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This thesis reports on the realization of the first experiments conducted with superfluid, strongly interacting Fermi gases of 6Li coupled to the light field of an optical cavity. In the scope of existing ultracold atomic platforms, this is the first time that a system with strong ground state fermionic correlations is operated in the framework of cavity quantum electrodynamics (cQED). From a condensed matter perspective, the system features a fully controllable microscopic Hamiltonian with control over both the strength of the ground state and light-matter interactions and the geometry of the latter. This contrasts with usual solid-state systems, in which the properties of the ground states are hardly tunable. As such our experiment is the perfect platform to simulate the physics of strongly correlated matter coupled to light fields. The manuscript is divided in three parts. The first part is dedicated to the presentation of technical details of the experiment and of measurement techniques we routinely employ to create and probe our strongly interacting gases coupled to light. We introduce the use of the cavity as a probing tool by presenting the achievement of strong light-matter coupling between the atomic ensemble and the cavity field. Similarly, we present a robust thermometry technique for the unitary Fermi gas, with which we measure temperature of the gases deep in the superfluid, quantum degenerate regime. In a second part we focus on the measurements of the strong atom-atom correlations which emerge from the energy spectrum of the atoms-cavity system. We start by laying down the theoretical basis needed for the understanding of the origin of the strong atom-atom interaction, and present its consequence on the many-body wavefunction of the gas by introducing the two-body contact as a universal thermodynamic quantity. We then report on the observation of strong light-matter coupling between pairs of atoms and the cavity field via photoassociation transitions. We describe the resulting light-matter coupling strength in terms of the two-body contact, imprinting many-body correlations onto cavity spectra for the first time. In the following experiment we study the optomechanical response of the gas in the dispersive regime. We observe distorted cavity transmission profiles, signatures of the nonlinear Kerr effect. The strength of the nonlinearity is governed by the density response of the gas, which we express via an operator product expansion in terms of the contact. In the last part, we investigate the effects of engineering long-range, photon-mediated interactions in the gas. We formally show how the system is expected, above a critical value for the strength of the long-range interaction, to undergo a phase transition to a density ordered state. The onset of density-ordering is observed by the superradiant properties of the ordered phase, and we show that it is also controlled by the density response of the unperturbed gas. In addition, we measure the divergence of the density wave susceptibility as the strength of the long-range interactions approaches the critical points: a striking feature of phase transitions. By measuring its temperature after the experiment, we prove that the gas remains superfluid.

This thesis presents the first cavity quantum electrodynamics experiments performed with a degenerate gas of $^6$Li with strong atom-atom interactions. The first part of this manuscript describes the design and the building of the apparatus that has been especially developed to bring together a high-finesse optical cavity and a strongly interacting Fermi gas. I described how the cavity and all the laser-cooling procedure can be realized in the same vacuum chamber, thus speeding up the production cycle of the degenerate Fermi gas. This new experimental apparatus is the first of its kind combining these two field of quantum physics. Placing a quantum gas of Fermions within an optical resonator gives a important technical advantages, allowing for the fast, all-optical production of a degenerate gas of $^6$Li. We apply an technique that make it possible to modify the longitudinal structure of the cavity trap to cancel its lattice structure. It increases the phase space density after the evaporative cooling leading to a ultracold gas at temperature lower than ten percent of the Fermi temperature. We describe how magnetic field allows us to tune the interatomic interactions, making use of the broad Feshbach resonance of $^6$Li at $832$ G and how we characterize the thermodynamic properties of the ultracold Fermi gas. The direct observation of phase separation for a spin-imbalanced Fermi gas between a fully paired region at the cloud center, surrounded by a spin-polarized shell experimentally proves the apparition of superfluidity at low enough temperature. The first experiment showing the strong coupling between the cavity photons and the strongly interacting Fermi gas is shown in this manuscript. The observation of large avoided-crossings when performing cavity transmission spectroscopy experiment are the experimental smoking gun of the strong light-matter coupling regime. We observe the expected scaling of the light-matter coupling strength with the number of atoms in the gas, proving the coherent coupling of the atoms with the cavity field. The thirs part of this manuscript presents the first cavity quantum electrodynamics experiment where a pairs of atoms couple to the cavity photons, forming a new dressed state: the pair-polariton. This dressed state inherits from its atomic part the characteristics of the many-body physics of the strongly interacting Fermi gas. We confirm experimentally that the properties of the short-range two-body correlation function, know as Tan's contact, can directly be measured optically, on the pair-polariton transmission spectrum. We observe the coherent coupling of the ground state Fermion pairs with the cavity photons and use the pair-polariton to perform single shot, real-time, weakly destructive measurement of the short range two body correlation function. This new measurement of Tan's contact allows to follow in-time the evolution of a single system, contrasting with existing techniques. The last part of this thesis will show experiment carried far in the dispersive regime, where both the cavity resonance and the probe laser frequency are far detuned from the atomic resonance. We will discuss how we can, in this regime, measure the atom number evolution in-time, with a weak destructivity. We show that optical non-linearity emerges and depends on the atom-atom interaction strength. Last we implement a pump not aligned with the cavity axis that allows to create long-range interactions between atoms.

The temperature dependence of Tan's contact parameter $C$ and its derivatives for spin gapped quantum magnets are investigated. We use the paradigm of Bose-Einstein condensation (BEC) to describe the low temperature properties of quasiparticles in the system known as triplons. Since the number of particles and the condensate fraction are not fixed we use the $\mu VT$ ensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree-Fock-Bogoliubov approximation level. We obtained the temperature dependence of $C$ and its derivative with respect to temperature and applied magnetic field both above and below $T_c$ of the phase transition from the normal phase to BEC. We have shown that $C$ is regular, while its derivatives are discontinuous at $T_c$ in accordance with Ehrenfest's classification of phase transitions. Moreover, we have found a sign change in $\partial C/\partial T$ close to the critical temperature. As to the quantum critical point, $C$ and its derivatives are regular as a function of the control parameter $r$, which induces the quantum phase transition. At very low temperatures, one may evaluate $C$ simply from the expression $C=m^2\mu^2/{\bar a}^{4}$, where the only parameter effective mass of quasiparticles should be estimated. We propose a method for measuring of Tan's contact for spin gapped dimerized magnets., Comment: 23 pages, 4 captioned figures

In this thesis, we investigate the properties of one-dimensional bosons in various types of systems, focusing on the phase transitions or crossovers between different quantum degeneracy regimes. Combining quantum Monte Carlo with other standard techniques such as exact diagonalization and thermal Bethe ansatz, we can compute the behavior of 1D bosons in different cases where the results are still lacking. First, in the case of harmonically trapped continuous bosons, we provide a full characterization of a quantity called Tan's contact. By computing the universal scaling function of it, we identify the behavior of the contact in various regimes of degeneracy for 1D bosons. We show that the contact exhibits a maximum versus temperature and that it is a signature of the crossover to fermionization in the strongly-interacting regime. Secondly, we study the localization and fractal properties of 1D ideal gases in shallow quasiperiodic potentials. The quasiperiodic system provides an appealing intermediate between long-range ordered and genuine disordered systems with unusual critical properties. While the tight-binding Aubry-Andr'e (AA) model has been widely studied, the shallow lattice case behaves differently. We determine the critical localization properties of the system, the critical potential, mobility edges and critical exponents which are universal. Moreover, we calculate the fractal dimension of the energy spectrum and find it is non-universal but always smaller than unity, which shows the spectrum is nowhere dense. Finally, we move to the study of the interacting case. With the quantum Monte Carlo calculations, we compute the phase diagram of Lieb-Liniger bosons in shallow quasiperiodic potentials. A Bose glass, surrounded by superfluid and Mott phases, is found. At finite temperature, we show that the melting of the Mott lobes is characteristic of a fractal structure and find that the Bose glass is robust against thermal fluctuations up to temperatures accessible in experiments., Dans cette thèse, nous étudions les propriétés des bosons unidimensionnels dans divers types de systèmes, en nous concentrant sur les transitions de phase ou les croisements entre différents régimes de dégénérescence quantique. En combinant la méthode de Monte Carlo quantique avec d'autres techniques standard telles que la diagonalisation exacte et l’ansatz de Bethe thermique, nous pouvons calculer le comportement des bosons à une dimension dans différents cas où les résultats font encore défaut. Tout d'abord, dans le cas de bosons continus piégés de manière harmonique, nous fournissons une caractérisation complète d'une quantité appelée contact de Tan. En calculant la fonction d'échelle universelle de cette quantité, nous identifions le comportement du contact dans différents régimes de dégénérescence pour les bosons 1D. Nous montrons que le contact présente un maximum en fonction de la température et qu’il s’agit d’une signature de la fermionisation du gaz dans le régime de forte interaction. Ensuite, nous étudions la localisation et les propriétés fractales des gaz idéaux 1D dans des potentiels quasi-périodiques peu profonds. Le système quasi-périodique constitue un intermédiaire intéressant entre les systèmes ordonnés à longue distance et les véritables systèmes désordonnés aux propriétés critiques inhabituelles. Alors que le modèle d'Aubry-André (AA) à liaison étroite a été largement étudié, le cas du réseau peu profond se comporte différemment. Nous déterminons les propriétés critiques de localisation du système, le potentiel critique, les bords de mobilité et les exposants critiques qui sont universels. De plus, nous calculons la dimension fractale du spectre d'énergie et nous constatons qu'elle est non universelle mais toujours inférieure à l'unité, ce qui montre que le spectre n'est dense nulle part. Enfin, nous passons à l'étude avec les interactions. Avec les calculs quantiques de Monte Carlo, nous calculons le diagramme de phase des bosons de Lieb-Liniger en potentiels quasi-périodiques peu profonds. On trouve un verre de Bose, entouré de phases superfluides et de Mott. À température finie, nous montrons que la fusion des lobes de Mott est caractéristique d'une structure fractale et constatons que le verre de Bose est robuste contre les fluctuations thermiques jusqu'à des températures accessibles dans les expériences.

The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k^−4 laws. The weights of these tails, denoted as Tan’s contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan’s contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattistheorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan’s contact as a function of the interaction, temperature and transverse confinement. These laws. Display displadisplay display interesting effects related to strong correlations and dimensionality.; L’objectif principal de cette thèse est l’étude théorique de mélanges quantiques fortement interagissants à une dimension et soumis à un potentiel externe harmonique. De tels systèmes fortement corrélés peuvent être réalisés et testés dans des expériences d’atomes ultrafroids. Leurs propriétés de symétrie par permutation non triviales sont étudiées, ainsi que leurs effets sur les corrélations. Exploitant une solution exacte pour des interactions fortes, nous extrayons des propriétés générales des corrélations encodées dans la matrice densité à un corps et dans les distributions des impulsions associées, dans les mélanges fermioniques et de Bose-Fermi. En particulier, nous obtenons des résultats substantiels sur le comportement à courtes distances, et donc les queues à haute impulsions, qui suivent des lois en k^−4 typiques. Les poids de ces queues, dénotés contacts de Tan, sont liés à de nombreuses propriétés thermodynamiques des systèmes telles que les corrélations à deux corps, la dérivée de l’énergie par rapport à la longueur de diffusion unidimensionnelle, ou le facteur de structure statique. Nous montrons que ces contacts universels de Tan permettent également de caractériser la symétrie spatiale des systèmes, et constituent donc une connexion profonde entre les corrélations et les symétries. En outre, la symétrie d’échange est extraite en utilisant une méthode de théorie des groupes, à savoir la méthode de la somme des classes (class-sum method en anglais), qui provient à l’origine de la physique nucléaire. De plus, nous montrons que ces systèmes suivent une version généralisée du fameux théorème de Lieb-Mattis. Souhaitant rendre nos résultats aussi pertinents expérimentalement que possible, nous dérivons des lois d’échelle pour le contact de Tan en fonction de l’interaction, de la température et du confinement transverse. Ces lois présentent des effets intéressants liés aux fortes corrélations et à la dimensionnalité.

The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k^−4 laws. The weights of these tails, denoted as Tan’s contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan’s contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattistheorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan’s contact as a function of the interaction, temperature and transverse confinement. These laws. Display displadisplay display interesting effects related to strong correlations and dimensionality.; L’objectif principal de cette thèse est l’étude théorique de mélanges quantiques fortement interagissants à une dimension et soumis à un potentiel externe harmonique. De tels systèmes fortement corrélés peuvent être réalisés et testés dans des expériences d’atomes ultrafroids. Leurs propriétés de symétrie par permutation non triviales sont étudiées, ainsi que leurs effets sur les corrélations. Exploitant une solution exacte pour des interactions fortes, nous extrayons des propriétés générales des corrélations encodées dans la matrice densité à un corps et dans les distributions des impulsions associées, dans les mélanges fermioniques et de Bose-Fermi. En particulier, nous obtenons des résultats substantiels sur le comportement à courtes distances, et donc les queues à haute impulsions, qui suivent des lois en k^−4 typiques. Les poids de ces queues, dénotés contacts de Tan, sont liés à de nombreuses propriétés thermodynamiques des systèmes telles que les corrélations à deux corps, la dérivée de l’énergie par rapport à la longueur de diffusion unidimensionnelle, ou le facteur de structure statique. Nous montrons que ces contacts universels de Tan permettent également de caractériser la symétrie spatiale des systèmes, et constituent donc une connexion profonde entre les corrélations et les symétries. En outre, la symétrie d’échange est extraite en utilisant une méthode de théorie des groupes, à savoir la méthode de la somme des classes (class-sum method en anglais), qui provient à l’origine de la physique nucléaire. De plus, nous montrons que ces systèmes suivent une version généralisée du fameux théorème de Lieb-Mattis. Souhaitant rendre nos résultats aussi pertinents expérimentalement que possible, nous dérivons des lois d’échelle pour le contact de Tan en fonction de l’interaction, de la température et du confinement transverse. Ces lois présentent des effets intéressants liés aux fortes corrélations et à la dimensionnalité.

Understanding the effect of interactions in quantum many-body systems presents some of the most compelling challenges in modern physics. Ultracold atoms have emerged as a versatile platform to engineer and investigate these strongly correlated systems. In this thesis, we study the properties of a mixture of Bose and Fermi superfluids with tunable interactions produced using ultracold vapors of ⁷Li and ⁶Li. We first study the hydrodynamic properties of the mixture by creating a counterflow between the superfluids. The relative motion only exhibit dissipation above a critical velocity that we measure in the BEC-BCS crossover. A numerical simulation of counterflowing condensates allows for a better understanding of the underlying mechanisms at play in the dynamics. In particular, this numerical study provides additional evidence that the onset of friction in our experiment is due to the simultaneous generation of elementary excitations in both superfluids. Finally, we consider the inelastic losses that occur via three-body recombination in our cold gases. This few-body process is intimately related to short-distance correlations and is thereby connected to the universal properties of the many-body system. This manifests as the apparition of an unusual dependence on density or temperature in the loss rate when increasing the interactions. We demonstrate this effect in two examples where interactions are resonant: the case of a dilute unitary Bose gas and the one of impurities weakly coupled to a unitary Fermi gas. More generally, our work shows that inelastic losses can be used to probe quantum correlations in a many-body system., La compréhension des effets des interactions dans un ensemble de particules quantiques représente un enjeu majeur de la physique moderne. Les atomes ultra-froids sont rapidement devenus un outil incomparable pour étudier ces systèmes quantiques fortement corrélés. Dans cette thèse, nous présentons plusieurs travaux portant sur les propriétés d’un mélange de superfluides de Bose et de Fermi créé à l’aide de vapeurs ultra-froides de ⁷Li et de ⁶Li. Nous étudions tout d'abord les propriétés hydrodynamiques du mélange en créant un contre-courant entre les superfluides. L'écoulement est dissipatif uniquement au dessus d'une vitesse critique que nous mesurons dans le crossover BEC-BCS. Une simulation numérique d’un contre-courant de deux condensats permet de mieux comprendre les mécanismes sous-jacents mis en jeu dans la dynamique. En particulier, l'étude numérique fournit des preuves supplémentaires que l'origine de la dissipation dans nos expériences est liée à l'émission d'excitation élémentaires dans chaque superfluide. Finalement, nous nous intéressons aux pertes inélastiques par recombinaison à trois corps qui peuvent limiter la stabilité de nos nuages. Ces pertes sont intimement liées aux corrélations à courte distance présentes dans le système et sont ainsi connectées aux propriétés universelles du gaz quantique. Cela se manifeste notamment par l’apparition de dépendances en densité ou en température inusuelles du taux de perte lorsque le système devient fortement corrélé. Nous démontrons cet effet dans deux exemples où les interactions sont résonantes, le cas du gaz de Bose unitaire et celui de notre mélange de superfluides Bose-Fermi. Plus généralement, nos travaux montrent que ces pertes inélastiques peuvent être utilisées pour sonder les corrélations quantiques dans un système en fortes interactions.