Diferentes tipos de sólitons têm sido observados em meios ópticos não-lineares, e seus comportamentos individuais descritos pela equação não-linear de Schrödinger e pela equação não-linear de Schrödinger generalizada, em diferentes dimensões e geometrias. Entretando, há situações onde muitos sólitons são gerados formando uma densa rede de sólitons. Nestes casos, é impossível desprezar as interações entre os sólitons e temos que considerar a evolução da estrutura como um todo. A teoria das ondas de choque dispersivas em meios fotorrefrativos e a teoria da difração não-linear de intensos feixes de luz propagando-se em meios fotorrefrativos com um fio refletor incorporado a esse meio foi desenvolvida, e verificamos que está em excelente acordo com nossas simulações numéricas. No caso da formação de sólitons em condensados, fizemos cálculos numéricos realísticos dentro da aproximação de campo médio usando a equação de Gross-Pitaevskii, incluindo também um potencial de confinamento, um potencial móvel e um potencial dipolar. A maioria dos resultados puderam ser comparados com experimentos recentes. / Different kinds of solitons have already been observed in various nonlinear optical media, and their behavior has been explained in the frameworks of such mathematical models as the nonlinear Schrödinger and generalized nonlinear Shrödinger equations for different dimensions and geometries. However, there are situations when many solitons are generated so that they can comprise a dense soliton train. In such situations, it is impossible to neglect interactions between solitons and one has to consider the evolution of the structure as a whole rather than to trace the evolution of each soliton separately. The theory of optical shock waves in photorefractive media and the theory of nonlinear diffraction of light beams propagating in photorefractive media with embedded reflecting wire was developed and agrees very well with our numerical simulations. In the condensate soliton formation case, we did numerical calculations in the mean field approach using the Gross-Pitaevskii equation, adding a trap potential and a moving potential and a potential of the dipole-dipole interaction. The main results were also checked by recent experiments.

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

Made available in DSpace on 2016-08-18T18:19:29Z (GMT). No. of bitstreams: 1 Kleber Anderson Teixeira da Silva.pdf: 275462 bytes, checksum: c0169a00ab68933c0fcb824d912b65e8 (MD5) Previous issue date: 2011-04-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Bose-Einstein condensate (BEC), also known as the fifth state of the matter, was predicted theoretically by Albert Einstein in 1925 and verified experimentally 70 years later in 1995. This dissertation addresses the effects of spontaneous broken of the CPT-symmetry and of the invariance Lorentz (also called simply the violation of Lorentz symmetry) in the Bose-Einstein condensation of an ideal bosonic gas in the limits nonrelativistic and will ultrarelativ´ıstico. The work is based on a model of fields theory described by a massive complex scalar field in the framework of spontaneous breaking of the CPT-symmetry and of the Lorentz invariance. First we study the CBE in the nonrelativistic limit starting from the non relativistic version of our model. Thus, the existence of the CBE imposes strong restrictions on some parameters governing the breach of Lorentz invariance (VIL). We observed that only the critical temperature is modified by the VIL. Also, we use experimental data to obtain limits upper for the coefficients that control the VIL. Because of our model describe consistently the CBE in the non relativistic limit We can use it to study the CBE ultrarelativıstica of an ideal bosonic gas loaded. Thus, we show that the construction of a partition function well defined to describe the relativistic ideal gas, imposes strong restrictions on two parameters that control the VIL. The analysis of the CBE in the ultrarelativistico limit shows that both the critical temperature as the chemical potential are affected by the spontaneous breaking of the invariance of Lorentz. / A condensação de Bose-Einstein (CBE), também conhecida como o quinto estado da matéria, foi prevista teoricamente por Albert Einstein em 1925 e verificado experimentalmente 70 anos depois, em 1995. Esta dissertação aborda os efeitos da quebra espontânea da simetria-CPT e da invariância de Lorentz (também chamada simplesmente de violação da simetria de Lorentz) na condensação de Bose-Einstein de um gás ideal bosônico nos limites não relativístico e ultrarelativístico. O trabalho é baseado em um modelo de teoria de campos descrito por um campo escalar complexo massivo no marco da quebra espontânea da simetria-CPT e da invariância de Lorentz. Primeiro estudamos a CBE no limite não relativístico partindo da versão não relativística do nosso modelo. Desse modo, a existência da CBE impõe restrições severas sobre alguns parâmetros que regem a violação da invariância de Lorentz (VIL). Observamos que somente a temperatura crítica é modificada pela VIL. Também, usamos dados experimentais para obter limites superiores para os coeficientes que controlam a VIL. Pelo fato do nosso modelo descrever de modo consistente a CBE no limite não relativíıstico podemos usá-lo para estudar a CBE ultrarelativística de um gás ideal bosônico carregado. Assim, mostramos que a construção de uma função de partição bem definida, para descrever o gás ideal relativístico, impõe restrições severas sobre dois parâmetros que controlam a VIL. A análise da CBE no limite ultrarelativístico mostra que tanto a temperatura crítica como o potencial químico são afetados pela quebra espontânea da invariância de Lorentz.

Nesta dissertação nós revisamos a teoria básica que descreve a junção Josephson bosônica para uma e duas espécies partindo do modelo de Bose-Hubbard. Em seguida explicamos como é possível gerar campos de calibe artificiais em um sistema de átomos neutros, como é o caso do condensado de Bose-Einstein. Finalmente, utilizando os conhecimentos teóricos desenvolvidos anteriormente nós buscamos os estados estacionários de um sistema de pseudospin 1/2 submetido a um campo de calibre não-Abeliano artificial, que torna a dinâmica da junção muito mais complexa e rica. Nós também exploramos um novo desbalanceamento de população que surge no sistema, devido a presença do campo de calibre, com características similares as do Macroscopic Quantum Self-Trapping. / In this dissertation we review the basic theory that describes the bosonic Josephson junction for one and two species using the Bose-Hubbard model. Afterwards, we explain how it is possible to generate artificial gauge fields for neutral atoms, like a Bose-Einstein condensate. Finally, using this theoretical background we search for stationary states of a pseudospin 1/2 system subject to a non-Abelian artificial gauge field which turns the dynamic of the junction much more complex and rich. We also explore a possible new populational imbalance that appears on the system due to the presence of the gauge field, with similar features as the Macroscopic Quantum Self-Trapping.

Neste trabalho descrevemos a produção e estudo de excitações topológicas em um condensado de Bose-Einstein em átomos de Rubídio-87. O condensado é produzido através de resfriamento evaporativo forçado por rádio-freqüência em uma armadilha puramente magnética do tipo QUIC. A armadilha magnética é carregada por um sistema de duplo-MOT. A temperatura de transição é de cerca de 150nK. Condensados puros com 1 - 2 × 10^5 átomos de Rb-87 são observados. Realizamos uma caracterização da amostra em relação às suas características fundamentais. Fração condensada, expansão anisotrópica, distribuição espacial e efeitos de temperatura finita são descritos. Com o objetivo de observar excitações coerentes do condensado entre os estados da armadilha, adicionamos um campo magnético do tipo quadrupolo esférico oscilante no tempo. Observamos, no entanto, a transferência de momento angular para a amostra com a formação de vórtices e arranjos de vórtices. Definimos regiões de amplitude que geram números de vórtices crescentes. Observamos a formação de estruturas de três vórtices não convencionais donde supusemos a possibilidade de excitação conjunta de vórtices e anti-vórtices. Observamos evidência de turbulência quântica, um estado onde os arranjos dos vórtices não são regulares nem as linhas de vórtices têm um eixo de rotação comum. / In this work we describe the production and investigation of topological excitations in a Bose-Einstein condensate in Rubidium-87 atoms. The condensate is produced through forced evaporative cooling by radio-frequency in a QUIC-type purely magnetic trap. The magnetic trap is loaded from a double-MOT system. Transition temperature is about 150nK. Pure condensates containing 1-2×105 87Rb atoms are observed. We performed the characterization of the sample in relation to its fundamental aspects. Condensed fraction, anisotropic expansion, spacial distribution and finite temperature effects are described. Aiming to observe coherent topological excitations of the condensate between two states of the trap, we added a spherical quadrupole magnetic fields oscillating in time. We observe, instead, angular momentum tranference to the sample and the formation of vortices and arrays of vortices. We define amplitude regions where an increasing number of vortices are observed. We observe the formation of non-usual three-vortex structures from which we infer the existence of vortices and anti-vortices together in the sample. We observe evidence of quantum turbulence, a state where non-regular vortex arrays appear as well as vortex lines have no preferred direction to form.

La dimensionnalité d’un système affecte fortement ses propriétés physiques ; les transitions de phasequi s’y déroulent ainsi que le type d’ordre qui y apparaît dépendent de la dimension. Dans les systèmesde basse dimension, la cohérence s’avère plus difficile à établir car les fluctuations thermiques etquantiques y jouent un rôle plus important. Le fluide de Bose à deux dimensions est particulièrementintéressant car, même si un ordre total est exclu, un ordre résiduel à « quasi-longue » portée s’établit àbasse température. Deux ingrédients ont un effet significatif sur l’état du système : (i) la taille finie d’unsystème réel permet de retrouver une occupation macroscopique d’un état à une particule ; (ii) les interactionsentre particules conduisent à l’apparition d’un type non-conventionnel de transition de phasevers un état superfluide.Dans cette thèse, nous présentons une étude expérimentale du gaz de Bose bidimensionnel (2D) utilisantdeux types de paysages énergétiques pour piéger nos atomes. Dans la première partie, nous utilisonsla dépendance spatiale de certaines propriétés locales d’un gaz inhomogène pour caractériser l’étatdu système homogène équivalent. Nous extrayons son équation d’état des profils de densité et noustestons son comportement superfluide en mesurant le chauffage induit par le mouvement d’une perturbationlocale. Dans la deuxième partie, nous observons et caractérisons l’émergence d’une cohérencede phase étendue dans un gaz 2D homogène, en particulier via le passage de trois dimensions à deux(croisement dimensionnel). Nous étudions l’établissement dynamique de la cohérence par un passagerapide du croisement dimensionnel et nous observons des défauts topologiques dans l’état superfluidefinal. Nous comparons nos résultats avec les prédictions du mécanisme de Kibble–Zurek. / The dimensionality of a system strongly affects its physical properties; the phase transitions that takeplace and the type of order that arises depend on the dimension. In low dimensional systems phasecoherence proves more difficult to achieve as both thermal and quantum fluctuations play a strongerrole. The two-dimensional Bose fluid is of particular interest as even if full order is precluded, a residual"quasi-long" range order arises at low temperatures. Then two ingredients have a significant effecton the state of the system: (i) the finite size of a real system enables one to recover of a macroscopicoccupation of a single-particle state; (ii) the interactions between particles lead to the emergence of anon-conventional type of phase transition toward a superfluid state.In this thesis, we present an experimental study of the two-dimensional (2D) Bose gas using two differentenergy landscapes to trap our atoms. In the first part, we use the spatial dependence of somelocal properties of an inhomogeneous gas to characterize the state of the equivalent homogeneous system.We extract its equation of state with a high accuracy from the gas density profiles and test itssuperfluid behavior by measuring the heating induced by a moving local perturbation. In the secondpart, we observe and characterize the emergence of an extended phase coherence in a 2D homogeneousgas in particular via a 3D-to-2D dimensional crossover. We investigate the dynamical establishment ofthe coherence via a rapid crossing of the dimensional crossover and observe topological defects in thefinal superfluid state. We compare our findings with the predictions for the Kibble–Zurek mechanism.

Ces travaux sont consacrés à l'étude d'un gaz d'excitons bidimensionnel créé dans un double puits quantique de semi-conducteurs. Grâce à un champ électrique appliqué perpendiculairement au plan des puits, les deux composants de l'exciton, l'électron et le trou, sont chacun confinés dans deux puits séparés. Ceci donne lieu à des excitons avec un fort moment dipolaire qui sont piégés et refroidis à très basse température. Dans ce manuscrit, nous étudions les propriétés du gaz d'excitons à travers la photoluminescence émise lors de la recombinaison des paires électron-trou, que nous analysons grâce à des mesures spectroscopiques et interférométriques. Ainsi, nous avons étudié la thermodynamique du gaz d'excitons dans un piège harmonique et établi son équation d'état. Nous avons dévoilé l'universalité de l'équation d'état ce qui a rendu possible la première démonstration de l'équilibre thermodynamique du gaz et a aussi permis d'évaluer la force des interactions entre excitons. En associant des mesures de cohérence spatiale et l'étude de la distribution de défauts de densité dans la photoluminescence des excitons, nous avons montré que dans une zone centrale du piège de rayon R < 3 µm, il s'opère conjointement une augmentation du niveau de cohérence et une diminution de la densité de défauts. Des travaux précédents ayant montré que ces défauts pouvaient constituer des défauts topologiques appelés vortex, nos observations constituent ainsi un ensemble de preuves de l'établissement d'un état superfluide dans la partie centrale du piège, en accord avec la théorie prédite par Berezinskii-Kosterlitz et Thouless. / This work is devoted to the study of a two dimensional exciton gas created in a semiconductordouble quantum wells. Thanks to an electric field applied perpendicular to these layers,the exciton constituents, namely, the electron and the hole, are confined each in one quantumwell so that excitons inherit a strong electric dipole. In this manuscript, we study the propertiesof such exciton gas trapped at very low temperature, through the photoluminescence emittedby electron-hole recombinations. We particularly study the emitted light using spectroscopic andinterferometric techniques. Thus, we explore the thermodynamics of a trapped gas and quantifyits equation of state. We unveil an important feature: the universality of the equation of state,which allows the first demonstration of thermodynamic equilibrium for excitons, and we realizea direct measurement of the excitons dipolar interaction strength. By probing both the spatialcoherence and the density of defects of the excitons photoluminescence, we show that it exists acentral region (R < 3 μm) in an exciton trap where a high degree of spatial coherence is conditionedby a decrease of the defects density. Our experiments thus provide a set of evidences pointingtowards a Berezinskii-Kosterlitz-Thouless transition for trapped 2D exciton gas.

Cette thèse porte sur l'étude de méthodes numériques pour l'analyse de deux modèles stochastiques apparaissant dans le contexte de la condensation de Bose-Einstein. Ceux-ci constituent deux généralisations de l'équation de Gross-Pitaevskii. Cette équation aux dérivées partielles déterministe modélise la dynamique de la fonction d'onde d'un condensat de Bose-Einstein piégé par un potentiel extérieur confinant.Le premier modèle étudié permet de modéliser les fluctuations de l'intensité du potentiel confinant et prend la forme d'une équation aux dérivées partielles stochastiques. Celles-ci conduisent en pratique à un échauffement du condensat, et parfois mêmeà son effondrement. Nous proposons dans un premier chapitre la construction d'un schéma numérique pour la résolution de ce modèle. Il est fondé sur une discrétisation spectrale en espace, et une discrétisation temporelle de type Crank-Nicolson. Nous démontrons que le schéma proposé converge fortement en probabilité à l'ordre au moins 1 en temps, et nous présentons des simulations numériques illustrant ce résultat. Le deuxième chapitre est consacré à l'étude théorique et numérique de la dynamique d'une solution stationnaire (pour l'équation déterministe) de type vortex. Nous étudions l'influence des perturbations aléatoires du potentiel sur la solution, et montrons que la solution perturbée garde les symétries de la solution stationnaire pour des temps au moins de l'ordre du carré de l'inverse de l'intensité des fluctuations. Ces résultats sont illustrés par des simulations numériques exploitant une méthode de Monte-Carlo adaptée à la simulation d'événements rares.Le deuxième modèle permet de modéliser les effets de la température sur la dynamique d'un condensat. Lorsque celle-ci n'est pas nulle, la condensation n'est pas complète et le condensat interagit avec les particules non condensées. Ces interactions sont d'un grand intérêt pour comprendre la dynamique de transition de phase et analyser les phénomènes de brisure de symétrie associés, comme la formation spontanée de vortex. Nous nous sommes intéressés dans les chapitres 3 et 4 à des questions relatives à la simulation de la distribution des solutions de cette équation en temps long. Le troisième chapitre est consacré à la construction d'une méthode d’échantillonnage sans biais pour des mesures connues à une constante multiplicative près. C'est une méthode de Monte Carlo par chaînes de Markov qui a la particularité de permettre un échantillonnage non-réversible basé sur une équation de type Langevin sur-amortie. Elle constitue une extension de Metropolis-Adjusted Langevin Algorithm (MALA). Le quatrième chapitre est quant à lui consacré à l'étude numérique de dynamiques métastables liées à la nucléation de vortex dans des condensats en rotation. Un intégrateur numérique pour la dynamique étudiée est proposé, ainsi qu'une méthode de Monte-Carlo adaptée à la simulation d'événements rares correspondant aux changements de configurations métastables. Cette dernière est basée sur l'algorithme Adaptive Multilevel Splitting (AMS). / This thesis is devoted to the numerical study of two stochastic models arising in Bose-Einstein condensation physics. They constitute two generalisations of the Gross-Pitaevskii Equation. This deterministic partial differential equation model the wave function dynamics of a Bose-Einstein condensate trapped in an external confining potential. The first chapter contains a simple presentation of the Bose-Einstein condensation phenomenon and of the experimental methods used to construct such systems.The first model considered enables to model the fluctuations of the confining potential intensity, and takes the form of a stochastic partial differential equation. In practice, these fluctuations lead to heating of the condensate and possibly to its collapse. In the second chapter we propose to build a numerical scheme to solve this model. It is based on a spectral space discretisation and a Crank-Nicolson discretisation in space. We show that the proposed scheme converges strongly at order at least one in probability. We also present numerical simulations to illustrate this result. The third chapter is devoted to the numerical and theoretical study of the dynamics of a stationary solution (for the deterministic equation) of vortex type. We study the influence of random disturbances of the confining potential on the solution. We show that the disturbed solution conserves the symmetries of the stationary solution for times up to at least the square of the inverse of the fluctuations intensity. These results are illustrated with numerical simulations based on a Monte-Carlo method suited to rare events estimation.The second model can be used to model the effects of the temperature on the dynamics of a Bose-Einstein condensate. In the case of finite temperature, the Bose-Einstein condensation is not complete and the condensate interacts with the non-condensed particles. These interactions are interesting to understand the dynamics of the phase transition and analyse the phenomena of symmetry breaking associated, like the spontaneous nucleation of vortices We have studied in the fourth and the fifth chapters some questions linked to the long time simulation of this model solutions. The fourth chapter is devoted to the construction of an unbiased sampling method of measures known up to a multiplicative constant. The distinctive feature of this Markov-Chain Monte-Carlo algorithm is that it enables to perform an unbiased non-reversible sampling based on an overdamped Langevin equation. It constitutes a generalization of the Metropolis-Adjusted Langevin Algorithm (MALA). The fifth chapter is devoted to the numerical study of metastable dynamics linked to the nucleation of vortices in rotating Bose-Einstein condensates. A numerical integrator and a suited Monte-Carlo methods for the simulation of metastable dynamics are proposed. This Monte-Carlo method is based on the Adaptive Multilevel Splitting (AMS) algorithm.

In this doctoral thesis, we theoretically investigate the propagation of sound waves in dilute Bose gases, in both the collisionless and hydrodynamic regimes. The study of sound wave is a topic of high relevance for the understanding of dynamical properties of any fluid, classical or quantum, and further provides insightful information about the equation of state of the system. In our work, we focus in particular on the two-dimensional (2D) Bose gas, in which the sound wave is predicted to give useful information about the nature of the superfluid phase transition. Recently, experimental measurement of sound wave in a uniform 2D Bose gas has become available, and we show that the measured data are quantitatively well explained by our collisionless theory. Finally, we study the mixtures of weakly interacting Bose gases, by developing a beyond mean-field theory, which includes the effects of thermal and quantum fluctuations in both the density and spin channels. Our new theory allows for the investigation of sound dynamics, as well as the fundamental problem of phase- separation.

This thesis explores the links between different ways of modelling the physical world. Finite difference numerical simulations may be used to encode the behaviour of physical systems, allowing us to gain insight into their workings and even to predict their behaviour. Similarly, one can investigate the properties of gravitational black holes through the use of analogue black holes, physical systems which share at least some part of the physics of the astronomical objects. Concentrating on black hole analogues using Bose-Einstein condensates, I show how simulations of these systems may be greatly assisted through the use of a proper absorbing boundary condition, the Perfectly Matched Layer. Such a boundary condition allows the effcient truncation of the computational domain, both saving computational time and increasing accuracy. I then apply this technique to the simulation of the supersonic flow of a Bose-Einstein condensate through a Laval nozzle, a black hole analogue, showing that such a flow should be stable and observable in the laboratory. Moving to a related system, I investigate the optical analogue of the Iordanskii force - the friction resulting from interaction between excitations in a superfluid's normal component and a superfluid vortex - through the simulation of such a vortex in a Bose-Einstein condensate illuminated by slow light, which is light whose group velocity is on the order of metres per second. The interaction of the slow light with the vortex should produce a momentum transfer due to the optical Aharonov-Bohm effect, exerting a force on the vortex. The coupled system of equations describing the condensate-slow light system is simulated, giving some surprising results.