Disordered photonics is the study of light in random media. In a disordered photonic medium, multiple scattering of light and coherence, together with the fundamental principle of reciprocity, produce a wide range of interesting phenomena, such as enhanced backscattering and Anderson localization of light. They are also responsible for the existence of modes in these random systems. It is known that analogous processes to Bose-Einstein condensation can occur in classical wave systems, too. Classical condensation has been studied in several contexts in photonics: pulse formation in lasers, mode-locking theory and coherent emission of disordered lasers. All these systems have the common theme of possessing a large ensemble of waves or modes, together with nonlinearity, dispersion or gain. In this work, we study light condensation and its connection with light localization in a disordered, passive dielectric medium. We develop a theory for the modes inside the disordered resonator, which combines the Feshbach projection technique with spin-glass theory and statistical physics. In particular, starting from the Maxwell’s equations, we map the system to a spherical p-spin model with p = 2. The spins are replaced by modes and the temperature is related to the fluctuations in the environment. We study the equilibrium thermodynamics of the system in a general framework and show that two distinct phases exist: a paramagnetic phase, where all the modes are randomly oscillating and a condensed phase, where the energy condensates on a single mode. The thermodynamic quantities can be explicitly interpreted and can also be computed from the disorder-averaged time domain correlation function. We launch an ab initio simulation campaign using our own code and the Shaheen supercomputer to test the theoretical predictions. We construct photonic samples of varying disorder and find computationally relevant ways to obtain the thermodynamic quantities. We observe the phase transition and also link the condensation process to the localization. Our research could be a step towards the ultimate goal: to build a ”photonic mode condenser”, which transforms a broadband spectrum to a narrow one - ideally, to a single mode - with minimal energy loss, aided solely by disorder.

Dans un milieu désordonné, une onde peut être localisée exponentiellement par des effets d'interférence. Ce phénomène de localisation d'Anderson conduit notamment à une annulation de la conductance d'un fluide quantique unidimensionnel. Des travaux théoriques ont cependant montré que l'application d'un champ électrique pouvait réduire, voire supprimer, cette localisation. Nous étudions ici l'effet d'une force sur la localisation d'une onde quantique de matière dans un système unidimensionnel. En lien direct avec les expériences d'atomes ultrafroids, qui permettent d'observer la localisation d'Anderson d'un paquet d'onde en étalement, ou bien l'effet du désordre sur le transport entre deux réservoirs, nous nous intéressons à deux systèmes : la diffusion et la transmission d'une particule. Afin d'étudier la transmission à travers un guide, nous étendons un formalisme de matrices de transfert à la présence d'une force, éventuellement inhomogène. Deux approches analytiques complémentaires nous permettent d'étendre les résultats au cas d'un désordre de tavelures tel que celui utilisé dans les expériences d'atomes ultrafroids. Nous montrons que la force peut être entièrement prise en compte à l'aide d'une renormalisation de la longueur du guide par un libre parcours moyen local de la particule. Pour un désordre blanc, la force conduit alors une localisation plus faible, algébrique, tandis qu'une délocalisation apparaît pour un désordre corrélé. Nous nous intéressons ensuite à la diffusion d'une particule, à l'aide d'une approche numérique. Nous mettons en évidence une délocalisation de la position à grande force sous la forme d'une croissance temporelle algébrique, dont l'exposant augmente avec la force. Nous montrons de plus que la localisation est systématiquement détruite dans un désordre corrélé. / A wave can be exponentially localized in a disordered medium, due to interference effects. This Anderson localization phenomenon leads to a cancellation of the conductance of a quantum fluid in 1D. However, theoretical works pointed out that an electric field may reduce or cancel this localization. We study here the effect of a force on the localization of a 1D quantum matter wave. Since both Anderson localization of an expanding wave packet and the effect of disorder on the transport between two reservoirs have been studied in ultracold atom experiments, we focus on two systems, namely the diffusion, or the transmission, of a particle.In order to calculate the transmission, we generalize a transfer matrix formalism to the presence of a, possibly inhomogeneous, force. The case of a speckle disorder as used in ultracold atom experiments is dealt with using two other analytical approaches. Our main is result is that the force can be entirely taken into account by renormalising the length with a local mean free path of the particle. For white-noise disorder, the force leads to a weaker, algebraic localization, whereas full delocalization appears for a correlated disorder. We then focus on the diffusion of a particle, using a numerical approach. A transition of delocalization of the particle for strong forces is shed into light through a power law increase of its position, whose exponent increases with the force. Moreover, we show that localization is systematically destroyed in a correlated disorder.

Ce manuscrit présente une étude expérimentale d'un gaz de 87Rb ultra-froid confiné à deux dimensions et en présence de désordre. Dans une première partie, nous mettons en place les outils expérimentaux développés pour manipuler les gaz confinés. Après un état de l'art sur l'apport de la communauté des atomes froids aux gaz de Bose 2D, nous détaillons notre expérience, en l'absence de désordre, qui par une comparaison fine avec des simulations Monte-Carlo quantique et grâce à une thermométrie en temps de vol très précise, a permis de quantifier l'apparition de la cohérence autour la transition de phase superfluide Berzinskii-Kosterlitz-Thouless (BKT). La seconde partie est dédiée à l'effet d'un potentiel désordonné généré optiquement et corrélé microscopiquement sur les propriétés de transport et de cohérence du gaz 2D en interaction. Cette partie suit la progression de l'expérience du régime de transport classique, dans lequel nous avons mesuré la dépendance du coefficient de diffusion classique en fonction de l'énergie de la particule, jusqu'au transport quantique, que nous avons atteint grâce à une ultime méthode de ralentissement. Sur la route entre ces deux régimes, nous observons un décalage vers les faibles entropies de l'établissement de la cohérence autour de la transition BKT provoqué par l'ajout adiabatique d'une quantité modérée de désordre ainsi que sa suppression pour un désordre de l'ordre de la température du nuage. Ce travail est une étape vers une étude expérimentale de la transition quantique vers le verre de Bose mettant en jeu à la fois désordre et interactions. / This manuscript shows an experimental study of a disordered ultra-cold 87Rb gas confined in two dimensions.In the first part, we set up the experimental tools used to manipulate the confined gases. After a state of the art in ultracold 2D Bose gases, we describe our disorder-free experiment from which, by a comparison with quantum Monte-Carlo calculations and thanks to an accurate time-of-flight thermometry, we are able to quantify the emergence of coherence around the Berezinskii-Kosterlitz-Thouless superfluid phase transition. The second part is devoted to the effect of a micrometer-range optically generated disordered potential on the transport and coherence properties of the interacting 2D gas. This part follows the experimental road from the regime of classical transport, in which we have measured the energy dependence of the classical diffusion coefficient, to the regime of quantum transport we have just reached thanks to a last cooling step. On the road between those two regimes, we observe a shift towards lower entropy of the emergence of coherence close to the BKT transition as a response of an adiabatic addition of a moderate amount of disorder. It is strongly suppressed for an amount of disorder of the order of the cloud temperature. This work is a first step to an experimental study of the quantum transition to the Bose-Glass phase involving disorder and interaction.

Many problems in applied mathematics and physics are formulated most naturally in terms of matrices, and can be solved by computing functions of these matrices. For example, in quantum mechanics, the coherent dynamics of physical systems is described by the matrix exponential of their Hamiltonian. In state of the art experiments, one can now observe such unitary evolution of many-body systems, which is of fundamental interest in the study of many-body quantum phenomena. On the other hand the theoretical simulation of such non-equilibrium many-body dynamics is very challenging. In this thesis, we develop a symbolic approach to matrix functions and quantum dynamics based on a novel algebraic structure we identify for sets of walks on graphs. We begin by establishing the graph theoretic equivalent to the fundamental theorem of arithmetic: all the walks on any finite digraph uniquely factorise into products of prime elements. These are the simple paths and simple cycles, walks forbidden from visiting any vertex more than once. We give an algorithm that efficiently factorises individual walks and obtain a recursive formula to factorise sets of walks. This yields a universal continued fraction representation for the formal series of all walks on digraphs. It only involves simple paths and simple cycles and is thus called a path-sum. In the second part, we recast matrix functions into path-sums. We present explicit results for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We introduce generalised matrix powers which extend desirable properties of the Drazin inverse to all powers of a matrix. In the third part, we derive an intermediary form of path-sum, called walk-sum, relying solely on physical considerations. Walk-sum describes the dynamics of a quantum system as resulting from the coherent superposition of its histories, a discrete analogue to the Feynman path-integrals. Using walk-sum we simulate the dynamics of quantum random walks and of Rydberg-excited Mott insulators. Using path-sum, we demonstrate many-body Anderson localisation in an interacting disordered spin system. We give two observable signatures of this phenomenon: localisation of the system magnetisation and of the linear magnetic response function. Lastly we return to the study of sets of walks. We show that one can construct as many representations of series of walks as there are ways to define a walk product such that the factorisation of a walk always exist and is unique. Illustrating this result we briefly present three further methods to evaluate functions of matrices. Regardless of the method used, we show that graphs are uniquely characterised, up to an isomorphism, by the prime walks they sustain.

Cette thèse est consacrée à l’étude mathématique de divers systèmes de particules classiques et quantiques, en milieu désordonné. Elle comprend quatre travaux publiés ou soumis. Dans le premier nous fournissons une nouvelle formule permettant de prouver la localisation d’Anderson en une dimension d’espace et de caractériser la décroissance des fonctions propres à l’infini. Le second contient l’une des premières preuves de la localisation pour une infinité de particules en intéraction, dans l’approximation d’Hartree-Fock. Le troisième est dédié au modèle d’Anderson soumis à une perturbation périodique en temps. Sous certaines conditions sur la fréquence d’oscillation nous prouvons l’absence de diffusion. Dans le dernier travail nous montrons la décroissancedes corrélations pour le modèle du Jellium en une dimension dans un fond inhomogène, en utilisant la distance de Hilbert sur les cônes et le théorème de Birkhoff-Hopf. / This thesis is devoted to the mathematical study of some systems of classical and quantum particles, in a disordered medium. It comprises four published or submitted works. In the first one we provide a new formula allowing to prove Anderson localisation in one space dimension and to characterise the decay at infinity of the eigenfunctions. The second contains one of the first proofs of localisation for infinitely many particles in interaction, in the Hartree-Fock approximation. The third work is dedicated to the Anderson model in a time-periodic perturbation. Under certain conditions on the oscillation frequency we prove the absence of diffusion. In the last work we show the decay of correlations for the one-dimensional Jellium model in an inhomogeneous background, using the Hilbert distance on cones and the Birkhoff-Hopf theorem