. Each particle can be labelled with an integer from 1 to N and therefore can be associated to a box of a standard Young tableau. The permutation symmetry of a state can be established by using the convention that the state is symmetric with respect to the exchange of particles associated to boxes in the same line and antisymmetric under exchanges involving boxes in the same column. This method allows us to characterize precisely the symmetry under particle exchange of a state with a non-trivial spin degree of freedom. As an example, we consider the ground state of a two-component Fermi gas. According to the Lieb-Mattis theorem [START_REF] Lieb | Theory of ferromagnetism and the ordering of electronic energy levels[END_REF], in the ground state the system assumes the most symmetric configuration possible. This implies that, when the number of spin up and spin down particles coincides, the permutation symmetry of this state is described by a SYT with two columns and the same number of boxes in each column. For instance when N = 4 the ground state of a two-component Fermi gas is associated to the tableau with the following shape:

. On the other hand, a Bose-Bose mixture in the ground state assumes the most antisymmetric configuration possible [START_REF] Decamp | Strongly correlated one-dimensional Bose-Fermi quantum mixtures: symmetry and correlations[END_REF][START_REF] Fang | Exact solution for the degenerate ground-state manifold of a strongly interacting one-dimensional Bose-Fermi mixture[END_REF]. Consequently, within this convention, the shape of the corresponding Young tableau is: .

2

One-dimensional quantum systems

In this chapter we summarize the physical properties of onedimensional quantum systems. We first introduce the most relevant experiments in the field and then we illustrate some of the theoretical methods usually employed to study onedimensional physics. We first focus on exact analytical methods for different systems and geometries and then we present some useful numerical techniques one can implement when the analytical solution is not available. One-dimensional (1D) systems realize suitable platforms to understand several quantum phenomena and the effects of interactions. For the most part, the reason relies on the large class of one-dimensional models that are either exactly solvable [START_REF] Franchini | An introduction to integrable techniques for one-dimensional quantum systems[END_REF]110] or can be well approximated using analytical or numerical methods. In one dimension, indeed, two-body interactions are markedly different from two or three dimensions. First of all, the particles have fewer neighbours as compared to higher dimensions, therefore the number of possible two-body scatterings is reduced. Moreover, in one-dimensional systems, correlations among the particles are enhanced: the degrees of freedom are constrained and therefore particles cannot avoid each other. The scattering processes occur when particles 'overpass each other', exploiting the tunnel effect. In particular, if we consider the scattering of two identical particles in one dimension, we have no means to distinguish whether two particles have scattered or not, whereas this is not the case in higher dimensions. This implies that, in one dimension, the effect of each collision is at most a phase shift of the state of the system. One-dimensional models have been theoretically studied since the first half of the last century [START_REF] Bethe | Zur theorie der metalle[END_REF][START_REF] Hubbard | Electron correlations in narrow energy bands[END_REF], principally as models of condensed matter systems. However, before the first experimental realizations of systems showing purely 1D features, these systems were mainly considered toy models. In more recent times, 1D systems received great interest due to the wide class of techniques one can implement and to the possibility to address a large number of fundamental questions (see [START_REF] Minguzzi | Strongly interacting trapped one-dimensional quantum gases: Exact solution[END_REF][START_REF] Si Mistakidis | Cold atoms in low dimensions-a laboratory for quantum dynamics[END_REF] and references therein). Moreover, one-dimensional models represent a suitable theoretical basis for quantum algorithms and to perform quantum computation [START_REF] Das | Colloquium: Quantum annealing and analog quantum computation[END_REF][START_REF] Hauke | Perspectives of quantum annealing: Methods and implementations[END_REF].

As we mentioned in Chapter 1, the first experimental realizations of 1D systems at intermediate interactions were performed in the early 2000s. After few years, the strongly interacting regime of 1D systems was also achieved, both at [START_REF] Kinoshita | Observation of a One-Dimensional Tonks-Girardeau gas[END_REF][START_REF] Moritz | Confinement induced molecules in a 1D Fermi gas[END_REF][START_REF] Paredes | Tonks-Girardeau gas of ultracold atoms in an optical lattice[END_REF][START_REF] Kinoshita | Local pair correlations in one-dimensional Bose gases[END_REF] and outof-equilibrium [START_REF] Kinoshita | A quantum Newton's cradle[END_REF]. We show in Fig. 2.1 the spatial density of a strongly interacting Bose gas oscillating in an anharmonic trap, realizing the so-called quantum Newton cradle.

Out-of-equilibrium and thermalization

One of the most important applications of 1D systems is the study of out-of-equilibrium phenomena [START_REF] Fabian | A short introduction to Generalized Hydrodynamics[END_REF][START_REF] Bertini | Finite-temperature transport in one-dimensional quantum lattice models[END_REF]. Indeed, the large number of exact [START_REF] Caux | Time evolution of local observables after quenching to an integrable model[END_REF] and approximate [START_REF] Bertini | Finite-temperature transport in one-dimensional quantum lattice models[END_REF][START_REF] Bastianello | Introduction to the special issue on emergent hydrodynamics in integrable manybody systems[END_REF] methods developed to analyze 1D systems allow one to obtain several results in this direction. One-dimensional models are a useful basis to understand quantum phenomena occurring in higher dimensions, once the microscopic mechanism has been understood in 1D, but also to reveal physical behaviours only occurring in one dimension. This is the case of quantum thermalization and equilibration of one-dimensional quantum systems [START_REF] Rigol | Thermalization and its mechanism for generic isolated quantum systems[END_REF][START_REF] Borgonovi | Quantum chaos and thermalization in isolated systems of interacting particles[END_REF]. For instance, one-dimensional lattices are widely used in the context of quantum chaos and quantum thermalization [128,[START_REF] Santos | Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization[END_REF]. The study of the dynamics in such integrable systems following a small perturbation allows one to perform precise numerical benchmarks of the theory underlying the thermalization of 1D quantum systems.

In quantum dynamics and on a microscopic level, we distinguish equilibration and thermalization as in the following. Consider an isolated system out-of-equilibrium in a Chapter 2. One-dimensional quantum systems

⟨ψ(t)| Ô |ψ(t)⟩ = ∑ kl C k C * l e i h (E l -E k )t ⟨φ l | Ô |φ k ⟩ = = ∑ k |C k | 2 ⟨φ k | Ô |φ k ⟩ + ∑ k̸ =l C k C * l e i h (E l -E k )t ⟨φ l | Ô |φ k ⟩ . (2.1)
Averaging such quantity for large value of t, makes the second term to vanish because of the fast-oscillating exponential and the expectation value of Ô does not depend on time. This phenomenon is called equilibration: the system has approached a stationary state. On the other hand, thermalization only occurs in systems which equilibrate and the following holds:

∑ k |C k | 2 ⟨φ k | Ô |φ k ⟩ = Tr[ Ôρ mc ], (2.2)
where ρ mc is the density matrix of the microcanonical, or thermal, ensemble and Tr[ Ôρ mc ] is the expectation value of Ô evaluated on this ensemble. If the system has some particular constraints, for instance integrability, it reaches equilibrium and the long time average of the expectation value of local observables converge to the one evaluated on the so-called generalized Gibbs ensemble ρ GGE [128,[START_REF] Vidmar | Generalized Gibbs ensemble in integrable lattice models[END_REF]. This ensemble, which is different from ρ mc , can be built starting from the additional conserved charges which add contraints on the Hilbert space. Explicitly, it reads:

ρ GGE = exp{-∑ m ϕ m Îm } Tr[exp{-∑ m ϕ m Îm }] , (2.3)
2.1. Introduction to one-dimensional models where Îm are the integrals of motion and ϕ m are a set of Lagrange multipliers. In a system where the integrals of motion are energy and number of particles, the generalized Gibbs ensemble reduces to the grand-canonical ensemble.

However, most quantum systems thermalize, even if they are isolated. The Eigenstate Thermalization Hypothesis (ETH) [START_REF] Deutsch | Quantum statistical mechanics in a closed system[END_REF][START_REF] Srednicki | Chaos and quantum thermalization[END_REF][START_REF] Rigol | Alternatives to Eigenstate Thermalization[END_REF] provides one possible explanation for this phenomenon. This can be formulated in two different versions, called strong and weak ETH. The former is based on the assumption that every eigenstate of the Hamiltonian dominating the dynamics of an isolated quantum system is in thermal equilibrium. This implies that ⟨φ k | Ô |φ k ⟩ = Tr[ Ôρ mc ] ∀k. In this way, Eq. (2.2) holds for any system which equilibrates. This version of the ETH is believed to hold for most of the quantum systems, while breaks down for the integrable ones and other classes of models [START_REF] Altman | Many-body localization and quantum thermalization[END_REF][START_REF] Nandkishore | Many-body localization and thermalization in quantum statistical mechanics[END_REF]. In these cases, usually one can introduce a weak ETH, which claims that most of the eigenstates of the Hamiltonian dominating the dynamics of an isolated quantum system are in thermal equilibrium. Therefore, the set of eigenstates that are not in thermal equilibrium may inhibit the thermalization of the system depending on the overlap between these states and the initial state of the system. In classical systems, the thermal state is reached when the dynamics starts from a state that has no memory of the previous states of the system. In the quantum regime, the notion of thermal equilibrium inherently belongs to any eigenstate of the Hamiltonian. The dephasing of the eigenstates occurring during the dynamics allows the thermalization to emerge. Figure from [START_REF] Rigol | Thermalization and its mechanism for generic isolated quantum systems[END_REF].

A third phenomenon, as opposed to equilibration and thermalization, which is relevant in quantum dynamics and has been intensively studied in 1D systems, is called prethermalization [START_REF] Mori | Thermalization and prethermalization in isolated quantum systems: a theoretical overview[END_REF]. Typically, this phenomenon occurs where the Hamiltonian of the system can be separated into a dominant part which equilibrates without thermalizing and a small perturbation which drives the system to thermalization. In this case, the system approaches a quasi-steady state, the prethermalized state, before definitely thermalizing. The time scales related to this phenomenon depend on the intensity of the perturbation and on how much the prethermalized and the thermal state are distinguishable.

Chapter 2. One-dimensional quantum systems

Strong interactions and fermionization of bosons

A remarkable phenomenon occurring in 1D ultracold gases is the so-called fermionization. It is a phenomenon that may occur both in bosonic and fermionic systems and it is related to the strongly repulsive regime. One can model the strong repulsion among the particles as an effective Pauli principle, where the particles repel each other. In particular, in the regime where such repulsions can be effectively considered infinite, the wavefunction of the system vanishes each time two particles occupy the same position. In Bose gases, this limit is called the Tonks Girardeau (TG) limit [START_REF] Girardeau | Relationship between systems of impenetrable bosons and fermions in one dimension[END_REF] and one can write the exact wavefunction of the system for different 1D confinement geometries. In this case, the system can be mapped, up to some mapping function, in a non-interacting spinless Fermi gas confined in the same geometry: the strong repulsion plays the role of the Pauli principle among the particles. The difference between the two systems emerges in the explicit expression of the wavefunction. Despite the spectra of the two models coincide, the two wavefunctions have to satisfy different symmetry constraints under particle exchange. Therefore, the wavefunction for the strongly repulsive, or fermionized, bosons has to be modified in order to be symmetric under particle exchange, whereas the wavefunction for a spinless free Fermi gas is completely antisymmetric. Explicitly, for N fermionized bosons, the wavefunction is the following:

Ψ T G (x 1 ...x N ) = N ∏ i< j sign(x i -x j ) Ψ Sl (x 1 ...x N ) (2.4)
where x 1 ...x N are the position of the particles and Ψ Sl is the completely antisymmetric Slater determinant of a gas of N non interacting fermions without spin in the corresponding external potential. The sign function is antisymmetric: the total wavefunction Ψ T G is therefore symmetric under every particle exchange. Such phase applied to the fermionic wavefunction alters the expectation value of some off-diagonal observables, such as the momentum distribution, which is different for TG bosons and non interacting fermions [START_REF] Lenard | Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons[END_REF]. It should be noted that such solution can be extended also to time dependent problem [START_REF] Girardeau | Dark solitons in a one-dimensional condensate of Hard Core bosons[END_REF] and therefore allows us to study the dynamics of such systems.

Remarkably, a similar procedure can be used to study the fermionization of a Fermi gas with spin, where the strong repulsions inhibit particles with different spin to occupy the same positions [START_REF] Girardeau | Effective interactions, Fermi-Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides[END_REF][START_REF] Girardeau | Theory of spinor Fermi and Bose gases in tight atom waveguides[END_REF].

Exact analytical mehtods for one-dimensional Fermi systems

In this section, we review two of the methods implemented to find exact solutions to 1D fermionic quantum models. The first one is called Bethe Ansatz and was introduced in 1931 from Bethe [START_REF] Bethe | Zur theorie der metalle[END_REF]. It allows finding the exact solution of some translationally invariant 1D models, on discrete and continuous geometries. Afterwards, we revise a similar method, that allows us to write the exact solution for systems of multi-component fermions in an external confinement, in the strongly interacting regime.

We stress that aside from the exact methods we describe here, there are several an-2.2. Exact analytical mehtods for one-dimensional Fermi systems alytical methods that provide an approximate solution for one-dimensional models. In particular, many of these methods [START_REF] Fabian | A short introduction to Generalized Hydrodynamics[END_REF][START_REF] Giamarchi | Quantum physics in one dimension[END_REF][START_REF] Cazalilla | One dimensional bosons: From condensed matter systems to ultracold gases[END_REF][START_REF] Doyon | Lecture Notes On Generalised Hydrodynamics[END_REF][START_REF] Alba | Generalized-hydrodynamic approach to inhomogeneous quenches: correlations, entanglement and quantum effects[END_REF][START_REF] Imambekov | One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm[END_REF] relies on low-energies or low-density approximations and they can be used to describe both the equilibrium and the out-ofequilibrium (including finite temperature) properties of one-dimensional systems.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Free Theory

Free Theory

x 1 = x 2 x 2 /L x 1 /L Figure 2
.3: Qualitative scheme to explain Bethe Ansatz for a system of N = 2 particles in the position x 1 and x 2 interacting via contact potential. In each coordinate sector x 1 ≷ x 2 the wavefunction of the system is the solution of the corresponding non-interacting theory.

The interaction can be implemented as a cusp condition (2.5) on each coordinate sector, when x 1 = x 2 .

Bethe Ansatz

The Bethe Ansatz is a very popular method for the solution of 1D models [START_REF] Oelkers | Bethe ansatz study of onedimensional bose and fermi gases with periodic and hard wall boundary conditions[END_REF][START_REF] Gaudin | The Bethe Wavefunction[END_REF][START_REF] Xi-Wen Guan | Fermi gases in one dimension: From Bethe Ansatz to experiments[END_REF][START_REF] Murray | The Bethe ansatz after 75 years[END_REF][START_REF] Fabian Hl Essler | The one-dimensional Hubbard model[END_REF].

It allows one to write the exact solution for the wavefunction of different models at any interaction strength and for an arbitrary number of particles. It reduces the complexity of the problem of the solution of the Schroedinger equation to the one of a system of coupled transcendental equations. Although Bethe Ansatz can be applied to different types of interactions [110], in this thesis, we will only consider systems subjected to contact interaction. Contact interaction is a zero-range interparticle interaction, where two atoms scatter only when they are in the same position. In the context of ultracold atoms, it describes the Van der Waals force [START_REF] Barry R Holstein | The van der Waals interaction[END_REF], which is a short-range interaction that can be both repulsive and attractive. Due to the very low temperatures, only the s-wave scattering is relevant: this allows us to approximate this interaction with a zero-range coupling [START_REF] Yu | Zero-range potentials and their applications in atomic physics[END_REF]. In particular, it is usually described using a Dirac delta multiplied by a real coupling constant V δ = g ∑ i< j δ (x ix j ). The sign of g determines whether the interaction is attractive or repulsive. This functional form is particularly suitable since it induces a discontinuity in the first derivative of the wavefunction [START_REF] David | Introduction to quantum mechanics[END_REF] of magnitude proportional to g. In detail, Chapter 2. One-dimensional quantum systems the potential V δ induces the following cusp condition for the wavefunction of the problem Ψ(x 1 ...x N ):

∂ ∂ x i - ∂ ∂ x i+1 Ψ| x i =x + i+1 - ∂ ∂ x i - ∂ ∂ x i+1 Ψ| x i =x - i+1 = 2m h2 g Ψ| x i =x i+1 , (2.5)
where m is the mass of the particles and h is the reduced Planck constant.

The basic idea of Bethe Ansatz is sketched in Fig. 2.3 for two particles in the position x 1 and x 2 . When x 1 ̸ = x 2 , the model is non-interacting and the wavefunction is the same as the one of the free theory. Therefore, in each coordinate sector x i ≤ x j one can write the wavefunction by diagonalizing the non-interacting Hamiltonian with an additional constraint on the boundary of the sector, where x i = x j . In particular, the constraint is given by Eq. (2.5). Therefore, in these cases, where the interaction can be implemented by imposing geometrical conditions on the wavefunction, one can write the exact solution for the Schroedinger equation at any interaction strength. In the next subsection, we will show a concrete example of a Bethe Ansatz solution for a Fermi gas confined on a 1D ring.

We want to stress that in this thesis we will focus on the so-called coordinate Bethe Ansatz at zero temperature. This method can be extended to finite temperature [START_REF] Stijn | Introduction to the thermodynamic Bethe ansatz[END_REF] or to algebraic approaches [START_REF] Ld Faddeev | Algebraic aspects of the Bethe ansatz[END_REF][START_REF] Vladimir E Korepin | Quantum inverse scattering method and correlation functions[END_REF][START_REF] Wang | Off-diagonal Bethe ansatz for exactly solvable models[END_REF], which are suitable to study larger class of models.

Gaudin-Yang model

In the following, we give a pedagogical example of a Bethe Ansatz solution for a spin- 1 2 Fermi confined in a one-dimensional ring-shaped potential. This model has been solved in 1967 [START_REF] Gaudin | Un systeme a une dimension de fermions en interaction[END_REF][START_REF] Yang | Some exact results for the many-body problem in one dimension with repulsive delta-function interaction[END_REF] using the Bethe Ansatz technique. Subsequently, has been named after the two authors who independently found the solution, as the Gaudin-Yang model. The sketches of the solution we show here can be found in [START_REF] Imambekov | Applications of exact solution for strongly interacting one-dimensional Bose-Fermi mixture: Low-temperature correlation functions, density profiles, and collective modes[END_REF][START_REF] Andrei | Solution of the Kondo problem[END_REF]. To better clarify the method, in Appendix B we provide the solution explicitly for the simple case N = 2 and M = 1.

Consider the following Hamiltonian describing N -M spin up and M spin down fermions:

H GY = - N ∑ i=1 ∂ 2 ∂ x 2 i + 2c ∑ i< j δ (x i -x j ), c = mg h2 (2.6)
The wavefunction must be antisymmetric separately under the exchange of M indexes and under the exchange of N -M indexes. The ansatz for the wavefunction is a piecewise function Ψ(x 1 ...x N ) defined in each coordinate sector Q as:

Ψ Q = ∑ P [QP]e i ∑ j k P( j) x Q( j) , 0 ≤ x Q(1) ≤ x Q(2) ≤ ... ≤ x Q(N) ≤ L, (2.7)
where L is the size of the system and the N quasi-momenta k j are called charge rapidities in Bethe Ansatz jargon. We want the wavefunction to satisfy two requirements:

(I) Continuity:

Ψ| x i =x - i+1 = Ψ| x i =x + i+1 ≡ Ψ| x i =x i+1 ;
(II) Cusp condition, given by Eq.(2.5).

Exact analytical mehtods for one-dimensional Fermi systems

We start from condition (I). Consider two pair of permutations Q, Q ′ and P, P ′ , acting respectively on the position of the particles and on the quasi-momenta, that differ only for a trasposition of the indexes Q(i), Q(i) + 1. We have

Ψ| x Q(i) =x - Q(i)+1 = Ψ| x Q(i) =x + Q(i)+1 ≡ Ψ| x Q(i) =x Q(i)+1 → → ([QP] + [QP ′ ])e i(k Q(i) +k Q(i)+1 )x Q(i) = ([Q ′ P] + [Q ′ P ′ ])e i(k Q(i) +k Q(i)+1 )x Q(i) ,
which yields:

[QP ′ ] = [Q ′ P] + [Q ′ P ′ ] -[QP] (2.8)
Within the same conventions for the permutations, we now consider the cusp condition.

∂ ∂ x i - ∂ ∂ x i+1 Ψ Q ′ | x i =x i+1 - ∂ ∂ x i - ∂ ∂ x i+1 Ψ Q | x i =x i+1 = 2cΨ| x i =x i+1 → → (ik i+1 -ik i ) [Q ′ P] -[Q ′ P ′ ] + [QP] -[QP ′ ] = 2c [QP] + [QP ′ ] (2.9)
Using Eq.(2.8), we get:

(ik i+1 -ik i ) [QP] -[Q ′ P ′ ] = c [QP] + [QP ′ ] , (2.10)
and consequently:

[QP] = Y i,i+1 P(i),P(i)+1 [QP ′ ], Y l,m i, j = - c + i(k i -k j )P lm c -i(k i -k j ) (2.11)
where P lm is the operator permuting the particles in the position x l and x m , ie

P lm [QP] = [Q ′ P].
The operator Y l,m i, j is the operator that permutes the momentum of the particles in the position x l and x m . This allows us to express each phase [QP] as a function of [QI], in each coordinate sector Q. Therefore, to fully specify the wavefunction we need to determine [QI] in the different coordinate sectors. We introduce the combined permutation operator X i j that interchanges both momentum and position of the particles:

X i j = P i j Y i j i j = - cP i j + i(k i -k j ) c -i(k i -k j ) . (2.12)
To recover the ring geometry, we impose periodic boundary conditions (PBCs) on the chain of length L such that Ψ(x 1 , ..

x i = L, ...x N ) = Ψ(x 1 , ..x i = 0, ...x N ), ∀i = 1...N.
We can write them using the operator X i j . The PBCs read:

X j+1, j X j+2, j ...X N, j X 1, j ...X j-1, j [QI] = e ik j L [QI], ∀ j = 1...N (2.13)
The statistics of the particles enters in the operator X i j through the permutation operator P i j . We also see that the only relevant permutation symmetry for the PBCs is the one of the identical coordinate sector. If [QI] is antisymmetric under the exchange of any particles, P i j = -1 and we recover the momentum quantization for spinless fermions e ik j L = 1, whereas for P i j = 1 (ie [QI] is fully symmetric under particle exchange) we have the Bethe equations for a spinless Bose gas [START_REF] Lieb | Exact analysis of an interacting Bose gas. i. the general solution and the ground state[END_REF]. In case of more complicated symmetry, [QI] ≡ ξ 0 follows the permutation properties of the corresponding Young Tableaux. For fermions, it is convenient to consider the problem in the conjugate representation (ie invert rows and columns of the corresponding Young tableaux). The reason behind this choice will Chapter 2. One-dimensional quantum systems be detailed later. In the case of SU(2) fermions, ξ 0 is antisymmetric with respect to the first M coordinates and with respect to the last N -M coordinate. Therefore, in this case, the conjugate representation ϕ is symmetric with respect to the first M indexes and to the last N -M. The PBCs read:

X ′ j+1, j X ′ j+2, j ...X ′ N, j X 1, j ...X ′ j-1, j ϕ = e ik j L ϕ, (2.14)
where

X ′ i, j = - -cP i j + i(k i -k j ) c -i(k i -k j ) ≡ a i j + b i j P i j , (2.15)
and we introduced the quantities a i j = k i -k j k i -k j +ic and b i j = ic k i -k j +ic . Let's consider the case where M = 1. We indicate by y the position of the spin down particle in the coordinate ordering. In this case we impose the following ansatz:

ϕ(y) ≡ F(Λ, y) = y-1 ∏ j=1 k j -Λ + ic/2 k j+1 -Λ -ic/2 (2.16)
Without loss of generality, let's restrict ourselves to the equation for the rapidity k N and compute:

X ′ N-1,N ϕ(y) = a N-1,N ϕ(y) + b N-1,N P N-1,N ϕ(y) = (2.17) 1 2 (1 -δ y,N-1 + 1 -δ y,N ) ϕ(y)+ + δ y,N-1 (a N-1,N ϕ(N -1) + b N-1,N ϕ(N)) + δ y,N (a N-1,N ϕ(N) + b N-1,N ϕ(N -1))
Where in the second line we use the property of P i j implementing symmetric exchanges and a N-1,N + b N-1,N = 1. We now focus on the last line of the previous equation. We have:

a N-1,N ϕ(N -1) + b N-1,N ϕ(N) = (2.18) a N-1,N N-2 ∏ j=1 k j -Λ + ic/2 k j+1 -Λ -ic/2 + b N-1,N N-1 ∏ j=1 k j -Λ + ic/2 k j+1 -Λ -ic/2 = (2.19) a N-1,N + b N-1,N k N-1 -Λ + ic/2 k N -Λ -ic/2 N-2 ∏ j=1 k j -Λ + ic/2 k j+1 -Λ -ic/2 = (2.20) = k N -Λ + ic/2 k N -Λ -ic/2 ϕ(N -1) ≡ µ N (Λ) ϕ(N -1) (2.21)
An analogous calculation for the second term yields:

a N-1,N ϕ(N) + b N-1,N ϕ(N -1) = µ -1 N-1 (Λ)ϕ(N). (2.22)
Where introduced the quantity µ j (Λ) = k j -Λ+ic/2 k j -Λ-ic/2 . As a consequence, we obtain:

X ′ N-1,N ϕ(y) = 1 2 (1 -δ y,N-1 + 1 -δ y,N ) ϕ(y)+ (2.23) + δ y,N-1 µ N (Λ) ϕ(N -1) + δ y,N µ -1 N-1 (Λ)ϕ(N).

Exact analytical mehtods for one-dimensional Fermi systems

We can then apply iteratively X ′ N-2 , X ′ N-3 ... and write the PBCs as:

X ′ 1,N X ′ 2,N ...X ′ N-1,N ϕ(y) = µ N (1 -δ yN )ϕ(y) + δ yN N ∏ j=1 µ -1 i ϕ(N) . (2.24)
We also have to impose PBCs on the spin component of the wavefunction ϕ(N + 1) = ϕ(1). This yields:

N ∏ j=1 k j -Λ + ic/2 k j -Λ -ic/2 = N ∏ j=1 µ j = 1, (2.25)
and, combining of Eqs (2.24) and (2.25):

       e ik j L = µ j = k j -Λ+ic/2 k j -Λ-ic/2 ∏ N j=1 k j -Λ+ic/2 k j -Λ-ic/2 = 1 (2.26)
where we used

∏ N j=1 µ j = 1 → ∏ N j=1 µ -1 j = 1.
The abovementioned procedure can be generalized to the case of M spin down particles. In this case, we impose the following ansatz:

ϕ(y 1 , y 2 ...y M ) = ∑ R A(R) M ∏ i=1 F(Λ R(i) , y i ).
(2.27)

Remarkably, this is the Bethe Ansatz for a spin chain, as originally proposed by Bethe in 1931 [START_REF] Bethe | Zur theorie der metalle[END_REF]. This second ansatz composes the so-called nested Bethe Ansatz, designed to solve systems having a non-trivial permutation symmetry. The structure is the same as the wavefunction (2.7): a weighted superposition of the product of single-particle solutions, that in this case is represented by the solution for a single particle with spin down in Eq. (2.16). In this case, the role of the charge rapidities is played by the M spin rapidities Λ i . The underlying idea of nested Bethe Ansatz is to propose the ansatz (2.7) to write the charge component of the wavefunction and then, in order to solve the spin sector, to use the same functional approach to solve a fictitious spin model, where the particles represent the sites of the chain. Imposing the PBCs on the spin chain allows us to fully reconstruct the spin component of the wavefunction. In this picture, we are able to justify the advantage of using the conjugate representation ϕ in place of the one with the right symmetry ξ 0 . As a matter of fact, the wavefunction of a spin system is symmetric under spin exchange, whereas this is not the case for the spin wavefunction of a fermionic system. In order to use the nested Bethe Ansatz approach and to restore the proper symmetry in the wavefunction, we need to symmetrize the spin wavefunction of the Fermi gas. After we find the solution for ϕ through the spin chain approach, when writing the wavefunction for the original problem we have to add a phase and restore the fermionic symmetry, since Eq. (2.7) depends on ξ 0 .

Using the ansatz (2.27) and following the same procedure showed for the case M = 1, one can obtain the general Bethe equations, ie the generalization of the system (2.26) for Chapter 2. One-dimensional quantum systems a generic M. These equations read:

e ik j L = M ∏ l=1 k j -Λ l + ic/2 k j -Λ l -ic/2
(2.28)

N ∏ j=1 k j -Λ k + ic/2 k j -Λ k -ic/2 = - M ∏ l=1 Λ l -Λ k + ic Λ l -Λ k + ic (2.29)
which encode the periodic boundary conditions for the charge and for the spin sector, highlighting the coupling between the two. Solving the system (2.29) allows us to obtain an explicit expression for the rapidities k j and Λ k and therefore to write the wavefunction. The ground state energy in this case can be consequently written as E GY = ∑ j k 2 j . It is often useful to write the Bethe equations in logarithmic form:

Lk j = 2πI j -2 M ∑ b=1 arctan 2 c (k j -Λ b ) (2.30) N ∑ j=1 2 arctan 2 c (Λ k -k j ) = 2πJ k + M ∑ l=1 2 arctan Λ k -Λ l c (2.31)
In this form, we introduced the quantum numbers I j and J k , which specify the state. They can be integer or semi-integer depending on the value of N and M [START_REF] Yang | Some exact results for the many-body problem in one dimension with repulsive delta-function interaction[END_REF][START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF].

For what concern the ground state, the nature of the solutions of the Bethe equations strongly depends on the sign of the coupling constant c. In the repulsive case, the rapidities k j are real numbers, whereas in the attractive case complex solutions arise: in this case the set of charge rapidities are called strings [START_REF] Woynarovich | Excitations with complex wavenumbers in a Hubbard chain. i. states with one pair of complex wavenumbers[END_REF][START_REF] Woynarovich | Excitations with complex wavenumbers in a Hubbard chain. ii. states with several pairs of complex wavenumbers[END_REF]. The complex solutions are two-by-two complex conjugated, such that the energy of the system is real. If we write explicitly k j = u j + iv j , we see that the energy associated to the state is E = ∑ j u 2 j -∑ j v 2 j . From this expression, we see that the contribution of the imaginary part is a negative binding energy term. Therefore, the presence of strings is related to the presence of bound states, which compose the ground state of the system for attractive interactions. In particular, for a Fermi gas, the absolute value of the imaginary part is the same for all the rapidities. One can obtain an explicit expression for the string in the thermodynamic limit, using the first of equations 2.29. Up to exponential correction in L, the string can be written as:

k j = Λ j ± ic 2 (2.32)
and therefore in this limit one has to solve only the equations to determine the real part of the strings, ie Λ j . For an extensive treatment of the strings in the context of Fermi gases, one can refer to [START_REF] Fabian Hl Essler | The one-dimensional Hubbard model[END_REF].

Lattice regularization: Fermi Hubbard In this section, we revise the discrete limit of the Gaudin-Yang model, the so-called Fermi Hubbard Hamiltonian. This model describes a Fermi gas confined on a 1D optical lattice and with on-site interaction. The Hamiltonian for a system of N particles on N s sites reads:

H FH = -J N s ∑ j=1 ∑ σ =↑,↓ c † j,σ c j+1,σ + H.c. +U N s ∑ j=1 n j,↑ n j,↓ (2.33)
where U is the particles interaction and J is the tunnel amplitude. This model is integrable using nested Bethe Ansatz [START_REF] Fabian Hl Essler | The one-dimensional Hubbard model[END_REF][START_REF] Lieb | Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension[END_REF]. The procedure is analogous to the one shown for the Gaudin-Yang model: the main effect of the lattice is to modify the relation dispersion in E FH = -2J ∑ j cos(k j ), where the unit of length is in this case the lattice spacing. This also implies different Bethe Equations:

e ik j N s = M ∏ l=1 sin(k j ) -Λ l + iU/(4J) sin(k j ) -Λ l -iU/(4J) (2.34) N ∏ j=1 sin(k j ) -Λ k + iU/(4J) sin(k j ) -Λ k -iU/(4J) = - M ∏ l=1 Λ l -Λ k + iU/(2J) Λ l -Λ k + iU/(2J) . ( 2

.35)

As one can see, the sets of Bethe equations are equivalent applying the trasformation k j → sin(k j ) and suitably renormalizing the coupling constant. We stress that in this case the string solutions have a different form in the thermodynamic limit, namely sin(k j ) = Λ j ± iU 4J . Notably, the Fermi Hubbard model reduces to the Gaudin-Yang in the limit of low filling, when the lattice system approaches a continuous one [START_REF] Fabian Hl Essler | The one-dimensional Hubbard model[END_REF].

Fermionized particles in an external potential

In this section, we describe a method for the exact solution of 1D models in the strongly repulsive regime, which also allows us to treat the presence of an arbitrary single-particle potential, provided that it is the same for any component of the system. This method, which has been developed both for multicomponent bosons and fermions [START_REF] Deuretzbacher | Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases[END_REF][START_REF] Volosniev | Strongly interacting confined quantum systems in one dimension[END_REF] allows us to write the exact wavefunction of the system when the interparticle interactions can be considered infinitely strong, but also to access the limit where the coupling is large but finite. Consider the following Hamiltonian, describing a spin- 1 2 Fermi gas of N atoms confined in a generic 1D single-particle potential V (x):

H SU = N ∑ i=1 p 2 i 2m +V (x i ) + g ∑ i< j δ (x i -x j ) ≡ H 0 + g ∑ i< j δ (x i -x j ), (2.36)
where we assume that V (x i ) to be the same for each particle. In the following, we consider the case where

V (x) = mω 2 0 2
x 2 is a harmonic trap of frequency ω 0 . In the limit g → ∞ we can write the wavefunction as [169]

Ψ(x 1 ...x N ) = ∑ P Θ(x P(1) ≤ x P(2) ≤ ...x P(N) ) a P Ψ A (x 1 ...x N ), (2.37)
where the sum on P runs over the N! possible permutations of the N particles, Θ(x P(1) ≤ x P(2) ≤ ...x P(N) is the characteristic function of the coordinate sector individuated by x P(1) ≤ x P(2) ≤ ...x P(N) and a P are sector-dependent phases. The wavefunction

Ψ A (x 1 ...x N )
is a fully antisymmetric product of eigenfunctions of H 0 . Therefore, in the following, Ψ A (x 1 ...x N ) is a Slater determinant of the first N eigenfunctions of the harmonic oscillator. We remark that the phases a P do not depend on the coordinates of the particles: they only contain information about the spin degree of freedom. On the other hand, Ψ A (x 1 ...x N ) only depends on the spatial coordinates and not on the spins of the particles. Consequently, in the wavefunction, we observe a full separation of the charge and of the Chapter 2. One-dimensional quantum systems spin degrees of freedom. This is typical of the infinite-repulsion regime and it occurs also for the Bethe wavefunction. The spectrum of Hamiltonian (2.36) when g → ∞ is divided in different manifolds of energy levels [START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF]. In the particular case we are considering, the energy difference between two consecutive manifolds is hω 0 . The energy levels belonging to the same manifold are all degenerate: they correspond to states with the same spatial component of the wavefunction Ψ A (x 1 ...x N ), but with different a P , which implies different spin ordering of the particles. In this limit, the coefficients a P depend on the spin ordering P and are fixed by the symmetry of the state. We can determine such phases by fixing a spin ordering in the identical sector P = I. By incorporating the phase a I in the normalization of the wavefunction, a generic phase a P depends on the number of particles we have to permute to obtain the ordering P starting from I. Due to the strong repulsive interactions and to the Pauli principle, each time we permute two particles we multiply the wavefunction by a factor (-1). We stress that in principle one has to determine N! phases in order to fully specify the wavefunction (2.40), one for each possible permutation of the N particles. However, the indistinguishability of the particles reduces this number. Consider indeed N particles, N ↑ spin up and N ↓ spin down. An exchange involving two particles with the same spin must have no visible effect on the wavefunction: the two states are indistinguishable. Therefore, the number of independent phases to compute is N! N ↑ !N ↓ ! . As we mentioned, starting from this solution, one can also access the strongly interacting regime, where the coupling constant is assumed to be large but finite. This can be done by determining perturbatively the corrections to the energy to the first order in 1/g. In this interacting regime the degeneracy between the states in the same manifold is lifted (see Fig. 2.4 for a qualitative scheme) and the energy spacings inside the same manifold 2.2. Exact analytical mehtods for one-dimensional Fermi systems are of the order 1/g. Therefore, in order to determine the ground state of the system to the first order in 1/g, one can compute the slope of the energy curve corresponding to each energy level, as a function of 1/g. The energy level associated to the largest slope, thus the one departing "faster" from the degeneracy point as g decreases, corresponds to the ground state for large and finite g. Remarkably, the calculation of the slope provides a complete ordering of the spectrum at large g, enabling to compute the proper sequence of all the excited states. As a consequence, solving the Schroedinger equation to the first order in 1/g is equivalent to maximizing the derivative of the energy with respect to 1/g, which can be done using the Feynman-Hellman theorem of perturbation theory.

Moreover, in this interacting regime, maximizing the slope of the energy curves is in turn equivalent to minimizing the energy of a fictitious spin chain [START_REF] Deuretzbacher | Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases[END_REF][START_REF] Volosniev | Engineering the dynamics of effective spin-chain models for strongly interacting atomic gases[END_REF], where particles play the role of the sites. In particular, Hamiltonian (2.36) to the first order in 1/g can be mapped in an isotropic Heisenberg antiferromagnet with site-dependent coupling and open boundary conditions:

H s = E F - N-1 ∑ i=1 J i I + N-1 ∑ i=1 J i P i,i+1 , (2.38)
where E F = N 2 hω 0 /2 is the energy of the system at 1/g = 0 and P i,i+1 = 1 2 (I + σ k • σ k+1 ) is the permutation operator for spin- 1 2 fermions. We report the explicit calculation of the mapping in Appendix D. The coefficients J i encodes the information about the external potentail of the original Hamiltonian (2.36): their explicit expression reads:

J i = 1 g ∞ -∞ dx 1 ...dx N δ (x i -x i+1 ) θ (x 1 < ... < x N ) ∂ Ψ A ∂ x i 2 .
(2.39)

Notably, this mapping holds for any manifold separately. The information about the manifold is contained in the coefficients J i , through the dependence on the antisymmetric wavefunction Ψ A , which fixes the occupied modes of the harmonic oscillator.

In order to write the wavefunction (2.40) to the first order in 1/g, one has to compute the phases a P in this interaction regime. These can be obtained by writing the Hamiltonian (2.38) on a basis where each vector is associated to each possible spin ordering P and then computing its spectrum. The projection of the ground state of (2.38) on the elements of the basis gives the weight of spin ordering P in the wavefunction, ie the phases a P to be inserted in (2.40). In order to write the wavefunction for an excited state belonging to the same energy manifold, one should consider the projections of the excited states of Hamiltonian (2.38) on each spin ordering to access the corresponding phases.

We also remark that in the limit ω 0 → 0, Hamiltonian (2.36) is the Gaudin-Yang Hamiltonian (2.6). In this case, the mapping on the spin chain still holds and all the J i coincide [START_REF] Volosniev | Strongly interacting confined quantum systems in one dimension[END_REF]. It is a known result from the Bethe Ansatz solution of (2.6) that to the first order in 1/g for large g, the spectrum of the Gaudin-Yang model can be mapped in an isotropic Heisenberg antiferromagnet [START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF][START_REF] Fv Kusmartsev | Magnetic resonance on a ring of aromatic molecules[END_REF]. This claim will be detailed more in Section 2.3.

As we mentioned, in the fermionic case, to the first order in 1/g, the ground state is an antiferromagnet. This is consistent with the Lieb-Mattis theorem [START_REF] Lieb | Theory of ferromagnetism and the ordering of electronic energy levels[END_REF], which claims that a spin- 1 2 fermionic system in the ground state assumes the most symmetric configuration possible.

Chapter 2. One-dimensional quantum systems

Dynamics

This method can also be applied to study the dynamics of fermionized particles confined in an external potential: we will provide an explicit application in Chapter 5. In the following, we sketch the details of the method for a generic dynamical protocol. As we mentioned, the wavefunction (2.40) is factorized in a term depending on the spin and in a term depending on the spatial coordinates. Therefore, in the general case, the timedependent wavefunction of the system can be written at g → ∞ as:

Ψ(x 1 ...x N ;t) = ∑ P Θ(x P(1) ≤ x P(2) ≤ ...x P(N) ) a P (t) Ψ A (x 1 ...x N ;t), (2.40)
where the coefficients a P (t) encode the time evolution of the spin sector and Ψ A (x 1 ...x N ;t) is responsable for the dynamics of the charge sector. Remarkably, the time-dependent antisymmetric function Ψ A (x 1 ...x N ;t) contains all the information about the external potential, therefore its functional form strongly depends on the quench performed on the external potential. A generic and non-trivial quench in the potential induces a time dependence in Eq. (2.39), such that J i → J i (t). However, the mapping on the spin Hamiltonian still holds: the time dependence of the coefficient implies a trasformation H s → H s (t), which depends on the quench. The time dependent antisymmetric function

Ψ A (x 1 ...x N ;t) is obtained at g → ∞ as eigenfunction of the Hamiltonian H 0 (t) = ∑ N i=1 p 2 i 2m +V (x i ,t) ,
where V (x i ,t) can be interpreted as a piecewise potential describing the quench. In order to specify the wavefunction to the first order in 1/g one has, in analogy with the static case, to determine the coefficients a P (t), at any time t. This can be done by evolving the initial state of the spin sector |χ(t = 0)⟩ using the Hamiltonian H s (t), where the nontrivial time evolution of the spatial sector is encoded in time-dependent coefficients J i (t).

One can solve the dynamics at any time to determine the state |χ(t)⟩ = e -iH s (t)t |χ(0)⟩.

In analogy with the static case, the projection of |χ(t)⟩ over the different spin ordering labelling the base of the Hilbert space gives the coefficient a P (t) corresponding to the coefficient of the spin ordering P to the time t. Once the wavefunction is fully specified at any time t, one can use it to calculate any relevant observable.

Bethe Ansatz vs fermionized solution

In this section, we provide a comparison between the two exact analytical methods we described so far. We mentioned that Bethe Ansatz provides the exact solution for 1D models at any interaction strength in a homogeneous potential (eg a ring). On the other hand, the fermionized exact solution holds only in the strongly interacting regime, but allows studying non translationally invariant systems (eg harmonic traps). However, this second solution also describes the strongly interacting regime of 1D homogeneous systems. In this section, we provide a link between the two methods. First, we provide the strong-coupling expansion of the Bethe equations (2.31) and show that to the first order in 1/c the energy of the system can be mapped in the one of an isotropic and antiferromagnetic spin chain. This will give us an expression of the coefficients J i as a function of the charge rapidities of the Bethe Ansatz. As we mentioned, in homogeneous systems all 2.3. Bethe Ansatz vs fermionized solution the coefficients coincide and J i ≡ J [START_REF] Volosniev | Strongly interacting confined quantum systems in one dimension[END_REF].

Using the fermionized solution to describe the strongly interacting regime of a homogeneous system might look redundant, provided the existence of an exact Bethe Ansatz solution which holds for any interaction strength. However, the advantages of the perturbative solution derive from the easier functional expression of the wavefunction. As we show in the following, the Bethe equations are non-trivial to the first order in 1/c, especially for what concerns the spin-rapidities Λ j . On the other hand, the fermionized solution can be specified by means of a numerically exact diagonalization of an isotropic spin chain. Moreover, the factorized form (in terms of spin and charge degrees of freedom) of the fermionized wavefunction allows one to obtain suitable expressions for the expectation values of the observables, for instance the correlations functions, which are hindered by the complicated form of the Bethe wavefunction. Lastly, if one wants to access approximate solutions for a mesoscopically large number of particles, it is possible to implement numerical methods (see Section 2.4) to diagonalize the spin Hamiltonian H s even for very large Hilbert spaces. On the other hand, the structure of the Bethe equations does not allow easy simplifications for large but finite number of particles in a finite system.

Consider equations (2.31). In this case, we have I j = M 2 (mod 1) and J a = N-M+1

2

(mod 1), N and M being the number of particles and the number of spin down particles respectively. Introducing the rescaled spin rapidities λ a = 2Λ a c , ∀a and performing a first-order expansion in 1/c we get:

Lk j = 2πI j + 2 M ∑ b=1 arctan λ b - 4 c ∑ b k j 1 + λ 2 b (2.41) 2N arctan λ a - 4 c ∑ j k j 1 1 + λ 2 a = 2πJ a + M ∑ b=1 2 arctan λ a -λ b 2 , (2.42)
where we used the trigonometric relation arctan(a + x) = arctan(a) + x 1+a 2 + o(x 2 ) that holds for small x. A straightforward substitution yields:

Lk j = 2πI j + 2π N ∑ a J a + 4 Nc ∑ j k j 1 1 + λ 2 a - 4 c ∑ b k j 1 + λ 2 b (2.43) 2N arctan λ a - 4 c ∑ j k j 1 1 + λ 2 a = 2πJ a + M ∑ b=1 2 arctan λ a -λ b 2 , (2.44)
The momentum is

P = ∑ j k ∞ j = 2π L ( ∑ j I j + ∑ a J a ) (2.45)
where we defined the rapidities k ∞ j = 2π L I j + 2π NL ∑ a J a . The quantum numbers I j and J a are chosen such that P = 0 in the ground state of the charge sector. We remark that the total momentum P does not contain any 1/c correction: it only depends on the properties of the gas at infinitely repulsive interactions.

The first-order correction to the energy is:

δ E F = ∑ j k 2 j -∑ j (k ∞ j ) 2 = - 8 Lc ∑ j (k ∞ j ) 2 ∑ b 1 1 + λ 2 b (2.46)
Chapter 2. One-dimensional quantum systems where we used the relation ∑ j k j = 0. We define

J = 4 Lc ∑ j (k ∞ j ) 2 : this provides δ E F = -2J ∑ b 1 1+λ 2 b
, which coincides with the energy of an antiferromagnetic Heisenberg chain, up to a constant shift [START_REF] Bethe | Zur theorie der metalle[END_REF][START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF]. We also remark that the Bethe equations (2.44) coincides with the Bethe equation of the isotropic spin chain [START_REF] Bethe | Zur theorie der metalle[END_REF][START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF].

As a useful remark, we provide a closed and analytical relation for the energy ∑ N j=1 (k ∞ j ) 2 in the ground state, as a function of the number of particles. Consider an odd number of particles: in the ground state we have ∑ a J a = 0 for any M, which implies that in this case k ∞ j = 2π L I j . The quantum numbers I j are integers. When N is odd they assume the following form in the ground state

I j = -N-1
2 + ( j -1), j = 1...N. This implies:

(k ∞ j ) 2 = 4π 2 L 2 - N -1 2 + j -1 2 = 4π 2 L 2 - N + 1 2 + j 2 = (2.47) = 4π 2 L 2 (N + 1) 2 4 + j 2 -j(N + 1) (2.48)
We recall the identities ∑ N j j = N(N+1) 2 and ∑ N j j 2 = N(1+N)(1+2N)

6

. Therefore:

N ∑ j (k ∞ j ) 2 = 4π 2 L 2 N(N + 1) 2 4 + N ∑ j j 2 -(N + 1) N ∑ j j = (2.49) = 4π 2 L 2 N(N + 1) 2 4 + N(1 + N)(1 + 2N) 6 - N(N + 1) 2 2 = = 4π 2 L 2 N(N 2 -1) 12 = π 2 L 2 N(N -1) 2 3 .
This expression gives an analytical closed form for the coefficient J to be used in the fermionized solution, when the system can be mapped on the isotropic spin chain.

Numerical methods

The analytical methods described in 2.2 give access to an exact, or sometimes very accurate, solution of some particular examples of 1D models. Nevertheless, a large part of the physical models is not integrable. Moreover, the analytical solution provides a functional form for the wavefunction, which can be fully specified by solving some transcendental equation, as in the case of Bethe Ansatz. Usually, this does not guarantee easy access to expectation values of the observables, ie to the outcomes of the experiments. In particular, when the system exceeds the regime of the few-body problems [START_REF] Sowiński | One-dimensional mixtures of several ultracold atoms: a review[END_REF], explicitly computing values to benchmark the theory is really complicated. Consequently, it may be necessary to use numerical methods, either approximate or numerically exact, to avoid these issues. One can use these methods to approach problems at the equilibrium, thus implementing minimization algorithms to find the ground state of a given Hamiltonian, and out-of-equilibrium. In this last case, there are several methods to solve the timedependent Schroedinger equation up to a fixed precision and access the time evolution of physical observables. In the following, we will revise some of the methods used for 1D quantum systems with short (or zero) range interactions.

Numerical methods

Statics

The most straightforward method to obtain the ground state of a given Hamiltonian is exact diagonalization. This method consists in writing the Hamiltonian in a computationally efficient form and then using some diagonalization algorithm to compute the whole set of eigenvalues, or just a subset of them [START_REF] Zhang | Exact diagonalization: the Bose-Hubbard model as an example[END_REF]. The advantage of this method is that provides a very accurate description of the system: once the whole spectrum (or some relevant part of it) is computed, one can obtain expectation values of the observables on the eigenstates of the Hamiltonian. The main challenge is to store efficiently the Hamiltonian in a matrix form since it is defined on the whole Hilbert space. In this case, if the Hilbert space is infinite-dimensional, one has to impose a cut-off on the number of states to be considered and the analysis cannot be exact. On the other hand, if the size of the Hilbert space is finite, one can access the whole spectrum. This implies that exact diagonalization is particularly used to study systems such as spin chain [START_REF] Weinberg | QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains[END_REF][START_REF] Weinberg | QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems. Part II: bosons, fermions and higher spins[END_REF] or systems on optical lattices [START_REF] Weiße | Exact diagonalization techniques[END_REF]. The limits of exact diagonalization arise as the system size increases: even for finite-dimensional Hilbert spaces, the size of the latter often grows exponentially with the size of the system. Even once one implements all the possible symmetries and simplifications available for the model, accessing many-body physics is generally very challenging.

Density Matrix Renormalization Group One of the most powerful methods that can be implemented to find the ground state of a low-dimensional strongly correlated Hamiltonian is the Density Matrix Renormalization Group (DMRG) [START_REF] Schollwöck | The density-matrix renormalization group[END_REF][START_REF] Schollwöck | The density-matrix renormalization group in the age of matrix product states[END_REF][START_REF] Steven | Density-matrix algorithms for quantum renormalization groups[END_REF][START_REF] Steven | Density matrix formulation for quantum renormalization groups[END_REF]. In essence, the DMRG algorithm provides an efficient truncation of the Hilbert space of the system and it's structured as follows. Assuming we want to find the ground state of a spin chain of N s sites, the DMRG algorithm first splits the system into two partitions, which are called the left and the right block. At each step, the state of the total system, called superblock, can be written as a tensor product of the states of the two blocks. In order to find the ground state of a single block, one usually needs to truncate the number of states in the Hilbert space, which grows exponentially with the system size, to a Chapter 2. One-dimensional quantum systems maximum value χ, called the bond dimension. In particular, the algorithm computes the reduced density matrix (RDM) of the single block on a truncated basis composed of the eigenvectors of the RDM itself associated to the largest χ eigenvalues. This choice is justified by the fact that the eigenvalues of the density matrix specify how much the corresponding eigenstates of the RDM (ie the projector on the different areas of the Hilbert space) contribute to the ground state of the single block.

Once the algorithm determines the ground state of the superblock, the size of one of the two blocks, for instance the left, is iteratively increased by one site at the expense of the other. At each step, the ground state of the new superblock is computed. When the left block reaches the targeted size of the system, the process is reversed and the right one starts to grow, while the left one shrinks back to the initial position. This procedure, which is called sweep, is repeated until the algorithm converges and the ground state of the whole system is variationally determined. This algorithm allows one to access the ground state of large 1D systems with great accuracy.

We want to stress that the DMRG algorithm relies on the approximation of the ground state of the system as a product state of the ground states of the two blocks. This approximation is particularly suitable when the two partitions are not very entangled. Since during the sweep the ground state is approximated for different partitions of the system, this property must hold regardless of the choice of the blocks. In this sense, DMRG is particularly effective to find the ground state of short-ranged Hamiltonians. In these systems, entanglement entropy follows (up to logarithmic corrections) the so-called area law (See for an extensive treatment [START_REF] Eisert | Colloquium: Area laws for the entanglement entropy[END_REF] and references therein), which implies that the entanglement entropy scales as the surface of the bipartition. In particular, in one dimension, being the surface of the system zero-dimensional, the entanglement entropy does not scale with the size of the system. The entanglement entropy is directly related to the number of states in the truncated basis: approximately we have that χ ≈ 2 S E , where S E is the entanglement entropy of one of the blocks. Therefore, an increase in the system size does not imply a drastic increase in the number of states required to suitably describe the system.

Dynamics

One of the most difficult tasks to address in strongly correlated quantum systems is the study of the out-of-equilibrium behaviour. In the same notation of Section 2.1.2, if we consider an isolated quantum system we can decompose any state |ψ⟩ on the basis of eigenvectors |φ k ⟩ of the Hamiltonian H. The time evolution of the state follows from the Schroedinger equation and reads:

|ψ(t)⟩ = e -i h Ht |ψ⟩ = ∑ k C * k C k e -i h E k t |φ k ⟩ . (2.50)
From this expression, we see that in order to fully describe the state |ψ(t)⟩ at any time, we need information about the whole spectrum of H. For systems where exact diagonalization routines fail, one has to implement approximate algorithms and efficiently truncate the Hilbert space. In the following, we briefly revise the Time-Evolving Block Decimation (TEBD) algorithm [START_REF] Vidal | Efficient simulation of one-dimensional quantum many-body systems[END_REF][START_REF] Vidal | Efficient classical simulation of slightly entangled quantum computations[END_REF], one of the classical methods to describe the dynamics of many-body quantum systems. This method is based on the approximation of the evolution operator e -i h Ht . Remarkably, this algorithm we present in the next paragraphs can be suitably embedded in DMRG codes: the minimization algorithm can be used to generate 2.4. Numerical methods the precise initial state of the dynamics and then one can simulate a quench dynamics by performing the quantum evolution with a different Hamiltonian. Moreover, the effective truncation of the Hilbert space provided by the DMRG can be exploited to reduce the complexity of the algorithms for the dynamics. A more detailed presentation of methods to study quantum dynamics is available in Refs. [START_REF] John Daley | Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces[END_REF][START_REF] Schollwöck | Methods for time dependence in DMRG[END_REF][START_REF] Gobert | Realtime dynamics in spin-1 2 chains with adaptive time-dependent density matrix renormalization group[END_REF][START_REF] Kramer | A review of the time-dependent variational principle[END_REF][START_REF] Jutho Haegeman | Time-Dependent Variational Principle for quantum lattices[END_REF]. Notably, some of these methods rely on very different approach than the one presented in the next paragraph, such as the approximation of the vector |ψ(t + δt)⟩ = e -i h Hδt |ψ(t)⟩ at each time step δt rather than the operator e -i h Ht .

Time-Evolving Block Decimation Consider a local Hamiltonian with nearest-neighbours interactions defined on a one-dimensional chain. We can split the chain into even and odd sites and write the Hamiltonian as:

H NN = ∑ j odd F j, j+1 ∑ j even G j, j+1 . (2.51)
The two local terms F j, j+1 and G j, j+1 in general do not commute if they share one site. Therefore, the time-evolution operator can be approximated at each time step δt using the Trotter formula as:

e -i h Hδt = ∏ j odd e -i h F j, j+1 δt ∏ j even e -i h G j, j+1 δt + O(δt 2 ), (2.52)
where the approximation holds to the first order in δt. In order to obtain more sophisticated results, one can implement the algorithm ad the following orders in δt: however the computational cost of the calculation increases. An iterative application of decomposition (2.52) provides the state at any time t with an error that has a lower bound the order chosen for the Trotter formula. A further error source comes from the truncation of the Hilbert space, which can be controlled by increasing the number of states χ we include in the description of the Hamiltonian.

3

Probing the BCS-BEC crossover with persistent currents

In this chapter, we study the persistent currents of an attractive Fermi gas confined in a tightly-confining ring trap and subjected to an artificial gauge field all through the BCS-BEC crossover. At weak attractions, on the BCS side, fermions display a parity effect in the persistent currents, ie their response to the gauge field is paramagnetic or diamagnetic depending on the number of pairs on the ring. At resonance and on the BEC side of the crossover we find a doubling of the periodicity of the ground-state energy as a function of the artificial gauge field and the disappearance of the parity effect, indicating that persistent currents can be used to infer the formation of tightlybound bosonic pairs. Our predictions can be accessed in ultracold atoms experiments through noise interferograms.

Introduction

A gas of weakly attractive spin-1 2 fermions can form bound pairs with opposite spin and condense into the BCS paired regime. On the other hand, particles with integer spin can display Bose-Einstein condensation. Despite BCS pairing and BEC being two distinct physical phenomena, they have been intensively studied as two different regimes that may occur in the same system. As introduced in Section 1.2.1, the evolution between these two regimes is called BCS-BEC crossover.

Recently, important progress has been achieved in the field, allowing coherent manipulation of atoms in trapping potentials with wide ranges of intensities and shapes, in an unprecedented precise manner [START_REF] Kwon | Strongly correlated superfluid order parameters from dc Josephson supercurrents[END_REF][START_REF] Zürn | Pairing in few-fermion systems with attractive interactions[END_REF]. In the context of atomtronics, as introduced in Section 1.1.6, by harnessing current states in an explicit way one can effectively widen the scope of quantum simulators and emulators to probe quantum phases of matter. Specifically, in the spirit of solid-state physics I-V (current-voltage) characteristics, the different many-body quantum regimes are characterized in terms of the current flowing through the cold atomic system. Here, we consider this approach to study the BCS-BEC crossover: we show how the persistent current in attracting fermionic systems confined in ring-shape lattice potential and pierced by an artificial gauge field provides a novel way to tell apart the BCS and BEC regimes.

Modeling the BCS-BEC crossover in 1D needs to be done differently than in the three-dimensional case [START_REF] Micnas | Superconductivity in narrow-band systems with local nonretarded attractive interactions[END_REF][START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF][START_REF] Chien | Superfuid-insulator transitions at noninteger filling in optical lattices of fermionic atoms[END_REF]. We follow the same strategy as in [START_REF] Fuchs | Exactly solvable model of the BCS-BEC crossover[END_REF]: we describe the system on the BCS side using a model of fermions with attractive contact interactions. At resonance, the system can be described as a Tonks-Girardeau gas of fermionized bosonic pairs [START_REF] Iida | Exact analysis of δ -function attractive fermions and repulsive bosons in one-dimension[END_REF], already introduced in Chapter 2. The BEC side of the crossover is described by a bosonic model for the pairs with contact interactions, to which the attractive fermionic model is continuously connected. By applying exact Bethe Ansatz methods corroborated by DMRG simulations, we access all the interacting regimes, ranging from weak to strong attractions.

Our main results are summarized in Fig. 3.1 and 3.2. We rely on the Leggett theorem introduced in Section 1.2.2 to demonstrate that the BCS-BEC crossover is marked by clear features of the periodicity of the persistent currents. Accordingly, assuming that the total number of particles is N = 2n, the persistent current of a gas of interacting spin- 1 2 fermions is predicted to be parity dependent: for even number n of pairs of fermions, the system behaves as a paramagnet with a non-vanishing persistent current at zero effective magnetic field; for odd n instead, the system behaves as a diamagnet (vanishing persistent current at zero field). This is the generalization of the parity effect described in Section 1.2.2 to particles with an internal spin degree of freedom.

For our specific problem, we find that while the persistent current displays clear parity dependence in the BCS regime, the latter is washed out for strongly attracting pairs. This indicates that at resonance and in the BEC regime, fermionic pairs behave as pointlike bosons, which are predicted not to show parity effects, as anticipated in Chapter 1. Finally, inspired by a procedure developed for bosonic condensates [START_REF] Corman | Quench-induced supercurrents in an annular Bose gas[END_REF][START_REF] Eckel | Interferometric measurement of the current-phase relationship of a superfluid weak link[END_REF][START_REF] Mathew | Self-heterodyne detection of the in situ phase of an atomic superconducting quantum interference device[END_REF][START_REF] Wright | Driving phase slips in a superfluid atom circuit with a rotating weak link[END_REF], we propose a protocol to evidence the parity effect in the persistent current from noise correlations, based on the self-heterodyne detection of the phase of the many-body wavefunction, which has already been sketched in Chapter 1. The correlators are all evaluated in r ′ = (R, 0) and t = 0.3 ω -1 0 , where ω 0 is the frequency of each lattice well. The noise correlator is expressed in unit of 1/R 2 . A circulating state is characterised by a spiral-like correlator, not symmetric by inversion with respect to the y = 0 axis. Chapter 3. Probing the BCS-BEC crossover with persistent currents

Introduction

The model

We consider a gas of degenerate fermions confined in a lattice ring of radius R and pierced by an effective gauge field. Such system is a paradigmatic example of atomtronic circuit [197,[START_REF] Dumke | Roadmap on quantum optical systems[END_REF].

To address the whole BCS-BEC crossover, we exploit the possibility to tune the interaction strength in ultracold atoms, eg across a confinement-induced resonance [START_REF] Olshanii | Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons[END_REF][START_REF] Valiente | Quasi-one-dimensional scattering in a discrete model[END_REF] as introduced in Chapter 1.

To describe the BCS side of the crossover, we use the one-dimensional attractive Fermi-Hubbard Hamiltonian presented in Section 2.2.1. In addition to that, we consider an artificial gauge flux added by means of Peierl's substitution [200]. After this transformation, for an even number N of particles of mass m on a lattice of N s sites, Eq. 2.33 reads:

ĤFH = -J N s ∑ j=1 ∑ σ =↑,↓ e i Ωc † j,σ c j+1,σ + H.c. +U N s ∑ j=1 n j,↑ n j,↓ (3.1)
where U < 0 and J is the tunnel amplitude. The normalized flux is Ω = 2πΩ N s Ω 0 , where Ω is the rotation frequency induced by the artificial gauge field and Ω 0 = h/mR 2 . As anticipated in Section 2.2.1,the Hamiltonian (3.1) is solvable by Bethe Ansatz (the presence of the flux does not affect the integrability). Since we consider attractive interactions, the ground state depends on charge rapidities k j with a non zero imaginary part. This indicates the formation of bound states: we can set k j,± = u j ± iv j . Excitations on top of the ground states are obtained either by pair breaking (gapped spin excitations), by pair re-arrangements (gapless charge excitations), or by combinations of thereof [START_REF] Andrei | Integrable models in condensed matter physics[END_REF]. We note that solutions deviate from the infinite-size system string form presented in Section 2.2.1 when UN s /J is not sufficiently large. This may happen on a lattice ring at very weak interactions. We provide a solution to this model at any interaction strength and in the total spin zero sector.

Solution of Bethe equations

In presence of the flux, the Bethe equations for the Fermi Hubbard model slightly differ from Eqs. (2.35). They read:

e iN s k j = e i Ω M ∏ l=1 -Λ l + sin(k j ) + iU 4 -Λ l + sin(k j ) -iU 4 N ∏ j=1 -Λ α + sin(k j ) + iU 4 -Λ α + sin(k j ) -iU 4 = M ∏ β =1 k̸ =m -Λ α + Λ β + iU 2 -Λ α + Λ β -iU 2 (3.2)
where M is the number of spin down particles. In order the Bethe equations to have a well defined solution, the rapidities must be distinct from each other. The charge rapidities fix the center of mass momentum and the energy eigenvalues, respectively P = ∑ N j=1 k j and E FH = -2J ∑ N j=1 cos(k j ). In order to solve Eq.(3.2) we implement an iterative method: we compute the solution at fixed interaction U 0 and then use this value as initial condition for a root-finding algorithm to determine the solution at increasing attractions. We choose as initial step the 3.2. The model solution for U = 0, which can be easily obtained analytically. Finally, to further benchmark our results, we compare the Bethe Ansatz solution and the DMRG calculation at different interaction regimes. In Fig. 3.3 we show an example of solutions of Bethe equations for the Hubbard model as a function of the dimensionless coupling constant U/J. For U = 0 and Ω = 0 the rapidities corresponding to the ground state of the system have a center of mass momentum P = 4π/N s for N = 4 and P = 0 for N = 6. Other branches of the excitation spectrum are obtained by choosing different values of the center of mass momentum.

Another type of solutions is shown in Fig. 3.4. They correspond to the energy branch centered in Ω/Ω 0 = 0 in Fig 3 .1, ie the branch responsible for the doubling of periodicity. Remarkably, in this case, the charge rapidities have an imaginary part only for finite interactions |U| > |U c |, where U c depends on the density of the gas, showing that this branch is not connected with the non-interacting solution. We also find that U c decreases at increasing system size.

In the small-|U| limit, the connection between the 1D-Hubbard model and BCS theory is well-established [START_REF] Xi-Wen Guan | Fermi gases in one dimension: From Bethe Ansatz to experiments[END_REF][START_REF] Marsiglio | Evaluation of the BCS approximation for the attractive Hubbard model in one dimension[END_REF].

The BCS regime corresponds to the limit where the binding energy is smaller than the kinetic energy of the particles, and the bound states are localized on spatial scales much larger than the average interparticle distances [START_REF] Batchelor | Exact results for the 1D interacting Fermi gas with arbitrary polarization[END_REF]. The known BCS exponential dependence of the energy gap on interaction was also obtained [START_REF] Ya | New method in the theory of a weakly ideal one-dimensional Fermi gas. correlation functions[END_REF].

For strongly attractive interactions, the Bethe Ansatz solution indicates the formation of a Tonks-Girardeau gas of fermionized bosons. Hence, in the spirit of [START_REF] Fuchs | Exactly solvable model of the BCS-BEC crossover[END_REF], we describe the BEC side of the crossover by a bosonic Hubbard model, where bosons correspond to 3.2. The model the fermionic pairs:

ĤBH = -J B N s ∑ j=1 e i Ωb † j b j+1 + H.c. + U B 2 N s ∑ j=1 n j (n j -1). (3.3)
Here J B is the tunnel amplitude for pairs and U B is the pair-pair interaction strength. We remark that in contrast with the Fermi-Hubbard model, the bosonic one is not integrable [START_REF] Choy | Failure of Bethe-Ansatz solutions of generalisations of the Hubbard chain to arbitrary permutation symmetry[END_REF][START_REF] Choy | Some exact results for a degenerate Hubbard model in one dimension[END_REF]: there is no exact solution for this model.

To address the whole BCS-BEC crossover, in our work we combine Bethe Ansatz calculations and DMRG simulations to calculate ground and excited state energy branches as a function of Ω. DMRG is also used to calculate noise correlators.

Fermi Hubbard versus Bose Hubbard models

In this section we discuss the relations between the coupling parameters in the Fermi and in the Bose Hubbard Hamiltonians. These are the lattice regularization of the Gaudin-Yang and the Lieb-Liniger models respectively. The first one describes a one-dimensional gas of N F fermions with contact interactions confined on a homogeneous ring of radius R. The Hamiltonian has been extensively studied in Chapter 2 and is solvable using Bethe Ansatz. In the notation of Chapter 2, introducing the density of the gas as n F = N F /(2πR), one can express the coupling constant in a dimensionless form γ F = c n F . The Lieb-Liniger model is an integrable model describing the same setup, but for a gas of N B bosons. The Hamiltonian can be written in a similar way [START_REF] Lieb | Exact analysis of an interacting Bose gas. i. the general solution and the ground state[END_REF]:

H LL = - h2 2m B N B ∑ i=1 ∂ 2 ∂ x 2 i + g B ∑ i̸ = j δ (x i -x j ). (3.4)
The coupling constant can be equivalently expressed in dimensionless unit using

γ B = m B g B h2 n B
, where n B = N B /(2πR) is the density of the Bose gas. It can be shown [START_REF] Gaudin | The Bethe Wavefunction[END_REF][START_REF] Iida | Exact analysis of δ -function attractive fermions and repulsive bosons in one-dimension[END_REF] that in thermodynamic limit the ground state energies of the two models can be mapped one into the other when γ F ≪ -1 and γ B ≫ 1. Such mapping also implies a rescaling of the densities and of the masses of the particles such that n B = n F /2 and m B = 2m, implying that γ B = 4|γ F |. Remarkably, such equivalence holds up to the second order in 1/γ F . When we discretize the Gaudin-Yang model to map it onto the Fermi Hubbard one, the dimensionless coupling parameter U/J is related to the one of the continuous theory by γ F = N s N F U J . An analogous relation holds for the Lieb-Liniger and the Bose Hubbard models [START_REF] Polo | Exact results for persistent currents of two bosons in a ring lattice[END_REF]. From the above considerations, we estimate the scaling of the dimensionless parameter in the mapping between the fermionic and the bosonic model. Taking into account the fact that on the bosonic side of the mapping we consider N/2 dimers of fermions with double mass 2m, we obtain U B /J B = 2|U|/J. For the finite lattice, corresponding to our ring geometry, no analogue proof as the one given in [START_REF] Gaudin | The Bethe Wavefunction[END_REF][START_REF] Iida | Exact analysis of δ -function attractive fermions and repulsive bosons in one-dimension[END_REF] is available. We have analytically checked the equivalence among ground state energies of the two models and the rescaling relations when N F = 4 and therefore N B = 2. Indeed, in this particular case the Bose-Hubbard model has a Bethe Ansatz solution that can be compared with the one for the Fermi Hubbard model [START_REF] Polo | Exact results for persistent currents of two bosons in a ring lattice[END_REF]. For the case N F = 6 and N B = 3 we have checked numerically that the two models at strong coupling have the same energies. This shows that the attractive Fermi Hubbard Hamiltonian is continuously connected to the Bose-Hubbard one.

Parity effects of the persistent current

We obtain the persistent current from the ground-state energy branch according to

I P = -Ω 0 ∂ E GS /∂ Ω.
For zero and weak interactions, the persistent current is a periodic function of the flux with period Ω 0 . The behaviour of the current markedly depends on the parity of the number of pairs N/2 (see Figs.3.1 and 3.2): for even N/2, the ground state energy displays a global maximum for Ω = 0 (paramagnetic behaviour); for odd N/2, instead, the ground state energy has a global minimum for Ω = 0 (diamagnetic behaviour). Such effect emerges by comparing the first two columns of Fig. 3.1 and 3.2. Remarkably, the parity dependence of the persistent current disappears for strong enough interactions: upon increasing interactions the ground-state energy displays a superlattice structure as the energy of the excited states decreases, leading to the doubling of its periodicity and the suppression of the parity effects in the persistent current at resonance. On the BEC side, the periodicity of the ground state energy is Ω 0 /2, corresponding to the quantum of flux of a pair, for any U B . We find that the doubling of periodicity of the current is a clear signature of the formation of pairs. This was originally suggested by Byers and Yang [START_REF] Byers | Theoretical considerations concerning quantized magnetic flux in superconducting cylinders[END_REF] for BCS superconducting rings. Our results elucidate their predictions by an exact calculation and extend it to the whole BCS-BEC crossover. The decrease of the energy of the excited states is also at the origin of the change of sign in the curvature of the ground state energy at zero flux observed in the disordered Fermi-Hubbard model for even N/2 [START_REF] Waintal | Persistent currents in one dimension: The counterpart of Leggett's theorem[END_REF]: disorder smooths the cusps at weak interactions yielding a negative curvature, while a positive curvature is found at strong attractions as in the clean case.

Readout: interferograms and noise correlations.

In the next, we detail the expansion protocol which has been outlined in Chapter 1. We study the co-expansion of the Fermi gas initially trapped in the ring with the one located at the ring's center, after the two confining potentials are suddenly switched off simultaneously [START_REF] Roscilde | From quantum to thermal topological-sector fluctuations of strongly interacting bosons in a ring lattice[END_REF][START_REF] Haug | Readout of the atomtronic quantum interference device[END_REF]. Interaction effects arising from inelastic collisions are expected 3.4. Readout: interferograms and noise correlations. to be negligible in our low density regime [START_REF] Kohstall | Observation of interference between two molecular Bose-Einstein condensates[END_REF]: for higher densities, a specific fast ramping protocol leading the system to the BEC side could be adopted to improve the visibility of the interferograms [START_REF] Greiner | Emergence of a molecular Bose-Einstein condensate from a Fermi gas[END_REF][START_REF] Martin W Zwierlein | Vortices and superfluidity in a strongly interacting Fermi gas[END_REF][START_REF] Valtolina | Josephson effect in fermionic superfluids across the BEC-BCS crossover[END_REF]. We also stress that we integrate the density profiles along the z axis and focus on the dependence on r = (x, y) in the ring plane.

Details on the interference for expanding ring and disk

The field operator of the whole system (ring plus center) is Ψ α (r,t) = w C (r,t)c C + ∑ N s j=1 w j (r,t)c j,α , where the Wannier function w Λ (r,t) is given by the following expression, for Λ = { j,C}:

w Λ (r,t) = 1 √ πσ 1 (1 + iω 0 t) exp - (r -r Λ ) 2 2σ 2 (1 + iω 0 t) = = 1 √ πσ (1 -iω 0 t) (1 + ω 2 0 t 2 ) exp - (r -r Λ ) 2 2σ 2 (1 + ω 2 0 t 2 ) exp iω 0 t 2σ 2 (1 + ω 2 0 t 2 ) (r -r Λ ) 2 (3.5)
where σ = h/mω 0 and r j are the initial width and the center of the j-th Wannier function respectively, ω 0 being the frequency of the bottom of each lattice well in the harmonic approximation. This allows for an explicit solution for the dynamics of the field operator following a sudden turn-off of the lattice [START_REF] Minguzzi | Exact coherent states of a harmonically confined Tonks-Girardeau gas[END_REF][START_REF] Chiofalo | Collective excitations of a periodic Bose condensate in the Wannier representation[END_REF]. We notice that the approximated expression (3.5) does not satisfy the completeness relation for the fermionic field at all times. Hence, the noise correlator is in the general case complex, however it becomes real at long times [START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF]. In the thesis we show the real part of the correlator at intermediate times. A comparison between imaginary, real part and the modulus of the correlator is presented in Fig. 3.5: we see that essentially the three functions carry the same physical information, justifying our choice.

In the experiment with weakly interacting bosonic condensates, a spiral interferogram emerges in a single co-expansion. In our theoretical approach, however, the particle density n(r,t) . = ∑ α=↑,↓ ⟨Ψ α (r,t) † Ψ α (r,t)⟩ is reconstructed as an expectation value, corresponding to an average over different realizations of the co-expansion protocol.

Since each co-expansion is characterized by a well-defined, yet randomly distributed, relative phase between the ring gas and the central system, the system's phase pattern is washed out in n(r,t). We shall see, instead, that non trivial phase information on the system is captured by the density-density correlator:

G(r, r ′ ;t) . = ∑ α,β =↑,↓ G α,β (r, r ′ ;t) (3.6) G α,β (r, r ′ ;t) . = ⟨Ψ † α (r,t)Ψ α (r,t)Ψ † β (r ′ ,t)Ψ β (r ′ ,t)⟩ (3.7)
For our combined center-ring system, such correlation can be broken down as R) , where the indexes R and C refer respectively to the ring and to the central site. Moreover, since ring and central gases are disconnected, we have |ψ(t = 0)⟩ = |ψ(t = 0)⟩ C ⊗ |ψ(t = 0)⟩ R . Assuming a free expansion for t ≥ 0, the ring-center correlations read

G(r, r ′ ,t) = G (C,C) + G (R,R) + G (C,
G (C,R) (r, r ′ ,t) = n (R) (r,t)n (C) (r ′ ,t) + n (R) (r ′ ,t)n C (r,t) + G(C,R) (r, r ′ ,t), where G(C,R) (r, r ′ ,t) = ∑ α ∑ i, j I i, j (r, r ′ ,t) δ i j -⟨c † i,α c j,α ⟩ (3.8)
Chapter 3. Probing the BCS-BEC crossover with persistent currents with I i j (r, r ′ ,t) . = w * C (r,t)w C (r ′ ,t)w * i (r,t)w j (r ′ ,t). From Eq. (3.8) we see that the interference between the center and the ring only affects the particles belonging to the same spin species. We note that such correlation can be accessed by measuring separately the full correlator G(r, r ′ ) and the exceeding terms depending on densities only. In order to enhance the visibility of the correlator, we subtract the term G0 (r, r ′ ,t) = w * C (r,t)w C (r ′ ,t) ∑ N s j=1 w j (r,t)w * j (r ′ ,t). This term corresponds to the one-body density matrix of a non-interacting Fermi gas for a completely filled lattice, ie when it forms a band insulator: can be separately measured and eliminated in post-processing. In the following, when not specified otherwise, we fixed r ′ = R in each plot to aid data visualization.

In the present Chapter, we will focus mostly on the interference term G(C,R) given by Eq. (3.8). In Chapter 4, we will monitor also the complete correlator G(r, r ′ ,t), and the spin resolved correlator G ↑,↓ . We shall see that at intermediate times G(C,R) contains direct information on the single-particle phase coherence and allows us to discern the regimes of the BCS-BEC crossover. Instead, at long times, G ↑,↓ (r, r ′ ,t) probes the many-body phase coherence in momentum space with k =

1 σ 2 ω 0 t to f r, k ′ = 1 σ 2 ω 0 t to f
r ′ . The details for the long times expansion will be given in Chapter 4. On the BEC side the first non-trivial correlator is the one associated to density of pairs n B [START_REF] Haug | Readout of the atomtronic quantum interference device[END_REF], ie n B (r,t)n B (r ′ ,t). We use a DMRG analysis of the correlator matrix. From the definition of G α,β (r, r ′ ;t) it emerges that for a lattice with N s sites, we have to calculate N 4 s correlators.

Time scale for spiral interferograms

We discuss in the following the time scales selected for the noise correlator. The spirallike pattern is expected to emerge at intermediate times after the release of the trap. Indeed, at very short times the interference pattern is very blurred, while at long times one recovers the momentum correlations in the initial state before the expansion, as extracted from the analysis of time-of-flight images in experiments [START_REF] Haug | Readout of the atomtronic quantum interference device[END_REF][START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF] and as detailed in Chapter 4. In Fig. 3.6 we show the interference pattern between the center and the ring for different values of time.

In the following, we refer to intermediate times as the interval when the expanding cloud is still far from the momentum distribution of a Fermi gas. In order to estimate the lower bound of the intermediate time regime, we consider Eq. (3.5). As far as the ring trap and the center are switched off, there's a non-zero interference term between the two subsystems: the wavefunctions have support in the whole real space. On the other hand, at short times, such interference only concerns the tails of the wavefunctions and therefore is very difficult to detect. In particular, we see that in Eq. (3.5) there is a Gaussian envelope that fixes the decay of the Wannier functions: at t = 0 and in the Gaussian approximation the Wannier function is localized on one site of the lattice. As t increases, the Gaussian envelope decays slower: the visibility of the interference patterns increases when the term σ 2 (1 + ω 2 0 t 2 ) is large enough to compensate the square distance (rr Λ ) 2 . We consider intermediate times the values of t for which (1 + ω 2 0 t 2 ) ≈ R 2 4σ 2 .

Readout of the BCS-BEC crossover

Readout of the BCS-BEC crossover

We study the interference pattern for systems with even and odd number of pairs in columns (c) and (d) of Figs. 3.1 and 3.2. In the small Ω case, at weak interactions we see a spiral image with a dislocation indicating non-zero current for even N/2, while there is no current for odd N/2 as the figure is symmetric by reflection along the y = 0 axis. The shape of the interferograms is due to the different contributions of the single-particle orbitals constituting the Fermi sphere, as we will show in detail in Chapter 4. In contrast, at strong interactions, the images for N/2 even or odd are both symmetric with respect to the y = 0 axis, indicating that the current vanishes regardless of the parity of the number of pairs. Chapter 3. Probing the BCS-BEC crossover with persistent currents We also study the correlation in the interference pattern for Ω slightly below the degeneracy point Ω = 1/2. Comparing the columns (c) and (d) of both Fig. 3.1 and 3.2, we see that at zero interactions the current is the same in the two cases, consistently with the fact that the two values of flux are on the same period of the ground state energy. At strong interactions, close to resonance, the images are more blurred by the reduced phase coherence. Nevertheless, we see that the correlation functions of column (d) display non-mirror symmetric, spiral-like features, indicating the presence of a current state and the doubling of the periodicity. Finally, on the BEC side, the coherence producing well defined spirals arises from the quasi-condensation of pairs.

Conclusions to Chapter 3

We studied the persistent current of a Fermi gas confined in a mesoscopic ring-shaped lattice subjected to an artificial magnetic field all through the BCS-BEC crossover. We described the system through attractive Fermi and repulsive Bose Hubbard models, using both the exact solution by Bethe Ansatz and DMRG. We demonstrated that the persistent current displays distinctive features in the various interaction regimes. At weak interactions (BCS regime), the persistent current is a periodic function of the single-particle flux quantum, displaying some modulations due to the superlattice structure of the groundstate energy. Such phenomenon indicates the onset of pairing correlations building up between up and down spins. The BCS regime is characterized by parity-dependent persistent currents: while for odd number of pairs the system has a diamagnetic response, for even number of pairs the system has paramagnetic response. Remarkably, the parity effect is washed out at resonance and in the BEC regime. In these regimes, the persistent current is periodic with a two-particle flux quantum, providing a clear signature of the formation of bound pairs. To experimentally monitor the features of the persistent current described above, we let the gas co-expand with a reference gas placed in the center and we study the noise correlations in the interference pattern. Quite remarkably, such interferometric analysis works also in the BCS regime, where the phase coherence ensured by the fermionic pairs is lower than in the BEC one.

Our work provides a clear evidence that the response of a system of mesoscopic size to an artificial gauge field is a relevant tool to study the BCS-BEC crossover. Our approach is fully general and readily applicable to other models, eg the boson-fermion one [START_REF] Ren | Bethe-ansatz analysis of near-resonant twocomponent systems in one dimension[END_REF] as well as to other systems, such as high T c superconductors [START_REF] Uemura | Bose-Einstein to BCS crossover picture for high-Tc cuprates[END_REF][START_REF] Chen | Applying BCS-BEC crossover theory to hightemperature superconductors and ultracold atomic Fermi gases[END_REF].

4

Single-particle versus many-body phase coherence in an interacting Fermi gas

In quantum mechanics, each particle is described by a complex-valued wave function characterized by amplitude and phase. When many particles interact with each other, cooperative phenomena give rise to a quantum many-body state with a specific quantum coherence. What is the interplay between single-particle's phase coherence and many-body quantum coherence? Over the years, such question has been object of profound analysis in quantum physics. Here, we demonstrate how the time-dependent interference formed by releasing an interacting degenerate Fermi-gas from a specific matter-wave circuit in an effective magnetic field can tell apart the two notions. Single-particle phase coherence, indicated by the first-order correlator, and many-body quantum coherence, indicated by the noise correlator, are displayed as distinct features of the interferogram. Single-particle phase coherence produces spiral interference of the Fermi orbitals at intermediate times.

Many-body quantum coherence emerges as long times interference. The interplay between single-particle coherence and many-body coherence is reflected in a stepwise dependence of the interference pattern on the effective magnetic field.

Introduction

Phase coherence is the ability of the quantum wavefunction to retain its phase information. For free particles, such notion is operatively inferred, for example, through the interference pattern in two slits experiments. When it comes to be referred to interacting quantum many particles systems, though, the notion of phase coherence is more involved. While each quantum particle can generally cooperate to establish a macroscopic coherence [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF][START_REF] Blundell | Magnetism in condensed matter[END_REF], bosons and fermions provide two different pictures. Bosonic systems can form a Bose-Einstein condensate and be described by a macroscopic wave function Φ ∼ e iφ , with a phase φ coinciding with the single-particle phase. With the spectacular advances on atom trapping and cooling such phase coherence has been studied with unprecedented degree of control and precision of physical conditions. Although cold atoms systems are made of large but finite number of particles (∼ 10 5 ), 'two slits' tests for a Bose-Einstein Condensate (BEC) provide a meaningful interference pattern as a result of self-averaging [START_REF] Andrews | Observation of interference between two Bose condensates[END_REF][START_REF] Castin | Relative phase of two Bose-Einstein condensates[END_REF]. Fundamental limits on fringes visibility are provided by phase diffusion [START_REF] Lewenstein | Quantum phase diffusion of a Bose-Einstein condensate[END_REF][START_REF] Kurkjian | Phase operators and blurring time of a pair-condensed Fermi gas[END_REF]. Bose Josephson effect is a direct test for BEC coherence [START_REF] Albiez | Direct observation of tunneling and nonlinear selftrapping in a single bosonic Josephson junction[END_REF][START_REF] Levy | The ac and dc Josephson effects in a Bose-Einstein condensate[END_REF][START_REF] Cataliotti | Josephson junction arrays with Bose-Einstein condensates[END_REF][START_REF] Smerzi | Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates[END_REF].

For fermionic systems, the Pauli principle prevents the occupancy of a single quantum level by particles with the same spin, and therefore the many-body coherence needs to be achieved from the single particle's coherence through more complicated means [START_REF] James | Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems[END_REF]. A frictionless flow, the persistent current we introduced in Section 1.2.2 , occurring in metallic or superconducting small rings provides a characteristic trait of particle's coherence [START_REF] Imry | Introduction to mesoscopic physics[END_REF][START_REF] Bleszynski-Jayich | Persistent currents in normal metal rings[END_REF][START_REF] Tinkham | Introduction to superconductivity[END_REF]. The nature of the many-body phase coherence in degenerate fermions, though, depends on particles interaction that can be tuned in cold atoms experiments from repulsive to attractive cases [START_REF] Inguscio | Ultra-cold Fermi gases[END_REF]. For repulsive interactions, the effects of the ferromagnetic correlations were observed [START_REF] Massignan | Polarons, dressed molecules and itinerant ferromagnetism in ultracold Fermi gases[END_REF][START_REF] Amico | Time-resolved observation of competing attractive and repulsive short-range correlations in strongly interacting Fermi gases[END_REF]. For attractive interactions, bound states of Fermi pairs can condense, experiencing the BCS-BEC crossover. Relevant information on the system coherence in the crossover can be extracted through the study of the Josephson effect [START_REF] Kwon | Strongly correlated superfluid order parameters from dc Josephson supercurrents[END_REF][START_REF] Valtolina | Josephson effect in fermionic superfluids across the BEC-BCS crossover[END_REF][START_REF] Luick | An ideal Josephson junction in an ultracold two-dimensional Fermi gas[END_REF].

While the features mentioned above do provide specific aspects for particle and manybody quantum coherence, the interplay between these two phenomena in interacting manybody systems remains unclear [START_REF] James | Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems[END_REF][START_REF] Anthony | On the concept of spontaneously broken gauge symmetry in condensed matter physics[END_REF]. In this chapter, we operatively track the interplay between the single-particle phase coherence and the many-body quantum coherence in the dynamics of an interacting Fermi gas. To this end, we study a degenerate interacting Fermi gas confined in a ring-shaped potential and pierced by an effective magnetic flux. Because of the effective magnetic flux, a current is imprinted on the degenerate gas. In analogy with bosonic protocols, such system is let to interfere with a second gas placed at the center of the ring. In the following, we consider the gas in the center to be composed by two fermions with opposite spins. We stress that the density of the gas in the center does not affect qualitatively our results. Here, we study the entire time evolution of such expansion. Despite the similarity in the schemes, we shall see that the fermionic interferograms are markedly different from the bosonic ones. We show that the particles' phase coherence is displayed in the intermediate times interference images displaying characteristic dislocations due to Fermi sphere effects. On the other hand, many-body coherence, exemplified by pairing correlations, and off-diagonal-long-range order (ODLRO), ie longranged spatial coherence [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF], emerge at long times in the noise correlators. The response of the many-body coherence to magnetic field is displayed as a step-wise dependence of the noise correlators on the magnetic field.

The model

The model

We consider the same model and the same notations we used in Chapter 3. Therefore we consider the Hamiltonian 3.1, which we report here for sake of convenience:

ĤFH = -J N s ∑ j=1 ∑ σ =↑,↓ e i Ωc † j,σ c j+1,σ + H.c. +U N s ∑ j=1 n j,↑ n j,↓ . (4.1)
We summarize the results we presented in Chapters 2 and 3. For U > 0, the groundstate many-body wavefunction is made of extended states characterized by real wave momenta [START_REF] Andrei | Integrable models in condensed matter physics[END_REF]. For U < 0, the ground state is characterized by bound pairs, with complex wave momenta. For weak attractions, the ground state of the system is a BCS-like state with the wavefunction of the pair decaying on distance larger than the mean interparticle separation. For stronger attractions, the bound states are formed by tightly bound particle pairs. Figure 4.1: Expansion protocol at different times. After the release of the trap, the particles in the center and on the ring interfere. We shall see the single-particle coherence emerge at intermediate times, while in the long-time limit this protocol provides information about the many-body quantum coherence.

In Fig. 4.1 we propose a scheme of the expansion protocol we examinate in this chapter. The details of the protocol at intermediate times have been presented in Section 3.4. We define the long times regime as t > t to f ≫ ω -1 0 . In this regime, from the second line of Eq.(3.5) one can readily compute the limit ω 0 t → ∞ while keeping the ratio r/(ω 0 t) finite and obtain w j (r,t to f ) ∝ e ik(r) •r j where k(r) = 1 σ 2 ω 0 t to f r. As a consequence of the expansion, the Wannier functions reduce to plane wave in real space. This implies that the expansion acts as a sort of Fourier transform in this time regime. Since, on the other hand, the correlation matrix ⟨c † i c j c † n c m ⟩ does not depend on time, the correlator can be Chapter 4. Single-particle versus many-body phase coherence in an interacting Fermi gas written as:

G α,β (r, r ′ ,t to f ) = N s ∑ k, j,n,m=0 e -ik •r k e ik •r j e -ik ′ •r n e ik ′ •r m ×⟨c † k, α c j,α c † n,β c m,β ⟩ ≡ ⟨n α (k)n β (k ′ )⟩, (4.2)
where k ≡ k(r) and k ′ ≡ k(r ′ ) and we introduced the momentum distribution as:

n α (k) ∝ N s ∑ i, j e ik •(r i -r j ) ⟨c † i, α c j, α ⟩. (4.3)
Therefore, at large times, the noise correlator in real space gives information about the momentum correlations of the system at t = 0.

Free fermion gas

In this section we provide the analytical results for the case U = 0. In particular, we want to evaluate

G(r, r ′ ,t) = N s ∑ i, j,n,m=0 w * i (r,t)w j (r,t)w * n (r ′ ,t)w m (r ′ ,t) × ⟨c † i c j c † n c m ⟩, (4.4)
where c i and c † i are the fermionic annihilation and creation operators and we have indicated by i = 0 the central site. It should be noted that, since in the following we will consider just one spin species, we dropped the spin indexes. The Wannier functions w j (r) are defined in Eq. (3.5). Imposing that the center and the ring are totally decoupled, the only terms surviving in (4.4) are encoded in Eq. (3.8). Moreover, in the same spirit of Chapter 3, in the following we subtract the term depending on the Kronecker delta G0 (r, r ′ ,t): we focus on the terms which are relevant for spiral interferograms.

The one-body correlator ⟨c † k c j ⟩ can be determined by solving the non-interacting Schroedinger equation under periodic boundary conditions. An explicit calculation yields:

⟨c † k c j ⟩ = ∑ {n} e -i 2π N s ( j-k)n , (4.5)
where {n} is the set of integer quantum numbers labelling the energy levels of the Fermi sphere, whose value depends on the applied flux Ω. Using Eq. (3.5), the interference term can be explicitly calculated:

G(CR) (r, r ′ ,t) -G0 (r, r ′ ,t) = = -|A(t)| 4 e -1 2σ 2 b 2 (t) (r 2 +r ′2 ) e -i ht mb 2 (t) (r 2 -r ′2 ) × ∑ k j e -1 2σ 2 b 2 (t) ((r-r j ) 2 +(r ′ -r k ) 2 ) e i ht mb 2 (t) ((r-r j ) 2 -(r ′ -r k ) 2 ) × ∑ {n} e -i 2π N s ( j-k)n ≡ -∑ {n} I n (r)I * n (r ′ ) (4.6)
4.3. Free fermion gas where b(t) = 1 + ω 2 0 t 2 and A(t) is the complex time-dependent amplitude of the Wannier functions. In the last equivalence, we factorize the summation over the indexes k and j. We see that, once r ′ is fixed, Eq. (4.6) will be a superposition of single-orbital functions I n (r), each weighted with a n-dependent coefficient. In the case of non-interacting bosons, a single quantum state, labelled by a quantum number n, is macroscopically occupied. As a consequencet he full correlator is proportional to the function I n (r)I * n (r ′ ), yielding a perfect spiral interference pattern whose number of branches gives access to the value of the current.

Time of flight

We now focus on the momentum distribution of the particles on the ring, that can be addressed through time of flight imaging using the same expansion protocol. As we mentioned, such observable can be accessed by measuring the density distribution of the expanding gas after at long times. Using the one-body correlation function evaluated in Eq. (4.5), Eq.( 4.3) can be written as:

n(k) ∝ 1 N s N s ∑ j=1 ∑ {n} e iR(k x cos θ j +k y sin θ j ) e 2πi N s jn 2 (4.7)
where we introduced (R, θ j ) as the polar coordinates of the j-th site of the ring and decompose k = (k x , k y ). Using that θ j = 2π N s j and defining J n (x, y)

. = ∑ N s j=1 e iR(k x cos( 2π N s j)+k y sin( 2π N s j)) e 2πi N s jn
the previous expression can be further simplified:

n(k) ∝ 1 N s ∑ {n} J n (k x , k y ) 2 . (4.8)
The functions J n (x, y) provide the discrete version of the Bessel function of order n. A remarkable property of such function is that |J n (x, y)| 2 is non-zero in the origin only

Free fermion gas

when n = 0, as shown in Fig. 4.2. From this expression it is evident that the structure of the momentum distribution strongly depends on the set of quantum numbers {n} and therefore on the flux acting on the system. In particular, in the ground state and for Ω = 0 such numbers are integers selected in order to minimize the current quantum number l 0 = ∑ {n} n, as shown in the example for N = 4 in the right panel of Fig. 4.2. The current quantum number l of the excited states is obtained by increasing each n of one unit. The momentum distribution for fermions on a lattice at Ω = 0 is expected to show a peak in the center k x = k y = 0, that disappears for large values of flux [START_REF] Amico | Quantum many particle systems in ring-shaped optical lattices[END_REF]. In order to observe the momentum distribution to vanish for k x = k y = 0, we have to excite all the energy levels of the Fermi sphere until n = 0 and correspondingly J 0 (k x , k y ) are excluded from the summation in Eq. (4.8). This is achieved after a shift in the energy levels equal to half of the size of the Fermi sphere, as we show in Fig. 4.3 for N = 4 particles.

Bosons vs. fermions

For bosons, the spiral interference pattern arises because of the simple coupling between the effective gauge field and the phase of the Bose condensate. The quantized circulation reflects the effective magnetic flux quantization. The complete phase structure of the bosonic field emerges as a characteristic spiral interferogram in the expanding density [START_REF] Corman | Quench-induced supercurrents in an annular Bose gas[END_REF][START_REF] Eckel | Interferometric measurement of the current-phase relationship of a superfluid weak link[END_REF][START_REF] Mathew | Self-heterodyne detection of the in situ phase of an atomic superconducting quantum interference device[END_REF] as well as in the noise correlator G(r, r ′ ;t) [START_REF] Haug | Readout of the atomtronic quantum interference device[END_REF] (see also Fig. 4.4).

For fermionic systems, the relation between the imparted phase and the momentum distribution is more involved. The difference traces back to the symmetry properties of the many-body wave functions of the two systems resulting in a different momentum distribution: while bosonic wave functions yield a momentum distribution peaked at k = 0 [START_REF] Wright | Driving phase slips in a superfluid atom circuit with a rotating weak link[END_REF][START_REF] Moulder | Quantized supercurrent decay in an annular Bose-Einstein condensate[END_REF][START_REF] Wright | Threshold for creating excitations in a stirred superfluid ring[END_REF], Fermi systems are characterized by a broader momentum distribution. Then, when fermionic particles are put in motion by an effective magnetic flux each momentum component of the distribution is characterized by a different phase factor. As a consequence, phases recombination occurs, and the time-of-flight image of the density results to be suppressed at |k| = 0 only after half of the Fermi sphere is displaced by the effective magnetic flux, similarly to what we show for non-interacting fermions in Sec.4.3. In the G(r, r ′ ;t) at intermediate expansion times specific dislocations are found in the interference pattern (see Fig. 2 d). This is again due to the distinct particle orbitals characterizing the fermionic state. As we showed in Section 4.3, for U = 0, such orbitals are strictly single-particle, and each of them yields a spiral-like interference. The dislocations, just N ↑ -1 (or equivalently N ↓ -1) in number, are due to the interference of the N ↑ (N ↓ ) independent orbitals. Remarkably, the dislocations are clearly visible at small and moderate interactions. By increasing interactions, we find that such feature disappears indicating that the system cannot be described in terms of independent quasi-particles.

Repulsive vs. attractive interactions

Below, we demonstrate that the long time expansion of our protocol's interference allows us to access to many-body coherence through the connected spin-resolved correlator G ↑↓ (r, r ′ ,t to f ) -⟨n ↑ (r,t to f )⟩⟨n ↓ (r ′ ,t to f )⟩ . Notably, we consider the connected part to eliminate the background from the total correlations. We shall see that the different nature of many-body state for repulsive and attractive interactions is clearly reflected in our pictures (see Fig. 4.5).

For U > 0 the ground state of the system is made of itinerant correlated particles. Therefore, the interferograms reflect the phase pattern imparted by the effective magnetic flux putting the system in a coherent motion. By the analysis of the long-time behavior of G ↑↓ (r, r ′ ;t), at increasing U we find that the correlation has a clear symmetry k = k ′ (see the caption of Fig. 4.5), reflecting the system's tendency to magnetic ordering occurring at large U/J [START_REF] Tasaki | Physics and mathematics of quantum many-body systems[END_REF][START_REF] Mathey | Noise correlations in low-dimensional systems of ultracold atoms[END_REF]. As function of the applied effective magnetic flux, the correlations are found to be displaced by a discrete amount reflecting the quantized particle circulation. This can be observed by looking at the diagonal displacement l/R of the origin x = x ′ = 0 as a function of the applied flux (see Fig. 4.6).

For U < 0, the system is characterized by the off-diagonal (quasi) long-range order due to fermionic pairing [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF]. In contrast with repulsive cases, for U < 0 we found that the Chapter 4. Single-particle versus many-body phase coherence in an interacting Fermi gas correlator G(r, r ′ ;t) displays a marked structure along the whole anti-diagonal x = -x ′ , reflecting the formation of pairs of smaller and smaller size at increasing |U|: while in the BCS regime the pairs correspond to enhanced (k F , -k F ) correlations at the Fermi sphere i.e. for wave vector |k| = k F , at larger interactions, where pairs of small spatial size are formed, correlations along the whole antidiagonal (k, -k) are predicted. Such approach is in line with [START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF][START_REF] Staudenmayer | Density correlations in ultracold fermi systems within the exact Richardson solution[END_REF]. For attractive or repulsive interactions, we find that the landscape along the antidiagonal depends on the number of particles with a markedly different scaling for U > 0 and U < 0. In line with Yang's criterion for the off-diagonal-long-range order [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF], we find that both the maxima and the minima of the momentum correlator along the antidiagonal scale the same way with N. Indeed, in the presence of quasi-ODLRO the momentum correlator is dominated by the pair-pair correlations, and scales as N α with 0 < α < 1 for any wavevector k. In Section 4.5.1, we rigorously proof this last statement.

For U > 0, instead, the maxima of the momentum correlator are independent on particle number, while the minima increase with N. As a result, the visibility

V (N,U) = Max[G ↑,↓ (x, -x;t)] -Min[G ↑,↓ (x, -x;t)] Max[G ↑,↓ (x, -x;t)] + Min[G ↑,↓ (x, -x;t] , (4.9)
plotted in Fig. 4.6, is independent of N for U < 0 and decreasing with N for U > 0. We note that the property clearly emerge already at small N, this providing a further evidence that ring geometries are well suited for minimizing finite size effects [START_REF] Fisher | Scaling theory for finite-size effects in the critical region[END_REF].

Visibility V (U, N) vs quasi-ODLRO

The concept of off-diagonal long range order allows one to quantify uniquely the properties of condensation or pairing in interacting quantum fluids [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF]. For paired fermions, such as superconductors and atomic Fermi superfluids, the relevant correlator is the twobody density matrix: ρ 2 (x, x ′ , y, y ′ ) = ⟨ψ † ↑ (x)ψ † ↓ (x ′ )ψ ↓ (y)ψ ↑ (y ′ )⟩ quantifying phase coherence between one pair centered at X = (x + x ′ )/2 and a second one at Y = (y + y ′ )/2. Existence of ODLRO implies that ρ 2 (x, x ′ , y, y ′ ) ≃ λ 0 (N)Φ * 0 (x, x ′ )Φ 0 (y, y ′ ) with Φ 0 natural orbital of ρ 2 with macroscopic eigenvalue λ 0 = O(N). Similarly, by analogy with bosonic systems [START_REF] Colcelli | Deviations from Off-Diagonal Long-Range order in one-dimensional quantum systems[END_REF], for quasi-ODRLO one expects that quantum fluctuations reduce the scaling to λ 0 (N) = O(N α ) with 0 < α < 1. All the above relations are readily generalized to lattice systems, by considering the discretized versions of ρ 2 and Φ 0 according to

ρ 2 ( j, l, m, n) = ⟨c † j,↑ c † l,↓ c m,↓ c n,↑ ⟩ and Φ 0 ( j, l) = Φ 0 (x j , x l ).
We show here how the G ↑↓ correlator at long times and its visibility allows precisely to address off-diagonal long-range order.

First, we remark that Eq. (4.2) is readily related to the two-body density matrix upon commutation of the order of the operators. If (quasi)ODRLO holds, along the antidiagonal

k ′ = -k one has n ↑ (k)n ↓ (-k) ≃ λ 0 (N)| Φ0 (k)| 2 (4.10)
where we have defined Φ0 (k) = ∑ j,l e ik •(r j -r l )Φ 0 ( j,l) and used that Φ 0 ( j, l) is an even function of jl.

The connected part of the correlator n ↑ (k)n ↓ (-k) contains essentially the same information of ODLRO as the full one since no ODRLO is found for fermionic systems in the disconnected terms [START_REF] Yang | Concept of Off-Diagonal Long-Range Order and the quantum phases of liquid He and of superconductors[END_REF].

Combining all the above considerations, we readily conclude that the visibility of the momentum correlator along the anti-diagonal k ′ = -k, in presence of quasi-ODLRO is Chapter 4. Single-particle versus many-body phase coherence in an interacting Fermi gas predicted to behave as

V (U, N) = Max| Φ0 (k)| 2 -Min| Φ0 (k)| 2 Max| Φ0 (k)| 2 + Min| Φ0 (k)| 2 (4.11)
ie, noticeably, it is independent on N.

Continuous limit of the Hubbard rings

In this section, we show that our results hold for the continuum limit of the Fermi-Hubbard model and they do not depend on the presence of the optical lattice. We consider Gaudin-Yang model as continuous limit of the Fermi-Hubbard Hamiltonian. In this section, we put Ω = 0. We define the density of fermions in the lattice as D = N/(N s ∆), ∆ being the lattice spacing. In the continuous limit one has that ∆ → 0, which implies that the filling factor ν = N/N s = D∆ must be accordingly small. In the continuous limit the fermionic operators must be rescaled:

c i,σ = √ ∆Ψ σ (x), n i,σ = ∆Ψ † σ (x)Ψ σ (x), x = ∆i.
Then, the Hubbard model reduces to the Fermi gas quantum field theory:

H FH = J∆ 2 H FG -2N, H FG = dx (∂ x Ψ † σ )(∂ x Ψ σ ) + cΨ † ↑ Ψ † ↓ Ψ ↓ Ψ ↑ , with c = U/(J∆). The quantum fields obey the anticommutation relations {Ψ σ (x), Ψ † σ ′ (y)} = δ σ ,σ ′ δ (x -y) and {Ψ † σ (x), Ψ † σ ′ (y)} = 0.
The Fermi gas field theory is the quantum field theory for the Gaudin-Yang model. Such statement can be demonstrated by writing the eigenstates of

H FG as |ψ(λ )⟩ = dzχ(z|λ )Ψ † σ (z 1 ) . . . Ψ † σ (z N )|0⟩.
Then, it can be proved that χ(z|λ ) must be eigenfunctions of the Gaudin-Yang Hamiltonian presented in 2.

The mapping sketched above makes our results for dilute lattices applicable to a 1d fermionic gas with delta-interaction. In Fig. 4.7 we show the time of flight and the interference term G(C,R) at intermediate times for a dilute systems of N = 4 particles on N s = 30 sites for U = 2. We observe the characteristic hole in the center of the momentum distribution due to the action of the flux, as described in the previous section for non-interacting fermions. Furthermore, in the second line, we recognize a dislocated spiral-like interference pattern, carrying information about the angular momentum of the system and marking the fermionic nature of the particles.

One-dimensional correlations

In this section, we study the correlation function in a 1d system in the reciprocal space. We shall see that such correlations display similar features of the noise correlations in the two-dimensional momentum distribution. The correlation function reads

G (1d) ↑,↓ (x, x ′ ;t to f ) ∝ N s ∑ l, j,m,n e i x R (l-j)+i x ′ R (m-n) ⟨c † l,↑ c j,↑ c † m,↓ c n,↓ ⟩ (4.12)
where the indexes i, j, l, m label the sites of the chain and R = 2πL is the radius of the ring, L being the length of the chain. Remarkably, the correlation matrix ⟨c † l,↑ c j,↑ c † m,↓ c n,↓ ⟩ We consider here a repulsive interaction U/J = 2. Consistently with the text, in the bottom plots we observe N ↑ -1 dislocations.

is the same as the one used in the two-dimensional expansion. We see in Fig. 4.8 that such correlator carries the same information as the two-dimensional one, with a peak along the anti-diagonal x = -x ′ for negative U revealing the formation of BCS-like pairs in the system. Because of the different geometry of the system with respect to the twodimensional case, we observe a symmetry breaking along the diagonal x = x ′ for non-zero flux. This is represented by a unidirectional shift of the anti-diagonal as a function of the flux that, as shown in Fig. 4.8, is still occuring in quantized steps. The connection between BCS pairing and the peaks along the antidiagonal of the noise correlator descends from the general expression of the BCS ground state. Consider indeed the state

|BCS⟩ = ∏ k (u k +v k b † k,↑ b † -k,↓ ) |0⟩
, where b † k,ρ is the creation operator for a particle with momentum k and spin ρ and |0⟩ is the BCS vacuum. By an explicit calculation we find the noise correlator to be:

∑ ρ,σ n ρ (q)n σ (q ′ ) = 2|v q | 2 |u q ′ | 2 δ (q -q ′ )δ ρ,σ + δ (q + q ′ )(1 -δ ρ,σ ) (4.13)
where the momentum density operator is defined as n ρ (q) = ⟨b † q,ρ b q,ρ ⟩. From this expression we see that, when ρ ̸ = σ , this correlation function shows a characteristic peak 4.8. Conclusions to Chapter 4 along q = -q ′ , with an envelope mediated by the amplitudes of the BCS state. We can also explicitly study the dependence of such correlator on the flux Ω. In order to do so, we introduce the latter using the Peierls substitution on the creation/annihilation operators in the real space ie b j,ρ → e i Ω j b j,ρ . = b j,ρ . The Fourier transform yields bq,ρ = 1 N s ∑ j=1 e iq j b j,ρ = 1 N s ∑ j=1 e i(q+ Ω) j b j,ρ = b q+ Ω,ρ . Therefore, the Peierls substition shifts all the momenta of the same flux-dependent amount. Inserting such result in Eq.(4.13), we don't affect the term proportional to δ ρ,σ , while the second term, proportional to the peak along q = -q ′ , get shifted by 2 Ω.

Conclusions to Chapter 4

In this work, we demonstrated how the interplay between the single-particle's phase coherence and the many-body phase coherence of an interacting Fermi gas can be probed with a single protocol inspired by heterodyne phase detection scheme. We described the ring-trapped gas through the Hubbard model with the local interaction U ranging from positive to negative values. We analysed the dynamics of the density-density correlators. Such quantities are studied here for the first time. They can be accessed, for example, in cold atoms experiments through state-of-the-art processing of the particles expansion. We note that also continuous (no lattice) ring-shaped degenerate gases can be accessed by our theory in the dilute lattice limit. For our protocol, we demonstrated that particle's phase information emerges in the intermediate times interference of the expansion; the many-body phase coherence can be tracked at longer times.

For Fermi systems the effective magnetic flux imparts the phase winding on a broad momentum distribution. We have shown that the relevant information of the phase of fermionic particles can be traced in the response of the orbitals. This analysis can be carried out in our protocol by suitably subtracting the correlations of non-interacting Fermi gas from the total correlations -Fig. 4.4. As remarkable spin-off of our analysis on the single orbital interference, we note that our results grant the access to the number of particles N ↑ (N ↓ ). Our prescription is that the number of dislocations obtained in our interferogram Fig. 4.4d is just N α -1 with α =↑ or ↓.

The proposed protocol provides a step forward towards the realization of the full counting statistics for the system's particle density fluctuations [START_REF] Staudenmayer | Density correlations in ultracold fermi systems within the exact Richardson solution[END_REF][START_REF] Belzig | Density correlations in ultracold atomic Fermi gases[END_REF].

For repulsive interactions, we found that the resulting long time image displays enhanced correlations along the diagonal k = k ′ (reflecting the magnetic correlations). By the application of the effective magnetic flux, the position of the peaks results to be displaced in discrete steps, reflecting the quantized circulation of current along the ring.

For attractive interactions a clear broad anti-diagonal k = -k ′ surfaces in the expansion. Such feature emerges at k = k F because of the fermionic pairing, leading to a manybody quantum coherence of the BCS type [START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF][START_REF] Staudenmayer | Density correlations in ultracold fermi systems within the exact Richardson solution[END_REF] . We find that anti-diagonal correlations arise also for strong attraction in which the pairs are tightly bound; in this case the peaks dissolve on a broad interval of k. As the counterpart of the effect found for repulsive fermions, the position of the anti-diagonal results to be displaced by the effective magnetic flux in a quantized fashion. The quantitative analysis shows that for repulsive/attractive interactions the visibility of the anti-diagonal correlations is characterized by a markedly different dependence on the number of particles. Such effect reflects the Yang's off-diagonal-long-range order scaling of the two-body density matrix.

With our work, we bring conceptually relevant aspects of many-body physics to the domain of what can be tested. Our analysis is very timely with the current stage of cold atoms quantum technology: Persistent current in toroidal cold fermionic atoms has been achieved in [START_REF] Cai | Persistent currents in rings of ultracold fermionic atoms[END_REF] and mesoscopic pairing was experimentally analysed in [START_REF] Holten | Observation of Cooper pairs in a mesoscopic two-dimensional fermi gas[END_REF].

5

Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas

We study the spin-mixing dynamics of a one-dimensional strongly repulsive Fermi gas under harmonic confinement. By employing a mapping onto an inhomogeneous isotropic Heisenberg model and the symmetries under particle exchange, we follow the dynamics till very long times. Starting from an initial spin-separated state, we observe superdiffusion, spin-dipolar large amplitude oscillations and thermalization. We report a universal scaling of the oscillations with particle number N 1/4 . Our study puts forward one-dimensional correlated fermions as a new system to observe the emergence of non-equilibrium universal features. Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas

Introduction

As stressed in Chapter 2, elucidating the dynamics of interacting Fermi gases is important for understanding a large variety of physical phenomena. The strongly out-ofequilibrium dynamics of interacting quantum systems is currently one of the most challenging open problems.

In this context, the spin dynamics deserves a specific focus. Spin currents can be easily damped by inter-particle collisions [START_REF] Polini | Spin drag and spin-charge separation in cold Fermi gases[END_REF] and the continuity equation for the spin density includes both orbital current and spin torque contributions [START_REF] Ralph | Spin transfer torques[END_REF]. Spin drag is another manifestation of interactions among the spin species, inducing spin-diffusive or non-dissipative dynamics depending on the interaction regimes [START_REF] Polini | Spin drag and spin-charge separation in cold Fermi gases[END_REF][START_REF] Amico | Theory of spin Coulomb drag in spinpolarized transport[END_REF][START_REF] Amico | Coulomb interaction effects in spinpolarized transport[END_REF][START_REF] Enss | Quantum mechanical limitations to spin diffusion in the unitary Fermi gas[END_REF][START_REF] Carlini | Spin drag and fast response in a quantum mixture of atomic gases[END_REF][START_REF] Valtolina | Exploring the ferromagnetic behaviour of a repulsive Fermi gas through spin dynamics[END_REF]. Ultracold atomic gases provide an ideal platform for exploring in isolated conditions the out-ofequilibrium spin dynamics [START_REF] Volosniev | Engineering the dynamics of effective spin-chain models for strongly interacting atomic gases[END_REF][START_REF] Rafael | Dynamics of spin and density fluctuations in strongly interacting few-body systems[END_REF][START_REF] Mestyán | Spincharge separation effects in the low-temperature transport of one-dimensional Fermi gases[END_REF]. In a three-dimensional geometry, the oscillatory dynamics of a strongly interacting Fermi gas with initially spatially separated spin components was studied in [START_REF] Sommer | Universal spin transport in a strongly interacting Fermi gas[END_REF]. The spin drag, spin diffusivity and spin susceptibility were obtained, and a universal limit for spin diffusivity at low temperature was reported for the unitary Fermi gas. A relevant question is what happens to the above quantities when reducing the dimensionality of the system to quasi one-dimensional, and what type of universality emerges. One-dimensional (1D) systems display specific features, as the enhancement of quantum fluctuations and correlations and, as we described in Chapter 2, they can be described by a wealth of theoretical and numerical methods.

We address this problem by following the dynamics of strongly repulsive fermions subjected to longitudinal harmonic confinement in a tight waveguide. As in the threedimensional case of Ref. [START_REF] Sommer | Universal spin transport in a strongly interacting Fermi gas[END_REF], we start from an initially imbalanced state with all spin up on the left and all spin down on the right of the harmonic trap, and we follow the damped oscillations of the magnetization. While the fully quantum dynamics at arbitrary interactions can be followed only at short times with a classical simulator, we focus here on the strongly correlated regime of very large interactions, close to the integrable point at infinite repulsions [START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF][START_REF] Fang | Exact solution for the degenerate ground-state manifold of a strongly interacting one-dimensional Bose-Fermi mixture[END_REF][START_REF] Volosniev | Strongly interacting confined quantum systems in one dimension[END_REF][START_REF] Harshman | Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap[END_REF][START_REF] Deuretzbacher | Exact solution of strongly interacting quasi-one-dimensional spinor Bose gases[END_REF][START_REF] Murmann | Antiferromagnetic Heisenberg spin chain of a few cold atoms in a one-dimensional trap[END_REF][START_REF] Boll | Spin-and densityresolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains[END_REF]. We apply the method we described in Section 2.2.2. In this regime, the dynamics of the charge and spin decouple, and the spin dynamics can be followed exactly till very long times by means of a mapping onto the one of an inhomogeneous, isotropic Heisenberg model.

An overview of the full spin dynamics is provided in Fig. 5.1, where three dynamical regimes arise. At short times, when the spin up and spin down clouds cross each other for the first time, we predict the emergence of a superdiffusive behaviour, compatible with Kardar-Parisi-Zhang (KPZ) universality, in striking difference from the diffusive one found in the three-dimensional counterpart [START_REF] Sommer | Universal spin transport in a strongly interacting Fermi gas[END_REF]. We thus identify 1D correlated fermions as a new system to observe the emergence of non-equilibrium universality, largely explored in homogeneous Heisenberg models [START_REF] Gobert | Realtime dynamics in spin-1 2 chains with adaptive time-dependent density matrix renormalization group[END_REF][START_REF] Ljubotina | Spin diffusion from an inhomogeneous quench in an integrable system[END_REF][START_REF] Ilievski | Superdiffusion in one-dimensional quantum lattice models[END_REF][START_REF] De Nardis | Anomalous spin diffusion in one-dimensional antiferromagnets[END_REF][START_REF] Ljubotina | physics in the quantum Heisenberg magnet[END_REF][START_REF] Gopalakrishnan | Kinetic theory of spin diffusion and superdiffusion in xxz spin chains[END_REF][START_REF] Dupont | Universal spin dynamics in infinitetemperature one-dimensional quantum magnets[END_REF][START_REF] Bastianello | Introduction to the special issue on emergent hydrodynamics in integrable manybody systems[END_REF] and experimentally evidenced in quantum magnets and in ultracold atoms on a lattice [START_REF] Vir | Kardar-Parisi-Zhang universality from soft gauge modes[END_REF][START_REF] Iversen | State transfer in an inhomogeneous spin chain[END_REF][START_REF] Wei | Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion[END_REF][START_REF] Scheie | Detection of Kardar-Parisi-Zhang hydrodynamics in a quantum Heisenberg spin-1/2 chain[END_REF].

At intermediate times the particles oscillate back and forth in the harmonic trap: we observe large-amplitude spin-dipole oscillations and we obtain the spin drag decay rate. We unveil a N 1/4 scaling in the oscillation frequency, implying a slow-down of the motion and the decrease of the zero-temperature spin drag rate as the particle number grows.

At long times, the oscillations are damped out and the system thermalizes to the diagonal ensemble [START_REF] Rigol | Thermalization and its mechanism for generic isolated quantum systems[END_REF]. From the analysis of the energy levels distribution we find that the system is weakly non-integrable. The proposed setup allows us to explore the conditions for the emergence of non-equilibrium universal behaviour in relation to the breaking of 5.2. Model and dynamics Figure 5.1: Left panel: spin up ρ ↑ in orange (light grey) and down ρ ↓ in violet (dark grey) spatial densities (in units of the inverse harmonic oscillator length ℓ -1, with ℓ = h/mω 0 ) as a function of position in the trap (in units of ℓ) at times ω 0 t = 0, 33, 200 from top to bottom. The two initially separated clouds start oscillating in the trap and eventually fully mix, approaching to a zero-magnetization state. Right panel: magnetization as a function of x (in units of ℓ) and t (in units of ω -1 0 ) for N = 12 fermions. The green line corresponds to center of mass d(t) of the magnetization. its integrability in one dimension.

Model and dynamics

We consider a one-dimensional SU(2) interacting Fermi gas confined in a harmonic trap. The Hamiltonian for such system has been introduced in Eq. (2.36):

H SU = N ∑ i=1 p 2 i 2m + mω 2 0 x 2 i 2 + g ∑ i̸ = j δ (x i -x j ), (5.1)
where N = N ↑ + N ↓ is the total number of particles and we take N ↑ = N ↓ , ω 0 is the frequency of the harmonic trap and we model the interspecies interaction using a delta potential of strength g. Hamiltonian (5.1) is characterized by the symmetry under exchange of particles having the same spin. For SU(2) fermions, the eigenstates can be classified by the irreducible representations of the permutation group (see eg. [START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF]). We focus on the strongly repulsive limit g → ∞: in this regime the model is exactly solvable: the wavefunction is given by Eq. (2.40). We remark that, in the g → ∞ limit, the spin and spatial ('charge') degrees of freedom are decoupled in the wavefunction. The time dependence of the wavefunction (2.40) can be extracted starting from [START_REF] Girardeau | Dark solitons in a one-dimensional condensate of Hard Core bosons[END_REF]: in the general case both the phases a P and the antisymmetric wavefunction Ψ A have a non-trivial time evolution. Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas As described in Section 2.2.2, we determine the phases a p in Eq.(2.40) to first order in 1/g by mapping the Hamiltonian (5.1) into an effective spin chain described by the Hamiltonian (2.38).

We classify the basis vectors of the Hilbert space associated to (2.38) according to the spin ordering on the chain (the so-called snippet basis [START_REF] Deuretzbacher | Exact solution of strongly interacting quasi-one-dimensional spinor Bose gases[END_REF]). For example, for N ↑ = N ↓ = 2 the vector |↑↑↓↓⟩ indicates that all the spins ↑ are placed in the left half of the chain.

The diagonalization of Eq. (2.38) allows us to calculate the a P and thus several observables such as the spin densities ρ ↑,↓ (x,t). This allows us to study the dynamics of the trapped system with an arbitrary initial state.

Strongly repulsive regime

We show here an intrinsic property of the strong-coupling expansion we are using to study the system: a variation of the coupling constant g to a new value g ′ induces a scaling of all many-body energy levels E n to new values E ′ n such that gE n = g ′ E ′ n , without affecting the eigenvectors. This follows from the expression of H s and of the coefficient J i and may be used as to identify the strongly interacting regime.

The above property implies that the frequencies ω driving the dynamics, i.e. the energy spacings, scale accordingly. As a consequence, by rescaling the time scale likewise, the dynamics doesn't depend on the actual value of the coupling constant g.

In Fig. 5.2 we show the center of mass of the magnetization d(t) as a function of the time by rescaling the time axis of a factor of 1/ g, being g = g/(hω 0 l). We observe that, for various values of the coupling constant, the spin dynamics has exactly the same features.

Model and dynamics

Figure 5.4: Top panel: integrated current δ j(t) in units of ℓ -1 from DMRG, as a function of time t (in units of ω -1 0 ) evaluated in the center of the trap x = 0 (solid lines), together with ballistic δ j ∼ t , KPZ δ j ∼ t 2/3 and diffusive behaviour δ j ∼ t 1/2 (dashed straight lines). Larger number of particles are associated with increasingly darker colors. Bottom panel: KPZ scaling of the magnetization m n (t), shown as a function of y = n/(ω 0 t) 2/3 . We show the comparison with the derivative of the suitably renormalized 1d KPZ scaling function g(y) [266].

Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas where δ q(l),↑ = 1 when the l-th spin in the ordering associated to |e q ⟩ is ↑ and zero otherwise.

Orbital current

We show here that the particle orbital current is zero and consequently the dynamics we describe is only due to spin torque.

The equation of motion for the density of spin ↑ particles reads (an analogous definition hold for spin ↓)

∂ n ↑ (x,t) ∂t = ∞ -∞ dx 1 ...dx N ∂ ∂t Ψ * N ↑ ∑ j=1 δ (x -x j )Ψ . (5.6)
Using the wavefunction reported in Eq. (2.40), and recalling that there is no time-dependence of the single-particle orbitals in the quench protocol here considered, we obtain

∂ n ↑ (x,t) ∂t = ∑ P,Q ∂ a * P ∂t a Q + a * P ∂ a Q ∂t ∞ -∞ dx 1 ...dx N N ↑ ∑ j=1 θ P θ Q δ (x -x j )|Ψ A | 2 , (5.7)
where we set θ P ≡ θ (x P(1) < x P(2) < ...x P(N) ). Remarkably, the product θ P θ Q is non-zero only when the permutations P and Q coincide. Consequently, the expression above can be simplified as in the following:

∂ n ↑ (x,
where in the last line we introduced ρ j (x) as the spatial density of the j-th particle in the trap [START_REF] Deuretzbacher | Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases[END_REF] in the coordinate sector individuated by the permutation P. From Eq. (5.8) we see that the only contribution to the equation of motion for the spin up density comes from the spin sector, whose time evolution is uniquely fixed by the Hamiltonian H s . The total density sector, that in the dynamical protocol we consider is constant in time, only contributes with a spatial envelope given by the densities of the particles in the harmonic trap.

Relevant observables

The experimentally accessible component of the spin vector is S z j , associated to the local magnetization

m(x,t) = N ∑ j=1 m j (t)ρ j (x), (5.9)
where m j (t) = ⟨χ(t)|S z j |χ(t)⟩ and |χ(t)⟩ = e -iH s t |χ(0)⟩ is the time-evolved spin state, obtained from the diagonalization of H s by exploiting all its symmetries. The magnetization is experimentally accessible by recording the population imbalance among ↑ and ↓ fermions, m(x,t) = ρ ↑ (x,t)ρ ↓ (x,t).

Short times

Another important observable for the dynamics is the spin current density j(x,t) = 1 2

N-1 ∑ j=1 j j (t)(ρ j (x) + ρ j+1 (x)), (5.10)
where j j (t) are obtained from the z component of Eq. (5.2), ie j j (t) = J j (σ x j σ y j+1σ y j σ x j+1 ). We detail in the following the spin-mixing dynamics in the various relevant time regimes. Systems up to N = 12 particles have been analyzed using exact diagonalization, while we used truncated Taylor series approximation of the time evolution operator [START_REF] Awad | Computing the action of the matrix exponential, with an application to exponential integrators[END_REF] and time-dependent DMRG [START_REF] Peotta | Short-time spin dynamics in strongly correlated few-fermion systems[END_REF] to study larger systems.

Short times

The short time dynamics, before the first spin oscillation, is shown in Fig. 5.3. We show the magnetization as a function of space and time, showing that the initially sharp magnetization interface spreads with time, till it starts feeling the effect of the confining potential. To identify the nature of the magnetization spreading, it is useful to study the time-integrated spin current density as in [START_REF] Ljubotina | Spin diffusion from an inhomogeneous quench in an integrable system[END_REF], δ j(t) = t 0 dt ′ j(0,t ′ ), where j(x,t) is defined in Eq.(5.10) and it is calculated at the center of the trap. We see that the integrated spin current displays a superdiffusive behaviour δ j(t) ∼ t η , with power law exponent η ∼ 0.638 [START_REF] Gauthier | Direct imaging of a digital-micromirror device for configurable microscopic optical potentials[END_REF]. This is compatible with the value η = 2/3 predicted for the homogeneous spin chain and clearly not ballistic nor diffusive. Remarkably, low-energy dynamics described by Luttinger liquid predicts ballistic behaviour [START_REF] Polini | Spin drag and spin-charge separation in cold Fermi gases[END_REF]: the deviation from this prediction discloses the marked out-of-equilibrium features of the physical system that are described in an exact way by our model. Using DMRG calculations we have checked that the KPZ region persists for larger numbers of particles (see Fig. 5.4) The deviation at later times from KPZ behaviour is due to onset of the oscillatory dynamics associated to the presence of the external trap. Within the KPZ region, we find that the magnetization profiles collapse onto each other if plotted as a function of x n /(ω 0 t) 1/z , z = 3/2 being the KPZ value for the dynamical critical exponent.

Intermediate times

We next focus on the intermediate time regime, when the particles undergo largeamplitude spin-dipole oscillations in the trap. We follow the center of mass oscillations of the magnetization,

d(t) = 1 N ∞ -∞
dx x m(x,t).

(5.11)

The time evolution of d(t) is shown in Fig. 5.5a for various values of the number of particles. We observe damped oscillations tending to a plateau corresponding to zero magnetization. At later times (not shown in the figure), the dynamics undergoes several Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas Figure 5.5: Left panel: center of mass of the magnetization d(t), in units of ℓ, as a function of time, in units of ω -1 0 , and scaled by a factor N 1/4 to evidence the universal behaviour of the oscillations. On the top right, d(t) for N = 12 fitted with a damped harmonic oscillator F(t) = f 0 e -γt cos(Ωt). partial revivals, as expected since we describe a closed quantum system. Quite remarkably, we find that the various curves for different particle numbers collapse one to another if we scale the time axis by a factor N α with α = 0.25. Using DMRG simulations up to N = 60 particles, we have tested that the scaling is robust at increasing particle numbers. The magnetization oscillations are well approximated by a damped harmonic oscillator F(t) = f 0 e -γt cos(Ωt)(see the right panel of Fig. 5.5). This allows us to obtain the oscillation frequency Ω for the various N values. We find that the damping rate γ doesn't depend on the number of particles. Combining the two values, we obtain the spin drag rate as Γ sd = Ω 2 /γ [START_REF] Amico | Theory of spin Coulomb drag in spinpolarized transport[END_REF][START_REF] Amico | Coulomb interaction effects in spinpolarized transport[END_REF][START_REF] Sommer | Universal spin transport in a strongly interacting Fermi gas[END_REF]269].

We also perform a spectral analysis of d(t) by introducing the spectral function A(ω) = ∞ 0 dt d(t)e iωt . In the left panel of Fig. 5.6 we show |A(ω)| and the Fourier transform of the fitted signal F(ω) = ∞ 0 dt F(t)e iωt as a function of the rescaled frequencies. The spectral function shows two main peaks centered around ±Ω univ . Several excitation frequencies contribute to the overall shape and the linewidth of F(ω). To estimate the scaling exponent α we evaluate the position P N (α) = ω P N α of the maximum of F(ωN α ) at positive frequencies, such that F(ω P N α ) = max ω>0 F(ωN α ), as a function of a scaling exponent α. As we show in the right panel of Fig. 5.6, the universal scaling is reached for α = 0.25.

The universal scaling observed in Fig. 5.5 allows us to estimate the spin-dipole oscillation frequency at larger N as Ω N ≃ Ω univ /N 1/4 , with Ω univ ≃ 0.19 ω 0 . Correspondingly, we find that the spin drag scales as Γ sd = Ω 2 univ /(γN 1/2 ), hence vanishing at large particle numbers, as also predicted in [START_REF] Polini | Spin drag and spin-charge separation in cold Fermi gases[END_REF] for low-energy excitations of the spectrum.

Intermediate times

Figure 5.6: On the left, modulus of A(ω), in units of ℓ/ω 0 , for different number of particles as a function of the universal frequencies ωN 1/4 /ω 0 , compared to the modulus of the Fourier transform F(ω) of the fit F(t) (dotted violet line). Colors codes are the same as in panel (a). On the right, position P N (α) of the peaks of F(ωN α ), in units of ω 0 , as a function of the scaling parameter α.

Parity of oscillation frequencies

We derive here the analytical expression for the magnetization, that illustrates the energy levels involved in the time evolution. The magnetization can be written as m(x,t) = ∑ s j=1 m j (t)ρ j (x). The probability of the i-th particle to have magnetization ±1 is:

m j (t) = ⟨χ(t)| S z j |χ(t)⟩ = s ∑ p=1 |a p (t)| 2 (δ p( j),↑ -δ p( j),↓ ), (5.12)
where |χ(t)⟩ = ∑ s p=1 a p |e p ⟩ is the spin component of the wavefunction, being |e p ⟩ the snippet basis. We also introduced δ p( j),↑(↓) that is equal to one if the j-th component of the p-th element of the basis is a spin up (down) and zero otherwise. We want to study the symmetry of Eq. (5.12), starting from the explicit expression for |a p (t)| 2 ,

|a p (t)| 2 = s ∑ l,k e -i(E k -E l )t f p,k f p,l ψ l0 ψ k0 , (5.13)
where f p,k = ⟨e p |ψ k ⟩ and ψ k0 = ⟨ψ k |χ(0)⟩, with |ψ k ⟩ eigenstates of H s . When N ↑ = N ↓ the Hamiltonian is symmetric under spin inversion: one can check that this implies f pk = ± f s-p+1,k ; , ∀k. The choice of the sign depends on the index k. Consequently, we split the sums in (5.13) according to the parity of the eigenstates. To do so, we divide all the possible indexes k, l in four sets {Λ ++ , Λ -+ , Λ +-, Λ --} such that if k, l ∈ Λ ab then f p,k = a f s-p+1,k and f p,l = b f s-p+1,l , a, b = +, -.

One can also show that the spin inversion symmetry induces the relation δ p( j),↑ = δ (s-p+1)( j),↓ ∀p. As a consequence, we can write Eq.(5.12) as in the following:

m j (t) = 4 s/2 ∑ p=1 ∑ l,k∈Λ +-
cos(ω k,l t) f p,k f p,l ψ l0 ψ k0 (δ p( j),↑δ p( j),↓ ).

(5.14)

Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas From Eq. (5.14) we see that the frequencies ω k,l = (E k -E l )/h, that have a non-vanishing contribution in the magnetization dynamics, are the ones connecting eigenstates with different parity. As immediate remark, we see that the frequencies ω k,k = 0 don't affect the time evolution consistently with the Fourier analysis presented in Fig. (5.6). The inset shows the level-spacing distribution of the whole unfolded spectrum and the Poisson distribution W P (∆ε) (green line). In all the panels, N = 14.

Long times

Finally, we study the long-time regime at which the damped dynamics becomes dominant and the system approaches a zero-magnetization state. Since the Hamiltonian (5.1) is not integrable at finite interaction strength, we expect some traces of chaoticity to emerge during the dynamics [START_REF] Ueda | Quantum equilibration, thermalization and prethermalization in ultracold atoms[END_REF][START_REF] Rigol | Thermalization and its mechanism for generic isolated quantum systems[END_REF]270]. In this case the system thermalizes to a state described by the diagonal ensemble, coinciding in our case with the microcanonical ensemble [START_REF] Luca D'alessio | From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics[END_REF]. We verify this by calculating the distance R(t) = dx |ρ ↑ (x,t)ρ ↑,MC (x)| 2 between the spin up density and its value in the microcanonical ensemble ρ ↑,MC (x). The results are presented in Fig. 5.7a. At times corresponding to the zero-magnetization plateau in Fig. 5.7, R(t) vanishes and the spin density approaches the steady state value. At later times, revivals occur and the system deviates from this configuration.

To further provide evidence for chaotic behaviour, we analyze the level-spacing distribution W (∆ε) [START_REF] Borgonovi | Quantum chaos and thermalization in isolated systems of interacting particles[END_REF][START_REF] Santos | Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization[END_REF][START_REF] Guhr | Random-matrix theories in quantum physics: common concepts[END_REF], constructed using the unfolded dimensionless energy levels [270,[START_REF] Gubin | Quantum chaos: An introduction via chains of interacting spins 1/2[END_REF][START_REF] Gubin | Quantum chaos: An introduction via chains of interacting spins 1/2[END_REF][START_REF]Unfolding is obtained by normalizing the energy levels such that the mean level spacing is equal to one everywhere in the spectrum[END_REF][275]. The spectrum of an integrable system follows a Poissonian distribution W P (∆ε)=e -∆ε , while a chaotic system is described by a Wigner-Dyson one W W D (∆ε)= π 2 ∆ε e -π To obtain the level-spacing distribution it is important to take into account the symmetries of the system [START_REF] Stone | Two simple systems with cold atoms: quantum chaos tests and non-equilibrium dynamics[END_REF], which in our case are the spatial parity and the symmetry under particle exchange. Our choice of basis vectors allows us to readily check the parity of the eigenstates. In order to identify the symmetry under particle exchange associated to a given Young tableau, we diagonalize the Heisenberg Hamiltonian in the basis of the permutational symmetry [START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF][START_REF] Nataf | Exact diagonalization of heisenberg SU(n) models[END_REF]. We then partition the energy levels according to the quantum numbers of the corresponding eigenstates. In the inset in Fig. 5.7b we show the distribution of all the unfolded level spacings, irrespectively of the symmetry constraints. In this case the chaoticity is hidden and the distribution is Poissonian. The level-spacing distribution of the largest subspace at fixed symmetry is shown in the main panel of Fig. 5.7b. We find that the level-spacing distribution is well described by the Brody distribution with parameter β = 0.22. This shows that large interactions destroy only partially the integrability of the infinite-repulsion model. A moderately chaotic behaviour also emerges from the study of the localization properties of the eigenstates of the Hamiltonian (5.1), as we show in the following section. Such intermediate regime is typical of integrable systems subjected to small perturbations [START_REF] Santos | Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization[END_REF]270,[START_REF] Santos | Integrability of a disordered Heisenberg spin-1/2 chain[END_REF].

Further probes of chaoticity

We provide here further arguments on the breaking of the integrability of the model induced by the strong coupling expansion to first order in 1/g. In the spirit of Ref.

[270], we calculated other quantities to probe the presence of chaoticity in the system. In particular, we focused on the eigenstates of the Hamiltonian and on their localization properties. In Fig. 5.8 we show from left to right respectively the inverse participation ratio I = ∑ s p=1 | f p, j | -4 , the overlap between consecutive probability distribution, Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas

O = ∑ s-1 p=1 | f p, j | 2 | f p, j | 2 and the Shannon entropy S H = -∑ s p=1 | f p, j | 2 ln| f p, j | 2 .
We compare the three quantities with the corresponding value predicted by the Random Matrix Theory, indicated by the green lines. By comparing the results with the ones of Ref.

[270], we see that the eigenstates show some similarities in terms of statistical properties, but are not completely sparse as they would be in the integrable system.

Conclusions to Chapter 5

We have studied the strongly out-of-equilibrium spin-mixing dynamics of repulsive 1D fermions under harmonic confinement, starting from an initial spatially separated spin configuration. Thanks to the mapping to an inhomogeneous Heisenberg model on an effective lattice in particle space, we have followed the real-space magnetization dynamics till very long times. At short times, as specific of one-dimensional systems and different from the three-dimensional strongly interacting Fermi gas, we observe superdiffusive behaviour of the magnetization profile in time. The system here considered is weakly not integrable, hence equivalent to the case where KPZ universality was reported in the short times dynamics [START_REF] Bastianello | Introduction to the special issue on emergent hydrodynamics in integrable manybody systems[END_REF][START_REF] De Nardis | Stability of superdiffusion in nearly integrable spin chains[END_REF][START_REF] De Nardis | Subdiffusive hydrodynamics of nearly-integrable anisotropic spin chains[END_REF]. Our observations call for the exploration of the universal properties of the corresponding spin model. At intermediate times, we have obtained damped spin-dipole oscillations characterized by a universal scaling of the oscillation time with N 1/4 , thus predicting a slow-down of the oscillation and decrease of spin drag at large particle numbers. At long times, we find that the system thermalizes to a diagonal ensemble state thanks to its moderately chaotic behaviour. All our conclusions hold exactly for strongly repulsive interactions to order 1/g. A study of itinerant fermions at arbitrary interactions and long times remains an open challenge. Our results show that harmonically trapped strongly interacting fermions are a promising platform for exploring the many facets of the non-equilibrium quantum dynamics.

6

Strongly-interacting multi-component bosons

In this Chapter, we study a two-component Bose gas confined on a ring. In this system, at difference from Fermi gases, the scatterings among particles with the same spin are allowed. One can tune the inter-and intra-species couplings and analyze the different phenomena arising in the different interaction regimes. In particular, we focus on two strongly interacting limits: the Tonks-Girardeau limit, where all the couplings are identical and infinitely large and the so-called symmetrybreaking regime, where the intra-species interactions are infinitely large (and repulsive), whereas the inter-species interactions are large but finite. We first provide a Bethe Ansatz solution for the wavefunction in this latter limit, then we study the different symmetries of the wavefunction in the two cases. Lastly, we study the persistent current induced in the ring following the application of an artificial gauge field. We compare the Tonks-Girardeau and the symmetry-breaking regimes and we compare the persistent current in the two cases with the one of the non-interacting Bose gas.

Introduction

In this Chapter, we describe some applications of the methods described in Chapter 2 to Bose-Bose mixtures. We start from a very general Hamiltonian describing a twocomponent Bose gas with N ↑ spin up and N ↓ particles:

H BB = ∑ σ =↑,↓ - N σ ∑ i ∂ 2 ∂ x 2 i,σ + 2c σ σ N σ ∑ i< j δ (x i,σ -x j,σ ) + 2c ↑↓ N ↑ ∑ i N ↓ ∑ j δ (x i,↑ -x j,↓ ), (6.1)
where we made explicit the dependence of the coupling constants and of the particles'positions on the spin. As opposed to Fermi mixtures, in spinor Bose gases the interaction occurs also among particles with the same spin. We call these interactions of intensity c σ σ = m h2 g σ σ intra-species interactions. The interactions among particles with different spin, ie inter-species, are regulated by the coupling constant c ↑↓ = m h2 g ↑↓ . Depending on the ratios between the three different coupling constant, this model exhibit different properties [START_REF] Angel García-March | Quantum correlations and spatial localization in onedimensional ultracold bosonic mixtures[END_REF]. In particular, when c ↑↑ = c ↓↑ = c ↑↓ , one obtains the SU(2) Bose gas. In this regime, the Bethe Ansatz solution has been provided [START_REF] Li | Exact results of the ground state and excitation properties of a two-component interacting Bose system[END_REF].

In the following, we first consider the Tonks-Girardeau limit c σ ρ → ∞, for ρ, σ =↑, ↓. One can write the exact solution in this interaction regime, following the same procedure as described in Sec. 2.1.3. Afterwards, we consider strong but finite inter-species interactions, leaving the intra-species ones in the fermionized limit c σ σ → ∞. In the same spirit of the method for fermionized particles introduced in Sec. 2.2.2, we perform a perturbative expansion to first order in 1/c ↑↓ and then we apply the same procedure as the one described in Sec. 2.2.2 to study some equilibrium properties of the gas. In particular we focus on the symmetry of the ground state and on the dependence on an artificial gauge flux. We consider a homogeneous geometry with periodic boundary conditions. In this regime, we also provide the Bethe Ansatz solution for the wavefunction.

Bethe Wavefunction

In this section we derive the Bethe Ansatz solution Ψ BB (x 1 ...x N ) for the Bose-Bose mixture in the case of large but finite inter-species interactions. We start from the cusp conditions deriving from the intra-and inter-species interactions. In the same spirit of Eq. (2.5), the cusp condition reads:

∂ ∂ x i,σ - ∂ ∂ x i+1,σ Ψ BB | x i,σ =x + i+1,σ - ∂ ∂ x i,σ - ∂ ∂ x i+1,σ Ψ BB | x i,σ =x - i+1,σ = 2c σ σ Ψ BB | x i,σ =x i+1,σ (6.2)
∂ ∂ x i,↑ - ∂ ∂ x i+1,↓ Ψ BB | x i,↑ =x + i+1,↓ - ∂ ∂ x i,↑ - ∂ ∂ x i+1,↓ Ψ BB | x i,↑ =x - i+1,↓ = 2c ↑↓ Ψ BB | x i,↑ =x i+1,↓ (6.3)

Bethe Wavefunction

where σ =↑, ↓. In the limit of c σ σ → ∞, the first cusp condition yields Ψ BB | x i,σ =x i+1,σ = 0. Therefore, the wavefunction in this limit vanishes each time two particles with the same spin occupy the same spatial position. On the other hand, the scattering of two particles with opposite spin yields a discontinuity in the derivative of the wavefunction proportional to c ↑↓ . These two properties are analogous to the scattering properties of a two-component Fermi gas: we can use the Bethe Ansatz solution described in 2.2.1 as a starting point to build the wavefunction Ψ BB . In particular, equations (2.11) and (2.12) still hold true.

We then consider the nested procedure, ie the solution for the spin sector. As described in Section 2.2.1, in order to write the ansatz for the case N ↓ = 1, ie the equivalent of Eq. (2.16), one has to impose symmetry constraints. For this purpose, in the Bethe Ansatz solution described in Section 2.2.1 we considered the conjugate representation of the spin wavefunction of the Fermi gas. This procedure allows one to map the spin component of the wavefunction into the Bethe wavefunction of a fictitious isotropic spin chain, which allows us to access the exact solution in the spin sector. Therefore, we impose the following ansatz for the single-spin-down-particle solution:

F B (Λ, y) = (-1) |Q| ↑↑ (-1) |Q| ↓↓ F(Λ, y), (6.4)
where we used the same notations as in Section 2.2.1 and F(Λ, y) is defined in Eq. (2.16). The quantities |Q| σ σ are the number of transpositions involving particles with the same spin that links the ordering Q and I. The two coordinate-sector-dependent phases we applied to the fermionic solution reflect the bosonic nature of the components of the mixture: despite the fermionization properties of the particles, the wavefunction must be symmetric with respect to the exchange of any particles (whereas this is not the case for a twocomponent Fermi gas, where the wavefunction must be antisymmetric under exchange of particles with same spin). The symmetrization procedure implemented in Eq. (6.4) reflects the same approach we described in Sec. 2.1.3 to perform the Bose-Fermi mapping to study the spinless TG gas, generalized to multi-component Bose gases. Within these considerations, the solution for generic M can be written as:

[QI] ≡ ξ 0B = ∑ R A R M ∏ i=1 F B (y R(i) , Λ R(i) ) = (-1) |Q| ↑↑ (-1) |Q| ↓↓ ∑ R A R M ∏ i=1 F(y R(i) , Λ R(i) ), (6.5
) where again we adopted the same notations as in 2.2.1. As a consequence, the Bethe wavefunction in this interaction regime can be written, in any coordinate sector Q, as:

Ψ BB (x 1 ...x N ) = ∏ i, j ∏ σ =↑,↓ sgn(x Q(i,σ ) -x Q( j,σ ) )Ψ Q (x 1 ...x N ) (6.6)
where the two sign functions reproduce the same effect as the phases (-1) |Q| σ σ and Ψ Q (x 1 ...x N ) is the Bethe wavefunction for the Gaudin Yang model obtained in Sec. 2.2.1. We remark that the phases adjusting the symmetry do not modify the energy spectrum. Therefore, the energy levels associated to Eq. (6.6) are the same as the ones of the Gaudin-Yang model. While it is useful to have an exact solution at hand, it is not easy to compute correlation functions from it. For this purpose, we restrict to the strongly interacting case.

Symmetry-breaking in the strongly repulsive regime model

In this section we study the symmetry of the ground state in the limit where c σ σ → ∞ and c σ σ ̸ = c ↑↓ , for large c ↑↓ . To do so, it is convenient to use the perturbative method described in Sec. 2.2.2 and map the system on a spin chain, to the first order in 1/c ↑↓ . The system can be mapped in the following spin chain:

H SB = E (c ↑↓ →∞) I -NJI -J N ∑ j=1 P j, j+1 + 2J N ∑ j=1 |s⟩⟨s|P j, j+1 |s⟩⟨s| (6.7)
where the last term is a shift performed only on the diagonal of P j, j+1 , ∀ j and |s⟩⟨s| is the projector on the corresponding subspace of the Hilbert space. By using the explicit definition of the permutation operator P j, j+1 in terms of Pauli matrices given in Sec. 2.2.2, this Hamiltonian can be written as:

H XXZ = -J N ∑ j=1 (σ x j σ x j+1 + σ y j σ y j+1 -σ z j σ z j+1 ) - 1 2 NJI, (6.8)
which is an anisotropic Heisenberg Hamiltonian, with an anisotropy coefficient ∆ = -1. This Hamiltonian can be mapped in an antiferromagnetic isotropic Heisenberg chain via the unitary transformation U XXZ = ∏ ℓ=even σ z ℓ [START_REF] Takahashi | Thermodynamics of one-dimensional solvable models[END_REF]. Therefore, the two Hamiltonians share the same spectrum. As we showed in Sec. 2.2.2 and 2.3, the spectrum of the antiferromagnetic Heisenberg chain corresponds to the one of a system of fermionized fermions. Therefore, in the limit we are considering, the energy levels of Hamiltonian (6.1) coincide with the ones of the two-component Fermi gas in the same interacting limit, whereas the symmetry of the wavefunction is different because of the transformation U XXZ linking the two effective spin Hamiltonians. Such transformation does not preserve the symmetry of the state under spin exchange: we expect the symmetry of the ground state to be different than in the fermionic case. One can benchmark this claim by computing the commutation relation between U XXZ and the class-sum operator of the permutation group, Γ 2 [?, ?]. This operator is defined via the summation of all the possible permutation operator acting on the Hilbert space, ie Γ 2 = ∑ N a,b=1 P a,b . This is consistent with the Bethe Ansatz solution we provide in Sec. 6.2: the two methods yield the same results for the energy spectrum and the symmetry of the wavefunction.

In the following, we detail precisely the symmetry of the ground-state wavefunction we are considering in terms of Young tableaux. In Sec 1.2.3 we showed that the groundstates of two-component Fermi and Bose gases are uniquely associated to a single Young tableau, which within our convention for the symmetry of the tableaux are respectively the most symmetric and antisymmetric configuration possible for a given balance of the mixture. This result is obtained by following the procedure sketched in Sec.1.2.3 and designed in [START_REF] Nataf | Exact diagonalization of heisenberg SU(n) models[END_REF], which allows one to write spin Hamiltonians on a basis where each element uniquely corresponds to a standard Young tableau. Therefore, in the limit where the method described in Sec. 2.2.2 holds, one can characterize the symmetry of atomic mixtures in terms of Young tableaux. In the two-component Fermi and Bose gases, the ground state has non-zero projection only on the subspaces spanned by tableaux of the same shape: they are states with well-defined symmetry. On the other hand, in the case we are currently analyzing (we will indicate it as symmetry-breaking (SB) in the following), the ground state has non-vanishing projection on subspaces generated by tableaux of 6.3. Symmetry-breaking in the strongly repulsive regime model different shapes. In particular, the ground state results in a superposition of the most symmetric and the most antisymmetric configurations, mixing the symmetry of the fermionic and of the bosonic ground states. In Appendix C we provide an explicit example of this phenomenon for N = 4 particles.

Short-distance correlations: the Tan's contact

In order to probe a quantitative difference between the two-component TG gas and the SB case, one can consider the short-distance correlations, which are encoded in the tails of the momentum distribution n BB (k). For a system with zero-range interactions, the momentum distribution decays as k -4 . The prefactor C = lim k→∞ n BB (k)k 4 is defined as the Tan's contact [START_REF] Tan | Large momentum part of a strongly correlated Fermi gas[END_REF].

For the TG gas, the Tan's contact is proportional to the energy slope close to c σ ρ → ∞, for ρ, σ =↑, ↓ [START_REF] Decamp | High-momentum tails as magnetic-structure probes for strongly correlated SU(κ) fermionic mixtures in one-dimensional traps[END_REF][START_REF] Decamp | Strongly correlated one-dimensional Bose-Fermi quantum mixtures: symmetry and correlations[END_REF]:

C T G = 2m 2 h4 dE (c B →∞) d(1/c B ) , (6.9)
where we defined c B ≡ c σ ρ , for ρ, σ =↑, ↓ as the unique coupling constant of the twocomponent TG gas and E (c B →∞) the corresponding ground state energy. As we show in Sec. 2.2.2, this quantity can be calculated by diagonalizing a suitable spin Hamiltonian.

In particular, for the TG case, the corresponding spin Hamiltonian is a ferromagnetic isotropic Heisenberg magnet [START_REF] Deuretzbacher | Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases[END_REF][START_REF] Deuretzbacher | Momentum distributions and numerical methods for strongly interacting one-dimensional spinor gases[END_REF], analogous to Eq. (2.38) under the transformation J i → -J i , where in this case we set J i = J because of the translational invariancec of the system (see Sec.2.3). Explicitly, this Hamiltonian reads:

H FERRO = E (c B →∞) I -NJI -J N ∑ j=1 P j, j+1 (6.10)
On the other hand, for the SB case, we have to be more careful with the definition of the contact C SB . The straightforward calculation of

dE (c ↑↓ →∞)
d(1/c ↑↓ ) , which can be obtained minimizing Hamiltonian (6.8), only takes into account the scatterings among particles with different spins. However, the momentum distribution contains also information about the inter-component interactions. One can obtain an explicit form for C SB by considering the Fourier transform of the Schroedinger equation associated to Hamiltonian (6.1) and computing the large momentum behaviour of the momentum distribution. Explicitly, we first split Hamiltonian (6.1) as follows:

H BB = H kin + ∑ σ =↑,↓ H int,σ σ + H int,↑↓ , (6.11)
where we have defined

H kin = ∑ σ =↑,↓ ∑ N σ i -∂ 2 ∂ x 2 i,σ , H int,σ σ = 2c σ σ ∑ N σ i ∑ N σ i< j δ (x i,σ -x j,σ ), and
H int,↑↓ = 2c ↑↓ ∑ N ↑ i ∑ N ↓ j δ (x i,↑ -x j,↓ )
. The first step is to make the Fourier Transform of the Schroedinger equation with respect to one of the coordinate, for instance, to x 1 . Let σ be the spin of such particle. In the large-momentum limit, using that lim k→∞ Ψ BB (k, x 2 , . . . , x N ) = 0, one gets

lim k→∞ h2 k 2 2m Ψ BB (k, x 2 , . . . , x N ) = g ↑↓ ∑ j,σ ′ ̸ =σ Ψ BB (x j,σ ′ , . . . , x j,σ ′ , . . . )e -ikx j,σ ′ + ∑ σ =↑,↓ g σ σ ∑ j,σ
Ψ(x j,σ , . . . , x j,σ , . . . )e -ikx j,σ . (6.12) Chapter 6. Strongly-interacting multi-component bosons By multiplying by the complex conjugate, one obtains the following asymptotic behaviour of the total momentum distribution n(k):

lim k→∞ k 4 n(k) = 2m 2 h4 g ↑↓ ⟨Ψ BB |H int,↑↓ |Ψ BB ⟩ + ∑ σ =↑,↓ g σ σ ⟨Ψ BB |H int,σ σ |Ψ BB ⟩ (6.13)
By applying the Hellmann-Feynman theorem to Eq. (6.13), with Ψ BB the symmetrybreaking many-body wavefunction, one can extract the explicit expression for the Tan's contact in the SB case:

C SB = 2m 2 h4 ⟨a SB P | H FERRO |a SB P ⟩ , (6.14)
where |a SB P ⟩ is the ground state of Hamiltonian (6.8). In other words, in order to get the right correction to the tails of the momentum distribution to first order in 1/c ↑↓ , one has to compute the expectation value of Eq. (6.10) as in the TG case, on the ground state of Hamiltonian (6.8). This is equivalent to minimize the whole energy of the system in the strongly interacting regime, on a state which is inherently constrained by the SB conditions c σ σ → ∞ and c σ σ ̸ = c ↑↓ .

In Fig. 6.1 we plot the ratio C SB /C T G as a function of N. We observe that C SB /C T G converges very rapidly to ∼ 0.9. Thus for N > 2, the contact is lower for the SB case than for the TG mixture. As reported for other multi-component mixtures 6.4. Persistent current of a Bose-Bose mixture [START_REF] Decamp | High-momentum tails as magnetic-structure probes for strongly correlated SU(κ) fermionic mixtures in one-dimensional traps[END_REF], the reduction of the symmetry also manifests itself in the lowering of the contact in this case.

Persistent current of a Bose-Bose mixture

We want now to study the response of the system described by Hamiltonian (6.1) to an external flux, in the strongly interacting limit.

In homogeneous systems, the artificial gauge flux only couples to the total momentum of the particles, shifting the latter by a quantity Ω [START_REF] Naldesi | Enhancing sensitivity to rotations with quantum solitonic currents[END_REF]. Moreover, since we are interested in studying the persistent current of the bosons on the ring, we restrict our analysis to the energy spectrum, which encodes all the information about the current (we remind that the current is defined as minus the derivative of the ground state energy with respect to the normalized external flux Ω, see Section 3.3). Since, as we mentioned in Sec. 6.3, the spectrum of the symmetry-breaking case coincides with the one of the two-component Fermi gas, we can use the results described in Sec. 2.3 to get the explicit dependence of the energy levels on the spectrum. We know from Legget's theorem introduced in Chapter 2 that the energy levels of a two-component free Fermi gas are a periodic function of the flux. In Chapter 3 we showed that for large attractive interactions the period of the energy with respect to the flux is halved, reflecting the formation of pairs. For large repulsive interactions, it is a known result [START_REF] Chetcuti | Persistent Current of SU(N) Fermions[END_REF][START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF] that for a Fermi-Fermi mixture the period is reduced by a factor of N, where N is the total number of particles in the mixture.

Remarkably, such results holds also for two-component fermionized bosons, in the limit c B → ∞. In this case, one can use the same strong-coupling expansion performed in Sec. 2.3 applied to the Bethe equations for the Bose mixture [START_REF] Li | Exact results of the ground state and excitation properties of a two-component interacting Bose system[END_REF], which read:

Lk B j = 2πI B j + 2 ∑ l arctan( k B l -k B j c B ) + 2 ∑ ν arctan( 2(k B j -Λ B ν ) c B ) (6.15) N ∑ j=1 2 arctan 2 c B (Λ B γ -k B j ) = 2πJ B + N ↓ ∑ ν=1 2 arctan Λ B γ -Λ B ν c B . ( 6.16)
The strong-coupling expansion yields: .17) where k B j and λ B n are respectively the charge and the renormalized spin rapidities of the Bethe wavefunction for the bosonic case in the limit we are considering, and I B j and J B j are the corresponding quantum numbers. Moreover, in analogy with the fermionic case, we assumed that in the ground state ∑ j k B j = 0. The second equation is the same as in the fermionic case, besides the quantum numbers J B j . Moreover, the first-order correction terms in the first equation have different In the inset, we show the energy levels of the non-interacting Bose gas. In both panels the red curve highlights the ground state. On the right, persistent current in the ground state in units of h2 mL 2 as a function of flux Ω for c B = 0 and c B → ∞. We observe the fractionalization of the period of the current of a factor of 1/N at large interactions. The plot has been realized for N = 3 particles and N ↓ = 2. signs. In the first equation, we also observe a term depending on k B lk B j : this emerges from the bosonic nature of the particles and it is present also in the spinless-bosons Bethe Ansatz solution provided by Lieb and Liniger [START_REF] Lieb | Exact analysis of an interacting Bose gas. i. the general solution and the ground state[END_REF].

Lk B j = 2πI B j + 2 ∑ l ( k B l -k B j c B ) - 2π N ∑ γ J B γ -N Ω + 4 c B k B j ∑ ν 1 1 + (λ B ν ) 2 2N arctan λ B γ = 2πJ B γ - N ↓ ∑ ν=1 2 arctan λ B ν -λ B γ 2 . ( 6
The momentum in the bosonic case is:

P B Ω = 2π L ∑ j I B j - 1 N ∑ γ J B γ -Ω . (6.18)
In the right panel of Fig. 6.2 we show the energy levels as a function of the flux of a strongly interacting Bose gas of N = 3 particles confined on a ring of R = 2πL, L being the circumference of the ring. Each parabola corresponds to a different excitation in the spectrum of the total momentum P B Ω . At zero interactions, the momentum only depends on the quantum numbers I B j and on the flux Ω. Therefore, an excitation in the momentum spectrum corresponds to a variation of the quantum numbers I B j which follows the selection rules we mentioned in Sec. 2.2.1 and 2.3. In particular, an excitation in the momentum spectrum is created by increasing or decreasing each I B j by one unit. We see that increasing by one I B j in Eq. (6.18) has the same effect as increasing Ω of one unit: this yields the periodicity of the current (and of the energy) at zero interactions. The fractionalization at large interactions is due to the term 1 N ∑ γ J B γ : a variation in the sum over this second set of quantum numbers in this interaction regime excites the total momentum. However, an increase of one unit in ∑ γ J B γ drives the system to a state with higher total momentum, that can be equivalently reached by increasing the flux Ω by 1/N. The quantum numbers J B γ are associated to spin excitations [START_REF] Andrei | Integrable models in condensed matter physics[END_REF]. In particular, in the ground state they assume integer or half-integer values depending on N and N ↓ .

Conclusions to Chapter 6

The creation of "holes" in the distribution of the J B γ , ie increasing one unit one of the quantum numbers, induces a spin excitations, so-called spinons [START_REF] Yu | Persistent current of a Hubbard ring threaded with a magnetic flux[END_REF]. For each value of the flux and at large interactions, the energy ground state is determined by minimizing the

I B j -1 N ∑ γ J B γ -Ω.
If we consider Ω ̸ = 0, in order to minimize the energy one has to excite either the angular momentum spectrum by modifying the set I B j or ∑ γ J B γ . This shows that at large interactions, the dependence of the energy on the flux, therefore the current, is influenced by the presence of spin excitations in the system, whereas this is not the case at weak interactions. In the following table, we show the explicit values of the quantum numbers we used to generate the left panel of Fig. 6.2. Tableau 6.1: Quantum numbers as a function of flux for N = 3 and N ↓ = 2. They have been used to produce Fig. 6.2

Ω I B 1 I B 2 I B 3 J B 1 + J B 2 [-1 , -5/6] -2 -1 0 0 ]-5/6 , -1/2] -2 -1 0 -1 ]-1/2 , -1/6] -2 -1 0 -2 ]-1/6 , 1/6] -1 0 1 0 ]1/6 , 1/2] 0 1 2 2 ]1/2 , 5/6] 0 1 2 1
]5/6 ,1] 0 1 2 0 ------In summary, the strong interactions modify the value of the total momentum allowed in the energy ground state (highlighted by the red line in the right panel of Fig. 6.2) at fixed flux. As a consequence, we see that the period of the ground state energy is reduced by a factor of N at large interaction, compared to the non-interacting case displayed in the inset of the right panel in Fig. 6.2. In the left panel of the same Figure, we show how this behaviour is also reflected in the persistent current.

Conclusions to Chapter 6

In this Chapter, we focused on some equilibrium properties of a two-component Bose gas confined on a ring. We considered strong repulsive interactions and we discern two different regimes. In the first one, both the inter-and the intra-species interactions are infinitely large, whereas in the second one we consider infinitely repulsive intra-species interactions and large but finite inter-species couplings. We called this last limit the symmetry-breaking case.

First, we provided the exact solution for the model at any value of the inter-species coupling constant c ↑↓ . We showed that the wavefunction is equivalent to the one of a two-component Fermi gas up to some unitary transformation which adjusts the symmetry under particle exchange. Consequently, the spectra of the two systems coincide and the energy levels of the symmetry-breaking case are the same as the one of the Gaudin Yang model.

Besides, we studied the symmetry of the ground state in the symmetry-breaking regime. As opposed to what happens with the Tonks-Girardeau limit of Bose-Bose mixtures and with the two-component Fermi gas, we show that in this case the ground state has not a well-defined symmetry. The ground state of the system in the symmetry-breaking interacting regime is not associated to a single Young tableau, but it mixes the symmetry of the ground states of a Fermi-Fermi mixture and of a Bose-Bose mixture with equal intraand inter-species interactions. We showed that this behaviour is reflected in the tails of the momentum distribution of the gas, by computing the Tan's contact in the different cases. This is consistent with the general theory of multi-component mixtures, predicting a lower Tan's contact for states associated to mixed symmetry.

At last, we studied the response of the system under the application of an artificial gauge field. In the symmetry-breaking case the response is trivially analogue to the one of Fermi-Fermi mixtures because of the equivalence of the two energy spectra. However, even in the Tonks-Giradeau limit of the Bose-Bose mixture we showed an effect qualitatively similar to what occurs in two-component Fermi gases. In particular, we studied the period of the persistent current as a function of the gauge flux: at strong interactions this quantity is reduced by a factor of N if compared to the non-interacting case. As occurs in Fermi gases, this fractionalization can be associated to the creation of spin excitations in the system. In order to better understand the difference between the two-component Fermi gas, the symmetry-breaking case and the Tonks-Girardeau regime, would be interesting to study the symmetry of the ground state as a function of the flux. This would characterize the fractionalization of the period of the persistent current in terms of Young tableaux and would provide more physical insights on the microscopic mechanism underlying this phenomenon.

Conclusions and outlook

Conclusions and outlook

In this thesis, we studied the coherence and the dynamics of ultracold atomic gases, with a particular focus on the Fermi gases.

The first part is dedicated to the coherence of Fermi gases. In particular, we considered a two-component Fermi gas confined on a ring-shaped potential threaded by an artificial gauge field. In Chapter 3, we focused on the persistent current induced in the ring and on their application to the analysis of the BCS-BEC crossover. The current exhibits a parity effect in the BCS regime which is washed out by increasing attractive interactions, as soon as we approach the BEC regime. In this purely bosonic regime, the parity effect disappears, consistently with the theoretical predictions for Bose gases. This provides a criterion to discern whether, at fixed interaction, the system is in the BCS regime, in the BEC regime or in some intermediate one. Moreover, we studied the period of the persistent current as a function of the gauge flux. This quantity exhibits a marked dependence on the attractive interaction. In particular, when the particles are strongly attracting each other, the period is reduced by a factor of 2, compared with the non-interacting case. In analogy with the superconducting mesoscopic rings, this can be associated to the formation of two-body bound states, ie fermionic pairs, in the ground state of the system. This also evidences the different nature of the ground state in the BEC regime, where the fermionic nature of the particles is hindered by the large attractive interactions. These results can be experimentally accessed by spiral interferometry of the gas, after the confinement potential is switched off. The system is let to co-expand together with a reference gas placed in the center of the ring. If the current in the ring is non-zero, the resulting interference pattern is a spiral whose number of branches is proportional to the value of the current flowing before the expansion. This enable us to probe the parity effect and to observe the halving of the periodicity.

In Chapter 4, we focused on the above-mentioned expansion protocol and we studied the phase of a two-component degenerate Fermi gas confined on a ring. At intermediate times after the confined potential is switched off, we again look at the current flowing in the ring following the application of an artificial gauge field. This time, we use this observable to access the coherence properties of the confined gas. In particular, we underlined the difference between the fermionic and the bosonic case. In the latter, the coherence of the state provides a well-defined spiral interferogram yielding information about the current flowing in the ring. On the other hand, the fermionic interference pattern exhibits signs of the lack of coherence of the Fermi wavefunction. In particular, at small interactions, one can observe the interference between the single-particle's orbitals composing the Fermi sphere, characterized by dislocations appearing in the spiral-shaped pattern. These dislocations allows us to access the number of particles of the gas in fewbody systems. As in the bosonic case, from the number of branches of the spiral, one can infer the value of the current flowing in the ring. At long times after the expansion, the density of the expanding cloud is proportional to the momentum distribution of the gas Conclusions and outlook confined on the ring, which can be observed by time-of-flight imaging. Therefore, one can access the momentum-momentum correlation function. We compared this quantity for attractive and repulsive interactions. In the first case, the formation of pairs is connected to an enhancement of the correlations among particles with opposite momenta. The dependence of the visibility of the correlations on the number of particles is linked to the presence of quasi-off-diagonal-long-range-order in the gas, associated to the formation of pairs. On the other hand, for repulsive interactions, we show an enhancement of the correlations among particles with coinciding momenta and there is no evidence of many-body quantum coherence.

These features stress the differences between attractive and repulsive interactions and reveal the many-body quantum phase of the Fermi gas, intended as the emergence of quantum coherence and pairing superfluidity. In the future, would be interesting to deepen the knowledge of the origin of the dislocations, which seems to be related to the nodes of the wavefunction, where the interference between the orbitals is completely disruptive.

In Chapter 5 we focused on the dynamics of a strongly interacting two-component Fermi gas trapped in a harmonic potential. We started from the integrable limit of the model, where the interparticle interactions are considered to be infinitely repulsive and the exact form of the wavefunction is known. We performed a strong-coupling expansion that holds for large but finite interactions. In this limit, we studied the exact time evolution of a state where all the spin up particles are placed in the left-hand side of the trap and the spin down are placed in the right-hand side. In this regime of interactions, a nontrivial dynamics occurs only in the spin sector, which is totally decoupled from the charge degrees of freedom. As a consequence, we focused on the time evolution of the magnetization and on the spin current at the center of the trap. At short times, the dynamics is superdiffusive and is dominated by the critical exponents of the Kardar-Parisi-Zhang universality class. We observe a temporal and spatial scaling both in the magnetization and in the integrated current in the center of the trap. At intermediate times, the spin up and spin down components oscillate in the trap and cross each other. We studied the details of the spin oscillations. We showed a damping, due to a spin-drag effect acting each time the two clouds scatter. The damping rate is linked to the rate of momentum loss of the particles. Moreover, we showed that the characteristic frequency of the spin oscillations exhibits a universal scaling in terms of number of particles with a scaling factor N 1/4 , where N is the number of atoms. We benchmarked the robustness of such scaling for various values of numbers of particles. Last, we focused on the long time dynamics. In order to study the equilibration properties of the gas, we probed the emergence of quantum chaotic behaviour. We quantitatively estimated the effect of our perturbative approach on the integrability of the model at infinite repulsion. We see that the perturbation that we apply only partially affects the integrability of the model, without inducing purely quantum chaos. Nevertheless, we showed that the system equilibrates to the thermal ensemble at long times and then undergoes revivals because of the finite size of the system. In outlook, it would be interesting to unveil the microscopic mechanism yielding the Kardar Parisi Zhang critical exponents at short times. Moreover, a physical explanation for the scaling exponent 1/4 of the characteristic frequencies of the spin oscillations is still an open point. Despite the spectral analysis confirming these peculiar features of the frequencies, it would be interesting to explain this effect in terms of the parameters of the trap. We stress that this particular scaling factor seems to be specific for the harmonic potential.

In Chapter 6, we focused on multi-component Bose gases. We considered two differ-Conclusions and outlook ent strongly interacting regimes. The first regime corresponds to a two-component Tonks-Girardeau gas, where both the inter-and the intra-species interactions are taken to be infinitely repulsive. The second regime, so-called symmetry-breaking case, corresponds to infinitely large repulsive intra-species interactions and large but finite interspecies interactions. We first obtained the exact wavefunction of the system in the symmetry-breaking regime. Afterwards, we determined the Young tableaux defining the symmetry of the ground state in this interaction regime. In particular, we showed that the ground state has not a well-defined symmetry. As opposed to the Tonks-Girardeau regime, where the ground state is uniquely associated to a single Young tableau, in the symmetry-breaking case the ground state is in a superposition of states with different symmetries. In particular, it mixes the symmetry of the Tonks-Girardeau ground state and the symmetry of the ground state of a strongly interacting two-component Fermi gas. Next, we showed how this lack of symmetry of the ground state affects the behaviour of the short-distance correlations, by looking to the behaviour of the tails of the momentum distribution, with particular regard to the Tan's contact in the two cases. Lastly, we studied the response of the system to the application of an artificial gauge field and we observed the reduction of the period of the persistent current as a function of the gauge flux by a factor of N, N being the total particle number. As future perspective, it would be interesting to classify in terms of Young tableaux the different energy levels responsible for the fractionalization of the current at large interaction. This would allow us to characterize the dependence of the symmetry of the ground state on the gauge flux. reads:

H s =               -5J √ 3J 0 0 0 0 √ 3J -3J 0 0 0 0 0 0 -3J J √ 3 -2 3 J 0 0 0 J √ 3 -7 3 J √ 2J 3 0 0 0 -2 3 J √ 2J 3 -8 3 J 0 0 0 0 0 0 0               . (C.2)
On the other hand, for Bose-Bose mixtures, the spin Hamiltonian providing the energy correction at large repulsive interaction, is Eq. (6.10). On this basis, it reads:

H FERRO =               -3J - √ 3J 0 0 0 0 - √ 3J -5J 0 0 0 0 0 0 -5J -J √ 3 2 3 J 0 0 0 -J √ 3 -17 3 J - √ 2 3 J 0 0 0 2 3 J - √ 2 3 J -16 3 J 0 0 0 0 0 0 -8J               . (C.3)
We see that these matrices are block diagonal. Each block has the dimension of the subspace spanned by basis vectors associated to tableaux of the same shape. This implies that the eigenvectors of Hamiltonians H s and H FERRO have non-zero projection only on subspaces spanned by vectors with the same symmetry. There's no symmetry mixing.

We consider now the symmetry-breaking spin Hamiltonian, given by Eq. (6.7). On the basis (C.1) explicitly one can write:

H SB =               -J -J √ 3 0 0 0 -2 2 3 J -J √ 3 -5 3 J 0 0 0 2 √ 2J 3 0 0 -3J J √ 3 -2 3 J 0 0 0 J √ 3 -7 3 J √ 2J 3 0 0 0 -2 3 J √ 2J 3 -8 3 J 0 -2 2 3 J 2 √ 2J 3 0 0 0 -16 3 J               . (C.4)
As opposed to the Fermi-Fermi and the Tonks-Girardeau Bose-Bose mixture, the Hamiltonian is not block diagonal. In particular we see that the sectors corresponding to the first two standard young tableaux of the basis (the most symmetric ones) and the last one (the most antisymmetric) are mixed. As a consequence, the eigenstates of H SB mix subspaces with different symmetry and therefore there is not a unique Young tableau describing their symmetry.

D

Appendix D : Mapping to the spin Hamiltonian: Bose-Bose and Fermi-Fermi mixtures

In this appendix, we recall how to build a spin Hamiltonian to minimise in order to study the strongly interacting limit of two-component atomic mixtures. The calculation is performed in [START_REF] Deuretzbacher | Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases[END_REF]. We report here for sake of convenience the Hamiltonian we consider introduced in Eq. (2.36):

H SU = N ∑ i=1 p 2 i 2m +V (x i ) + g ∑ i< j
δ (x ix j ) ≡ H 0 + g ∑ i< j

δ (x ix j ), (D. [START_REF] Gauthier | Direct imaging of a digital-micromirror device for configurable microscopic optical potentials[END_REF] where in this case we do not make any further hypothesis on the statistics of the particles involved. We recall also that in the limit g → ∞ the wavefunction is given by Eq. (2.40):

Ψ(x 1 ...

Mots-clés :

Gaz quantiques ultra-froids, gaz de Fermi, systèmes unidimensionnels

Contents 1 . 1 1 . 2

 1112 Cold atoms confined in an external potential . . . . . . . . . . . . . 1.1.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . 1.1.2 Atom trapping and manipulation . . . . . . . . . . . . . . . . . 1.1.3 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Atom imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Ultracold atoms as quantum simulators . . . . . . . . . . . . . 1.1.6 Atomtronics . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultracold gases of fermions . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Fermi gases on a ring . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Large-spin systems and SU(r) symmetry . . . . . . . . . . . . . 1.1. Cold atoms confined in an external potential 1.1 Cold atoms confined in an external potential 1.1.

Figure 1 . 1 :

 11 Figure 1.1: Portrait of Bose and Einstein realized using an optical trapped Bose-Einstein condensate of Rubidium atoms. The last panel shows the optical depth (OD) of the condensate after 27 ms of expansion time. The dotted line is a Gaussian fit. Figure from [1].

Figure 1 . 2 :

 12 Figure 1.2: Panel (a): Superposition of two standing waves realizing a two-dimensional optical lattice. The atoms are trapped in tightly confined one-dimensional tubes. Panel (b): Three-dimensional optical lattice. Figure from [17].

Figure 1 . 3 :

 13 Figure 1.3: Number of particles (top) and scattering length (bottom) of a BEC of Sodium atoms as a function of an external magnetic field. The resonance occurs when the scattering length diverges and the loss of particles is maximum. Figure from [33].

Figure 1 . 4 :

 14 Figure 1.4: Velocity distribution of the expanding cloud of Rb-87 in three different cases: (A) right before the condensation, (B) just after the Bose-Einstein condensation occurs and (C) when the noncondensate fraction is nearly zero. The blue and white colors are associated to the condensate, all the other colors represent the noncondensate fraction. The only parameter varying in the three panels is the temperature. Figure from [2].

Figure 1 . 5 :

 15 Figure 1.5: In situ imaging of a BEC of Rubidium atoms trapped on a optical lattice with period 640 nm. Figure from [46].

Figure 1 . 6 :

 16 Figure 1.6: Interference pattern between two BECs after the expansion in real space. In panel (a) the current is zero. In panels (b) and (c) the current has the same intensity, but opposite direction. Finally, in panel (d) the intensity of the current is further increased: we observe a spiral with two branches. Figure from [66].

Figure 1 . 7 :

 17 Figure 1.7: Interference pattern I for a single particle on a lattice of 10 sites. On the left we show the single contribution of three different sites. To each contribution (a), (b) and (c) we associate the function profiles (labelled by (d), (e) and (f) respectively) along the radius connecting the center and the corresponding site on the ring. We observe a shift of the maxima of the fringes towards the center of the ring, yielding the rotational symmetry breaking. On the right, we show the final interference pattern. The black dots indicate the sources on the ring and in the center. The calculation yielding these figures is reported in Appendix A.

Figure 1 . 8 :

 18 Figure 1.8: Qualitative phase diagram of BCS-BEC crossover as a function of temperature (expressed in unit of the Fermi energy) and of the inverse of the scattering length (in unit of the Fermi momentum). Figure from [80].

Figure 1 . 10 :

 110 Figure 1.10: Qualitative differences among bosons, spinless fermions and SU(r) fermions. Figure from [94].

Chapter 1 .Figure 1 . 11 :

 1111 Figure 1.11: Panel (a): different examples of Young tableaux for N = 4. Panel(b): all the possible standard Young tableaux for a particular fixed shape of the tableau.

2. 1 .

 1 Introduction to one-dimensional models 2.1 Introduction to one-dimensional models 2.1.1 Why one-dimensional?

Figure 2 . 1 :

 21 Figure 2.1: On the left, spatial density of an out-of-equilibrium gas of strongly interacting bosons oscillating in an anharmonic trap. The oscillations are undamped for a very long time. This is similar to what happens classically with the well-known Newton craddle (right). Both figures from [121].

Figure 2 . 2 :

 22 Figure 2.2: Comparison between classical and quantum thermalization, according to ETH. In classical systems, the thermal state is reached when the dynamics starts from a state that has no memory of the previous states of the system. In the quantum regime, the notion of thermal equilibrium inherently belongs to any eigenstate of the Hamiltonian. The dephasing of the eigenstates occurring during the dynamics allows the thermalization to emerge. Figure from[START_REF] Rigol | Thermalization and its mechanism for generic isolated quantum systems[END_REF].

Figure 2 . 4 :

 24 Figure 2.4: Scheme of the low-excitations spectrum for the Hamiltonian (2.36) as a function of -1/g. When 1/g goes to zero, all the states belonging to the same manifold are degenerate in energy. As far as g decreases, the degeneracy in each manifold is lifted. Fig. from [169].

Figure 2 . 5 :

 25 Figure2.5: Scheme of a bipartition of the system implemented by DMRG algorithm. Once the ground state of the superblock is obtained, the left block increases its size: the system is in a new configuration (indicated by the dotted line) and the ground state of the new superblock is the following target of the algorithm.

Contents 3 . 1 45 3. 2 48 3. 2 . 1 48 3. 2 . 2 Fermi 51 3. 3

 3145248214822513 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of Bethe equations . . . . . . . . . . . . . . . . . . . Hubbard versus Bose Hubbard models . . . . . . . . . . Parity effects of the persistent current . . . . . . . . . . . . . . . . . 52 3.4 Readout: interferograms and noise correlations. . . . . . . . . . . . 52 3.4.1 Details on the interference for expanding ring and disk . . . . . 53 3.4.2 Time scale for spiral interferograms . . . . . . . . . . . . . . . 54 3.5 Readout of the BCS-BEC crossover . . . . . . . . . . . . . . . . . . 55 3.1. Introduction

Chapter 3 .Figure 3 . 1 :

 331 Figure 3.1: Column (a): Energies vs Ω for N = 4, N s = 8 from Bethe Ansatz (blue line) and DMRG (yellow triangles).In each column, the first three panels from the top correspond to U/J = 0, -2, -6 respectively. The last row describes the BEC regime for U B /J B = 1 (here only DMRG data are available since the model is not integrable). Column (b): persistent current as a function of the flux for the same set of parameters, obtained as the derivative of the ground-state energy in column (a). Columns (c) and (d): noise correlator for N = 4, N s = 10 for Ω = 0.1 and Ω = 0.4 respectively, indicated by red circles in column (a). The correlators are all evaluated in r ′ = (R, 0) and t = 0.3 ω -1 0 , where ω 0 is the frequency of each lattice well. The noise correlator is expressed in unit of 1/R 2 . A circulating state is characterised by a spiral-like correlator, not symmetric by inversion with respect to the y = 0 axis.

Figure 3 . 2 :

 32 Figure 3.2: Energies, persistent current and noise correlators as in Figure 3.1 here studied for N = 6, and for U/J = 0, -5, -10 (from top to bottom) [196]. The last line refers to the bosonic case for U B /J B = 1. All the other parameters are the same as in Fig. 3.1.

Figure 3 . 3 :

 33 Figure 3.3: Solutions of the Bethe equations as a function of interaction for N s = 8, N = 4 and M = 2 (left) and N s = 8, N = 6, M = 3 (right). The N = 4 column corresponds to the branch centered in Ω = 0.5 in column (a) of Fig. 3.1, while the N = 6 panels corresponds to the one centered in Ω = 0 in the same column of Fig. 3.2.

Figure 3 . 4 :

 34 Figure 3.4: Solution of the Bethe Ansatz equations: charge and spin rapidities (dimensionless) as a function of interaction U (in units of J) for N = 4 and N s = 8, and scaling of U c /J with number of lattice sites N s . For U < U c the imaginary part of the rapidities, thus the binding energy, vanishes. This solution corresponds to the energy branch yielding the doubling of the periodicity at large |U|.

Figure 3 . 5 :

 35 Figure 3.5: Comparison between real, imaginary part and modulus of the density-density correlator, calculated for N = 4 , N s = 10, U = 0, Ω = 0.4 and time t = 0.3 ω -1 0 . The three panels display similar topological features.

Figure 3 . 6 :

 36 Figure 3.6: Interference pattern between ring and the central site at different times for N = 4, N s = 10, U = -2. The flux is Ω = 0.4 (column a) and Ω = 0.1 (column b).

Contents 4 . 1

 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Free fermion gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 4 . 2 :

 42 Figure 4.2: On the left, profile J n (x, 0) 2 for different n. We see that when n = 0 the function has a finite value in the origin x = y = 0. As n increases, the function collapses to zero in this point. On the right, we show the configuration of the quantum numbers for N = 4 spinless fermions. In the ground state, they are selected in order to minimize the value of the current.

Chapter 4 .

 4 Single-particle versus many-body phase coherence in an interacting Fermi gas

Figure 4 . 3 :

 43 Figure 4.3: Time of flight for N = 4 non interacting fermions on N s = 10 sites. We observe the central peak to disappear when ll 0 ≥ 2 ie after a shift in the energy levels equal to half of the size of the Fermi sphere.

4. 5 .

 5 Repulsive vs. attractive interactionsBosons

Figure 4 . 4 :

 44 Figure 4.4: Intermediate times density-density correlations. The Bose and the Fermi case are compared. In the upper panels, the overall correlation G(r, r ′ ,t) for ω 0 t = 3 is presented. In (c) and (d), the interference G(C,R) (r, r ′ ,t) between the ring and the central site is displayed (see also text). Calculation performed by DMRG on a system of N = 14 particles on N s = 20 sites, with interaction U/J = 0.2 and Ω = 1.4.

Figure 4 . 5 :

 45 Figure 4.5: Long times density-density correlator G ↑,↓ (r, r ′ ;t to f ). The correlation for r = (x, 0), r ′ = (x ′ , 0) and t to f = 100 ω -10 is evaluated. For weak attractive interaction we observe BCS-like correlations at x = -x ′ ≃ ht to f k F /m. At strong attractive interactions tightly bound pairs are revealed by the enhancement of the correlations along the whole x = -x ′ diagonal. In the third panel, the square indicates the size of the Fermi sphere at x, x ′ = ±ht to f k F . The calculations are performed using the DMRG method with N = 14, N s = 20 and Ω = 1.4.

4. 5 .Figure 4 . 6 :

 546 Figure 4.6: Left panel: displacement l/R of the correlation peak of Fig.4.5 as a function of the flux for N = 6, U/J = -5.4 and U/J = 0.5. Right panel: visibility V (N,U) of the correlation peak as a function of interactions for Ω = 0 and various values of number of particles. We observe a markedly different N scaling for repulsive or attractive interactions. In both panels N s = 10.

Figure 4 . 7 :

 47 Figure 4.7: Time of flight (top) and G(C,R) (bottom) for a dilute system of N = 4 particle on N s = 30 sites for various values of angular momentum (left panels ll 0 = 1, right panels ll 0 = 2).We consider here a repulsive interaction U/J = 2. Consistently with the text, in the bottom plots we observe N ↑ -1 dislocations.

Figure 4 . 8 :

 48 Figure 4.8: The panel (a) shows the shift of the antidiagonal x = -x ′ as a function of the applied flux Ω for U/J = -5.4 and U/J = 0.5. In the panels (b), (c), (d), we show the one-dimensional correlator G ↑,↓ (x, x ′ ;t) Eq. 4.12 at long expansion time t to f and with flux Ω = 1.4. We observe that similar structures as in the two-dimensional case emerge. The peaks surfacing at U = 0.4 are located at k = ±k F . The figure refers to a system of for N = 14 and N s = 20.

Contents 5 . 1 78 5. 2 79 5. 2 . 1 81 5. 2 . 3

 5178279218123 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model and dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . Strongly repulsive regime . . . . . . . . . . . . . . . . . . . . 80 5.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.4 Orbital current . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.5 Relevant observables . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Short times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 Intermediate times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 5 . 2 :

 52 Figure 5.2: Center of mass of the magnetization d(t), in units of ℓ, for N = 8 particles, as a function of the rescaled time ω 0 t/ g, for various values of the dimensionless coupling constant g. The curves corresponding to different g are not resolved since they collapse one onto the other.

dx 1

 1 ...dx N θ P δ (xx j )|Ψ A | 2 =

Figure 5 . 7 :

 57 Figure 5.7: (a) Distance R(t) (in units of ℓ -1 ) as a function of time (in units of ω -1 0 ). The inset shows a zoom of the area indicated by the rectangle. (b) Level spacing distribution W (∆ε) for the unfolded spectrum in a sector at fixed symmetry. The orange and the blue curves show respectively the Wigner-Dyson W W D (∆ε) and the Brody distribution W B (∆ε) with β = 0.22. The inset shows the level-spacing distribution of the whole unfolded spectrum and the Poisson distribution W P (∆ε) (green line). In all the panels, N = 14.

5. 5 .

 5 Long times level of chaoticity encoded in the spectrum, through the Brody distribution[START_REF] Brody | Random-matrix physics: spectrum and strength fluctuations[END_REF]:W B (∆ε) = (β + 1)b∆ε β e -b∆ε β +1 , (5.15) where b = {Γ[(β + 2)/(β + 1)]} β +1 and Γ is the Euler Gamma function. The Brody distribution reduces to the Poisson or Wigner-Dyson ones for β = 0 or 1 respectively.

Figure 5 . 8 :

 58 Figure 5.8: From left to right respectively: (a) inverse partecipation ratio I r (b) overlap between the probability distributions of neighboring eigenstates O and (c) Shannon entropy S H as functions of the energy, in units of hω 0 , for a system of N = 14 particles. The green lines indicate the corresponding value predicted by Random Matrix Theory.

Contents 6 . 1

 61 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Bethe Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Symmetry-breaking in the strongly repulsive regime model . . . . . 96 6.3.1 Short-distance correlations: the Tan's contact . . . . . . . . . . 97 6.4 Persistent current of a Bose-Bose mixture . . . . . . . . . . . . . . . 99

Figure 6 . 1 :

 61 Figure 6.1: The ratio C SB /C T G as a function of N = N ↑ + N ↓ for balanced mixtures (the line is a guide to the eye). In the inset we show n(k)k 4 , in units of C N = N 2 (N 2 -1)/L 3 , as a function of kL/(2π) for the case of a TG gas (stars) and a SB one (triangles) of N = 4 + 4 bosons. The horizontal lines indicates the values of C T G /C N (continuous line) and C SB /C N (dashed line).

Chapter 6 . 6 . 2 :

 662 Strongly-interacting multi-component bosons On the left, energy levels in units of h2 mL 2 as a function of Ω for c B → ∞.

Table of contents

 of C Appendix C : Symmetry analysis of multi-component mixtures: N=4 XXVIII D Appendix D : Mapping to the spin Hamiltonian: Bose-Bose and Fermi-Fermi mixtures XXX

et dynamique des gaz atom- iques ultrafroids de Fermi Coherence and dynamics of ultracold atomic Fermi gases Résumé Dans

  x N ) = ∑ P(1) ≤ x P(2) ≤ ...x P(N) ) a P Ψ A (x 1 ...x N ), (D.2) where θ (X) is the characteristic function of the ordering X and Ψ A is a fully Slater determinant of eigenfunctions of H 0 . On first order in 1/g, we have to solve the following eigenvalue problem:To solve the second part of the Schroedinger equation, we have to write the matrix elements of 1 g lim g→∞ g 2 dH SU dg on the basis of eigenvectors of H: ..dx N δ (x ix j )θ (x 1 ... ≤ x N )θ (x P(1) ...≤ x P(N) ) a * ⊮ a P |Ψ A | 2 (D.6)Remarkably, we decomposed the permutations of the N elements starting from an identity sector, which corresponds to the ordering x 1 ≤ x 2 ≤ ...x N without loss of generality. The integral is non zero only on the intersection between two coordinate sector θ (x 1 ... ≤ x N ) and θ (x P(1) ...≤ x P(N) ). Such intersection is x 1 ≤ ... ≤ x 2 ≤ x N ∩ x P(1) ≤ ... ≤ x P(2) ≤ x P(N)is non-zero only when P = I and when P is the trasposition (i, i + 1), ∀i = 1...N. As a consequence, the previous equation reads:g 2 dx 1 ..dx N δ (x ix i+1 )θ (x 1 ... ≤ x N ) a * I a P |Ψ A | 2 ≡ (D.7) P,I + δ P,(i,i+1) )J i (D.8)and consequently, the operatorial form of the Hamiltonian is:P i,i+1 ,(D.9)where P i,i+1 is the trasposition operator for two-component systems and the choice of the sign depends on the statistics of the state Ψ under the permutation (i, i + 1) (+ for fermions,for bosons in this case). cette thèse, nous étudions la cohérence et la dynamique des gaz quantiques atomiques ultra-froids, avec un regard particulier sur les gaz de Fermi. Pour ce qui concerne la cohérence, nous étudions un gaz de Fermi confiné sur un anneau en presence de un champ de jauge artificiel. Nous montrons tout d'abord que le courant persistant qui circule dans l'anneau peut être utilisé comme sonde pour les différents régimes du BCS-BEC crossover. Nous proposons un protocole de lecture de nos résultats, basé sur la co-expansion du gaz piégé sur l'anneau et d'un gaz de référence placé au centre de l'anneau lui-même. Ensuite, nous montrons qu'en utilisant le même protocole d'expansion, nous pouvons extraire des informations sur la phase du gaz quantique. Aux temps intermédiaires, le nuage en expansion fournit des informations sur la cohérence de premier ordre du gaz de Fermi. Dans les systèmes à plusieurs corps, nous pouvons également accéder au nombre de particules dans le système. Aux temps longs, nous extrayons des informations sur la cohérence à plusieurs corps du gaz. Par la suite, nous nous concentrons sur la dynamique d'un gaz de Fermi à deux composants en forte interaction, piégé dans un potentiel harmonique. Nous considérons un état séparé en spin, puis nous suivons la dynamique jusqu'aux temps longs. Aux temps courts, la dynamique est superdiffusive avec des exposants dynamiques de la classe d'universalité de Kardar-Parisi-Zhang. Aux temps intermédiaires, la dynamique est dominée par des oscillations de spin amorties dévoilant un effet de traînée de spin. Aux temps longs, nous observons une thermalisation. Finalement, nous étudions un gaz de Bose multicomposant dans le régime de forte interaction. Nous fournissons d'abord la solution exacte de la fonction d'onde et nous discutons de sa symétrie. Ensuite, nous étudions la réponse du gaz à un champ de jauge artificiel, avec une attention particulière à la période du courant persistant en fonction du flux de jauge.

	⟨Ψ|	1 g	lim g→∞	g 2 dH SU dg		|Ψ⟩ =		(D.5)
	1 g dx 1 ∑ lim g→∞ g 2 ∑ i< j ∑ P
						-	1 g	lim g→∞	g 2 dH SU dg	= -∑
			lim g→∞	H SU Ψ +	1 g	lim g→∞	dH SU d(1/g)	Ψ = H SU Ψ +	1 g	lim g→∞	dH SU d(1/g)	Ψ =	(D.3)
			E F Ψ -	1 g	lim g→∞	g 2 dH SU dg	Ψ	(D.4)
												XXX

P Θ(x i (δ P,I + δ P,(i,i+1) ) 1 g lim g→∞ ≡ ∑ i (δ i J i I ± ∑ i J i XXXI Cohérence

3.6 Conclusions to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 56

∆ε2 . We interpolate between the two regimes, thus quantifying the

6.5 Conclusions to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 101

Dynamics

In this chapter we follow the fermion dynamics starting from the initially strongly outof-equilibrium state |χ(t = 0)⟩ = |↑↑↑ ... ↓↓↓⟩, as in Ref. [START_REF] Sommer | Universal spin transport in a strongly interacting Fermi gas[END_REF], where the spins up and down are separated in the two opposite sides of the trap.

Since the harmonic trap is unchanged, the spatial part of the wavefunction (2.40) is constant during the motion, hence J i are constant in time. The time evolution involves only the spin degrees of freedom and can be obtained using the effective spin chain Hamiltonian (2.38). Recalling that the spin operators are related to the permutation operator by the relation P k,k+1 = 1 2 (I + σ k • σ k+1 ), Hamiltonian (2.38) can be written in terms of spin operators as H H = ∑ N-1 j=1 J j σ j • σ j+1 , but in particle space, ie each lattice site is associated to a particle index. The equation of motion for the spin operator S j = 1 2 (σ x j , σ y j , σ z j ) for the j-th particle reads dS

where µ = x, y, z and τ j = J j-1 σ j-1 + J j σ j+1 is the torque acting on a fixed particle due to the coupling with the neighbouring ones. We recall that the Hamiltonian H s has been defined in Eq. (2.38). In the following, we derive Eq. ( 5.2) and we show that, in our system, the spin dynamics is entirely due to spin torque.

Continuity equation

We demonstrate here Eq. (5.2). Consider indeed

Decomposing the commutator and using the relation

from which we readily obtain Eq. (5.2). In order to have a better understanding of the dynamics, we write the three components of the last equation on the snippet basis, which is our computational basis. To do so, we recall the action of the Pauli matrices on the single-spin Hilbert space. From their explicit expression one sees that both σ x j and σ y j induce a spin flip on the j-th site, the latter also imprinting a spin-dependent phase. On the other hand, σ z j doesn't invert the spin on the site j and its action it's equivalent to the identity if the j-th spin is up and induces a phase shift of π if this spin is down. Consequently, as the components x and y of Eq. (5.4) involve respectively products of σ y j σ z j+1 and σ x j σ z j+1 , they modify the total spin of the state.

From the above considerations and since we are working at fixed total spin, we can assess that the expectation value of dt are vanishing on the basis we are considering. This is not the case for dS z j dt , whose expectation value provides access to the expression of the spin current in terms of the permutation operator:

Chapter 5. Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas

A

Appendix A : Spiral interferograms for coherent states

In the following, we provide the analytical derivation for the interference of two condensates freely expanding in real space. We consider one condensate trapped on a ring-shaped optical lattice of N s sites and a second one placed in a blob in the center of the ring. We show that the interference pattern between the center and each site of the ring results in linear fringes. Moreover, we consider a site-dependent phase imprinted on the condensate trapped in the ring. This shifts the maxima of the linear fringes: the total interference pattern, arising from the combination of the single linear fringes, is a spiral whose number of branches is proportional to the imprinted phase. This calculation underlies Fig. 1.7.

As we mentioned in Section 3.4, the exact solution for a condensate expanding from a harmonic trap in the Gaussian approximation is given by Eq. (3.5). Using the same notation of Eq. (3.5), we set b(t) ≡ σ 1 + ω 0 t 2 . In the following we also neglect the normalization amplitude, to lighten the notation. We indicate as Ψ C (x, y) and Ψ R j (x, y) the wavefunction of the condensates respectively in the center and of the j-th site of the ring. Explicitly, they read:

where (x j , y j ) are the coordinates of the j-th site of the ring and n is the imprinted phase.

The interference between the two condensates is given by [START_REF] Jesacher | Spiral interferogram analysis[END_REF]:

and, if we neglect the two density-like terms, the interference pattern reads explicitly

We plot this last expression in the panels (a), (b) and (c) of Fig. 1.7 for j = 1, 2, 3 respectively. We also see that the imprinted phase depends on the site: it is proportional to the index of the site. To obtain XXIII Annexe A. Appendix A : Spiral interferograms for coherent states the spiral, we have to consider the global interference of the center with every site of the ring:

This function is plotted in the rightmost panel of Fig. 1.7 for a lattice of N s = 10 sites.

XXIV

B

Appendix B : Two-particle Bethe Ansatz solution

In this Appendix we show the explicit calculation for N = 2 fermions confined on a ring. We consider one spin up and one spin down particle, in order to have non-trivial interactions. The solution for an arbitrary number of particles is presented in Section 2.2.1. Consider the Gaudin-Yang Hamiltonian for N = 2 particles and M = 1. The spatial positions of the particles are x 1 and x 2 . From Eq. 2.6 one gets:

In this case, there are just two coordinate sector

In each coordinate sector, there are two possible permutations of the charge rapidities k 1 and k 2 : in the following we refer to them as I P and P.

We impose the following ansatz for the wavefunction:

where the Heaviside theta Θ(X) specifies the coordinate sector. First, we impose the continuity of the wavefunction on the boundary between the two coordinate sectors, ie

. This implies:

The cusp condition, encoding the contact interaction, reads:

XXV Annexe B. Appendix B : Two-particle Bethe Ansatz solution which explicitly implies:

Applying the continuity constraint we can simplify this last relation:

We see that when we permute the rapidities of the two particles, the wavefunction acquires a phase that depends on [Q,I P ] [I Q ,I P ] , which in Section 2.2.1 has been indicated as the operator permuting the position of two particles. The phase [Q,I P ] [I Q ,I P ] maps the coordinate sector I Q in Q. Explicitly, we can write the wavefunction as:

where the amplitude in the second line has been obtained by solving Eq. (B.6) for the coefficient [Q, P] as a function of [Q, I].

We now impose the periodic boundary conditions Ψ 2 (x 1 = 0, x 2 ) = Ψ 2 (x 1 = L, x 2 ), L being the circumference of the ring. Applying a straightforward substitution of the explicit expression for the wavefunction, one finds that periodic boundary conditions hold if:

Remarkably, if we consider the triplet state the wavefunction is antisymmetric under the exchange of the two particles. As a consequence, the contact interaction contribution is always null, since the wavefunction vanishes when x 2 = x 1 . The antisymmetry under particle exchange implies [Q,I P ] [I Q ,I P ] = -1 and Eq. (B.8) yields to the quantization of the momenta of the non interacting theory, ie e ik 1 L = e ik 2 L = 1. On the other hand, if we consider the singlet state, which is the ground state of the system according to the Lieb-Mattis [START_REF] Lieb | Theory of ferromagnetism and the ordering of electronic energy levels[END_REF] theorem, we have [Q,I P ] [I Q ,I P ] = 1. This latter property descends from the symmetry of the wavefunction under particle exchange in the singlet state. The periodic boundary conditions in this case yield:

XXVI

In order to fully specify the wavefunction at any given interaction c, one has to solve this system to compute the charge rapidities k j . As a final remark, we notice that in the case N = 2 there is no need to introduce the spin rapidities Λ j , even though we consider a multi-component system. This is due to the fact that in the two-particle sector, the contribution of the spin is trivial and the nested structure of the Bethe Ansatz described in Section 2.2.1 is not necessary to write the exact solution of the model.

C

Appendix C : Symmetry analysis of multi-component mixtures: N=4

In this appendix we study explicitly the symmetry of the ground state of a system of different multi-component mixtures in the strongly interacting regime. We start from the description of strongly interacting regimes of multi-component mixtures using spin chain that we introduced in Sec. 2.2.2. We used the formalism introduced in Section 1.2.3 to characterize the symmetry of a state in terms of Young tableaux. In particular, we justify the claim made in Section 6.3 about the symmetry-breaking case, where we stated that the ground state as a not well-defined permutation symmetry, as opposed to the ground states of Fermi-Fermi mixtures and the Tonks-Girardeau limit of the Bose-Bose mixtures.

Consider a system of N = 4 particles, with N ↑ = N ↓ = 2. Using the method described in [START_REF] Nataf | Exact diagonalization of heisenberg SU(n) models[END_REF], we can write a basis for the Hilbert space associating each vector to a standard Young tableau. Moreover, we can write the explicit expression of the permutation operator P i,i+1 for each i on this basis. In this case, identifying each vector with the corresponding tableau, the basis is composed by the 4! 2!2! = 6 following tableaux : The matrix associated to a Fermi-Fermi mixture is given by Eq. (2.38), where all the coupling J i ≡ J since we consider a homogeneous system. On the basis C.1, this matrix

Abstract

In this thesis, we study the coherence and the dynamics of ultracold atomic quantum gases, with a particular regard on Fermi gases. For what concerns the coherence, we study a Fermi gas confined on a ring threaded by an artificial gauge field. First, we show that the persistent current flowing in the ring can be used as probe for the different regimes of the BCS-BEC crossover. We propose a readout protocol for our results, based on the co-expansion of the gas trapped on the ring and a reference gas placed in the center of the ring itself. Next, we show that using the same expansion protocol we can extract information about the phase of the quantum gas. At intermediate times, the expanding cloud provides information about the first order coherence of the Fermi gas. In few-body systems, we can also access the number of particles in the system. At long times, we extract information about the many-body coherence coherence of the gas. Subsequently we focus on the dynamics of a strongly interacting two-component Fermi gas trapped in a harmonic potential. We consider a spin-separated state and then we follow the dynamics until long times. At short times, the dynamics is superdiffusive with dynamical exponents of the Kardar-Parisi-Zhang universality class. At intermediate times, the dynamics is dominated by damped spin oscillations unveleing a spin-drag effect. At long times, we observe thermalization. Ultimately, we study a multicomponent Bose gas in the strongly interacting regime. We first provide the exact solution for the wavefunction and we discuss its symmetry. Afterwards, we study the response of the gas to an artificial gauge field, with particular regard to the period of the persistent current as a function of the gauge flux.