This thesis is devoted to the study of excitations in atomic and polariton Bose-Einstein condensates (BEC). These two specimens are prime examples for equilibrium and non equilibrium BEC. The corresponding condensate wave function of each system satisfies a particular partial differential equation (PDE). These PDEs are discussed in the beginning of this thesis and justified in the context of the quantum many-body problem. For high occupation numbers and when neglecting quantum fluctuations the quantum field operator simplifies to a semiclassical wave. It turns out that the interparticle interactions can be simplified to a single parameter, the scattering length, which gives rise to an effective potential and introduces a nonlinearity to the PDE. In both cases, i.e. equilibrium and non equilibrium, the main model corresponding to the semiclassical wave is the Gross-Pitaevskii equation (GPE), which includes certain mathematical adaptions depending on the physical context of the consideration and the nature of particles/quasiparticles, such as additional complex pumping and growth terms or terms due to motion. In the course of this work I apply a variety of state-of-the-art analytical and numerical tools to gain information about these semiclassical waves. The analytical tools allow e.g. to determine the position of the maximum density of the condensate wave function or to find the critical velocities at which excitations are expected to be generated within the condensate. In addition to analytical considerations I approximate the GPE numerically. This allows to gain the condensate wave function explicitly and is often a convenient tool to study the emergence of excitations in BEC. It is in particular shown that the form of the possible excitations significantly depends on the dimensionality of the considered system. The generated excitations within the BEC include quantum vortices, quantum vortex rings or solitons. In addition multicomponent systems are considered, which enable more complex dynamical scenarios. Under certain conditions imposed on the condensate one obtains dark-bright soliton trains within the condensate wave function. This is shown numerically and analytical expressions are found as well. In the end of this thesis I present results as part of an collaborative effort with a group of experimenters. Here it is shown that the wave function due to a complex GPE fits well with experiments made on polariton condensates, statically and dynamically.

The density amplitude of an isolated quantum vortex line in superfluid 4He is calculated using a generalized Gross-Pitaevskii (G-P) equation. The generalized G-P equation for the order parameter extends the usual mean-field approach by replacing the interatomic potential in the ordinary G-P equation by a local, static T matrix, which takes correlations between the particles into account. The T matrix is a sum of ladder diagrams appearing in a diagrammatic expansion of the mean field term in an exact equation for the order parameter. It is an effective interaction which is much softer than the realistic interatomic Morse dipole-dipole potential from which it is calculated. A numerical solution of the generalized G-P equation is required since it is a nonlinear integro-differential equation with infinite limits. For the energy denominator in the T matrix equation, a free-particle spectrum and the observed phonon-roton spectrum are each used. For the fraction of particles in the zero-momentum state (Bose-Einstein dondensate) which enters the equation, both a theoretical value of 0.1 and an experimental value of 0.024 are used. The chemical potential is adjusted so that the density as a function of distance from the vortex core approaches the bulk density asymptotically. Solutions of the generalized G-P equation are not very dependent on the choice of energy denominator or condensate fraction. The density profile is a monotonically increasing function of the distance from the vortex core. The core radius, defined to be the distance to half the bulk density, varies from 3.7 A to 4.7 A, which is over three times the experimental value of 1.14 A at absolute zero.

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<p>L^p is supposed to be L with a superscript lower case 'p.'</p> / <p>This thesis investigates a 1-dimensional Gross-Pitaevskii (GP) equation from the viewpoint of a system of Hamiltonian partial differential equations (PDEs). A theorem on Birkhoff normal forms is a particularly important goal of this study. The resulting system is a perturbed system of a completely resonant system, which we analyze, using several forms of perturbation theory.</p> <p>In chapter two, we study estimates 011 integrals of products of four Hermite functions, which represent coefficients of mode coupling, and play an important role in the proof of the Birkhoff normal form theorem. This is a basic problem, which has a close relationship with a problem of Besicovitch, namely the behavior of the L^p norms of L² -normalized Hermite functions.</p> <p>In chapter three we carefully reconsider the linear Schrodinger equation with a harmonic potential, and we introduce a family of Hilbert spaces for studying the GP equation, which generalize the traditional energy spaces in which one works. One unexpected fact is that these function spaces have a close relationship with the former works for the tempered distributions, in particular the N-representation theory due to B. Simon, and V. Bargmann's theory, which uncovers relationship between the tempered distributions and his function spaces through the so-called Segal-Bargmann transformation. In addition, our function spaces have a nice relationship with the Sobolev spaces. In this chapter, a few other questions regarding these function spaces are discussed.</p> <p>In chapter four the proof of the Birkhoff normal form theorem on spaces we have introduced are provided. The analysis is divided into two cases according to the regularity of the related function space. After proving the Birkhoff normal form theorem, we made an analysis of the impact of the perturbation on the main part of the GP system, which we remark is completel:y resonant.</p> / Doctor of Philosophy (PhD)

We consider a gas of indistinguishable bosons, confined to a ring of radius R, and interacting via a pair-wise cosine potential. This may be thought of as the quantized Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis as a generalization of Antoni and Ruffo’s classical model. This thesis contains three parts: In part one, the dynamics of a Bose-condensate are considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum effects due to the quantum pressure are found to substantially alter the system’s dynamics, and can serve to inhibit a pathological instability for repulsive interactions. The non-commutativity of the large-N , long-time, and classical limits is discussed. In part two, we consider the GGPE studied above and seek static solutions. Exact solutions are identified by solving a non-linear eigenvalue problem which is closely related to the Mathieu equation. Stationary solutions are identified as solitary waves (or solitons) due to their small spatial extent and the system’s underlying Galilean invariance. Asymptotic series are developed to give an analytic solution to the non- linear eigenvalue problem, and these are then used to study the stability of the solitary wave mentioned above. In part three, the exact solutions outlined above are used to study quantum fluctuations of gapless excitations in the HMF model’s symmetry broken phase. It is found that this phase is destroyed at zero temperature by large quantum fluctuations. This demonstrates that mean-field theory is not exact, and can in fact be qualitatively wrong, for long-range interacting quantum systems, in contrast to conventional wisdom. / Thesis / Doctor of Philosophy (PhD) / The Hamiltonian Mean Field (HMF) model was initially proposed as a simplified description of self-gravitating systems. Its simplicity shortens calculations and makes the underlying physics more transparent. This has made the HMF model a key tool in the study of systems with long-range interactions. In this thesis we study a quantum extension of the HMF model. The goal is to understand how quantum effects can modify the behaviour of a system with long-range interactions. We focus on how the model relaxes to equilibrium, the existence of special “solitary waves”, and whether quantum fluctuations can prevent a second order (quantum) phase transition from occurring at zero temperature.

Dans cette thèse, nous étudions différents aspects mathématiques et numériques des équations de Gross-Pitaevskii et de Schrödinger non linéaire. Nous commençons (chapitre 1) par introduire différents modèles à partir des systèmes physiques que sont les condensats de Bose-Einstein et les impulsions lumineuses dans les fibres optiques. Cette modélisation conduit aux équations aux dérivées partielles stochastiques suivantes : l'équation de Gross-Pitaevskii stochastique et l'équation de Schrödinger non linéaire avec dispersion aléatoire. Ensuite, dans le second chapitre, nous nous intéressons au problème de l'existence et l'unicité d'une solution de ces équations. On montre notamment que le problème de Cauchy a une solution pour l'équation de Gross-Pitaevskii stochastique avec rotation grâce à la construction de la solution associée au problème. Nous abordons ensuite dans le troisième chapitre le problème du calcul des états stationnaires pour l'équation de Gross-Pitaevskii. Nous développons une méthode pseudo-spectrale de discrétisation du Continuous Normalized Gradient Flow, associée à une résolution itérative préconditionnée des sous-espaces de Krylov. Le quatrième chapitre concerne l'étude de schémas pseudo-spectraux pour la dynamique de l'équation de Gross-Pitaevskii et de Schrödinger non linéaire. On procède à une étude numérique de ces schémas (schéma de splitting de Lie et de Strang, ainsi qu'un schéma de relaxation). De plus, on analyse le schéma de Lie dans le cadre de l'équation de Schrödinger non linéaire avec dispersion aléatoire. Finalement, nous présentons, dans le cinquième chapitre, une boîte à outils Matlab (GPELab) développée dans le but de fournir les méthodes numériques que nous avons étudiées / The aim of this Thesis is to study various mathematical and numerical aspects related to the Gross-Pitaevskii and nonlinear Schrödinger equations. We begin (chapter 1) by introducing a few models starting from the physics of Bose-Einstein condensates and optical fibers. This naturally leads to introducing a stochastic Gross-Pitaevskii equation and a nonlinear Schrödinger equation with random dispersion. Next, in the second chapter, we analyze the existence and uniqueness problem for these two equations. We prove that the Cauchy problem admits a solution for the stochastic Gross-Pitaevskii equation with a rotational term by constructing the solution associated with the linear. The third chapter is concerned with the computation of stationary states for the Gross-Pitaevskii equation. We develop a pseudo-spectral approximation scheme for the Continuous Normalized Gradient Flow formulation, combined with preconditioned Krylov subspace methods. This original approach leads to the robust and efficient computation of ground states for fast rotations and strong nonlinearities. In the fourth chapter, we consider some pseudo-spectral schemes for computing the dynamics of the Gross-Pitaevskii and nonlinear Schrödinger equations. These schemes (the Lie's and Strang's splitting schemes and the relaxation scheme) are numerically studied. Moreover, we proceed to a rigorous numerical analysis of the Lie scheme for the associated stochastic PDEs. Finally, we present in the fifth chapter a Matlab toolbox (called GPELab) that provides computational solutions based on the schemes previously introduced in the Thesis

Cette thèse est dédiée à l'étude des fluctuations non-linéaires dans les condensats de Bose-Einstein à deux composantes. On présente dans le premier chapitre la dynamique de champ moyen des condensats à deux composantes et les différents phénomènes typiques associés au degré de liberté spinoriel. Dans ce même chapitre, on montre que la dynamique des excitations se sépare en deux modes distincts : un mode dit de densité correspondant au mouvement global des atomes à l'intérieur du condensat et un mode dit de polarisation correspondant à la dynamique relative entre les deux espèces constituant le condensat. Ce calcul est généralisé dans le deuxième chapitre où l'on montre que le mode de polarisation persiste en présence d'un couplage cohérent entre les deux composantes. En particulier on analyse la stabilité modulationnelle du mode en déterminant, à l'aide d'une analyse multi-échelle, la dynamique des excitations non-linéaires. On montre alors que les excitations de polarisation, au contraire des excitations de densité, souffrent d'une instabilité de Benjamin-Feir. Cette instabilité est stabilisée aux grandes impulsions par une résonance onde longue - onde courte. Enfin dans le dernier chapitre, on dérive de façon non-perturbative la dynamique de polarisation proche de la limite de Manakov, dynamique quise révèle être régie par une équation de Landau-Lifshitz sans dissipation. Les équations de Landau-Lifshitz appartiennent à une hiérarchie d'équations intégrables (hiérarchie Ablowitz-Kaup-Newell-Segur) et on étudie les solutions à une phase à l'aide de la méthode d'intégration finite-gap ; on détermine notamment à l'aide de cette méthode un nouveau type de soliton pour les condensats à deux composantes. Finalement, profitant de l'intégrabilité du système, on résout le problème de Riemann à l'aide de la théorie de modulation de Whitham et on montre que les condensats à deux composantes peuvent propager des ondes de raréfaction ainsi que des ondes de choc dispersives ; on décrit notamment la modulation de ces ondes de choc par la propagation d'ondes simples et d'ondes de contact d'invariants de Riemann. / This thesis is devoted to the study of nonlinear fluctuations in two-component Bose-Einstein condensates. In the first chapter we derive the mean field dynamics of two-component condensates and we present the distinctive phenomena associated to the spinorial degree of freedom. In the same chapter, we show that the dynamics of the excitations is divided in two distinct modes: a so-called density mode which corresponds to the global motion of the atoms, and a so-called polarization mode which corresponds to the relative motion between the two species composing the condensate. The computation is generalized in the second chapter in which we demonstrate that the polarization mode remains in presence of a coherent coupling between the two components. In particular we study the modulational stability of the mode and we determine through a multi-scaling analysis the dynamics of non-linear excitations. We show that the excitations of polarization undergo a Benjamin-Feir instability contrary to the density excitations. This instability is then stabilized in the short wavelength regime by a long wave - short wave resonance. Finally in the last chapter, we derive in a non-perturbative way the polarisation dynamics close the Manakov limit.In this limit, the dynamics proves to be governed by a Landau-Lifshitz equation without dissipation. Landau-Lifshitz equations belong to a hierarchy of integrable equations (Ablowitz-Kaup-Newell-Segur hierarchy) and we derive the single-phase solutions thanks to the finite-gap method; in particular we identify a new type of soliton for the two-component Bose-Einstein condensates. Finally, taking advantage of the integrability of the system, we solve the Riemann problem thanks to the Whitham modulation theory and we show that the two-component condensates can propagate rarefaction waves as well as dispersive shockwaves; we describe the modulation of the shockwaves by the propagation of simple waves and contact waves of Riemann invariants.

<p>Ova doktorska teza se bavi proučavanjem i razvojem paralelnih<br />algoritama za transponovanje distribuiranih trodimenzionalnih<br />struktura podataka, kao i implementacijom ovih algoritama u okviru<br />C/OpenMP/MPI programske paradigme. Razvijena implementacija je<br />primenjena na rešavanje nelinearne parcijalne diferencijealne<br />jednačine Šredingerovog tipa (Gros-Pitaevski jednačina) korišćenjem<br />Krenk-Nikolson metoda, a u okviru teze je predstavljen ciklus razvoja<br />odgovarajućeg softvera, kao i rezultati testova validnosti i merenja<br />performansi dobijenih na računarskom klasteru.</p> / <p>This thesis studies and develops parallel algorithms for transposing<br />distributed three-dimensional data structures, and describes their technical<br />implementation in C/OpenMP/MPI programing paradigm. The developed<br />implementation is applied for solving of nonlinear partial differential equation<br />of the Schroedinger type (Gross-Pitaevskii equation) using Crank-Nicolson<br />method. The thesis presents the corresponding software development cycle,<br />as well as results of validity tests and performance measurements obtained<br />on a computer cluster.</p>

Dans cette thèse, nous étudions théoriquement plusieurs effets liés à la localisation d'Anderson, dans le contexte des atomes froids. Dans les systèmes d'atomes froids, le désordre est généralement créé à l'aide d'une figure de tavelure optique. Dans la première partie de la thèse, nous discutons des spécificités de ces potentiels optiques, et nous nous intéressons en particulier aux propriétés spectrales. Les expériences usant de l'interaction lumière-matière offrent d'intéressantes possibilités. Dans ce cadre, nous considérons dans une deuxième partie de la thèse l'étalement d'un paquet d'ondes atomique, initialement lancé avec une vitesse non nulle dans un potentiel désordonné. Nous trouvons qu'après un mouvement balistique, le centre de masse du paquet subit une rétro-réflection et retourne lentement à sa position initialle, se comportant comme un boomerang. Nous introduisons ensuite les interactions inter-atomiques dans une troisième partie. Nous considèrons des gaz dilués de bosons condensés, et traitons les interactions au niveau champ moyen. Plusieurs situations sont étudiées numériquement, en particulier le boomerang quantique, et l'étalement dynamique -- à la fois en impulsion et en énergie -- d'ondes de matière préparées en ondes planes. Dans la dernière partie de la thèse, nous montrons que des modèles chaotiques offrent des perspectives intéressantes pour l'étude de la localisation d'Anderson. D'une part, nous présentons des éléments probants en faveur d'un kick rotor sans spin dans l'ensemble symplectique. D'autre part, le réexamen de modèles communément étudiés de kick rotors quasi-périodiques révèle des résultats intrigants. / This thesis theoretically investigates several effects related to Anderson localization, focusing on the context of disordered and chaotic cold-atomic systems. In cold-atomic systems, optical speckle patterns are often used to create the disorder. The resulting potentials felt by the atoms differ from Gaussian random potentials, commonly assumed in the description of condensed-matter systems. In the first part of the thesis, we discuss their specificities, with an emphasis on the spectral properties. Atom-optics experiments offer interesting possibilities, such as the possibility to directly probe the atoms inside the disordered potential. In view of these possibilities, we consider in the second part of the thesis the spreading of matter wave packets initially launched with a non-zero velocity. We find that after an initial ballistic motion, the packet center-of-mass experiences a retroreflection and slowly returns to its initial position, mimicking a boomerang. Atom-atom interactions are then introduced in a third part. We consider dilute condensed bosonic gases, and treat the interactions at the mean-field (Gross-Pitaevskii) level. Various situations are studied numerically, in particular the quantum boomerang scenario, and the dynamical spreading both in momentum and energy of matter waves prepared as plane waves. In the last part, we show that chaotic models offer interesting prospects for the study of Anderson localization. On the one hand, we present strong evidences in favor of a spinless kicked rotor in the sympletic ensemble. On the other hand, a second look at a commonly studied quasi-periodically modulated kicked rotor reveals intriguing results.

Le travail réalisé dans le cadre de cette thèse s'articule en deux volets. Le premier porte sur l'étude de l'effet des interactions entre atomes au sein d'un interféromètre atomique, dont la source est un condensat de Bose-Eintein. Nous présentons un modèle analytiquepermettant d'obtenir des expressions simples pour le déphasage induit par celles-ci. Ce modèle est comparé à des simulations numériques résolvant les équations de Gross-Pitaevskii couplées, et présente un excellent accord. Le second concerne la conception et la construction d'un nouveau dispositif expérimental visant à obtenir un condensat de Bose-Einteindans le but de réaliser des mesures de haute précision par interférométrie atomique. / The work performed during this thesis comprises two orientations. The first one is the study of the effect of interactions between atoms in an atom interferometer which source of atoms is a Bose-Einstein condensate. We present an analytical model allowing to obtain simple expressions for the phase shift induced by them. This model is compared to numerical simulations solving the coupled Gross-Pitaevskii equations and presents a good agreement. The second one is the design and construction of a new experimental set-up for the production of a Bose-Einstein condensate to perform high precision measurements with the use of atom interferometry.