Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-03Bitstream added on 2014-06-13T20:27:56Z : No. of bitstreams: 1 holz_sm_me_ift.pdf: 1325216 bytes, checksum: 3c1a2e906f591704a593a9670e903eee (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Investigamos as soluções estacionárias e dinâmicas da equação de Gross-Pitaevskii generalizada para sistemas atômicos com um potencial confinante e termos não conservativos associados à dissipação e à alimentação atômica, visando a descrição de condensados de Bose-Einstein. Consideramos os casos de comprimentos de espalhamento negativos (interações atrativas) e positivos (interações repulsivas) entre dois átomos. Verificamos como a variação dos parâmetros associados aos termos não conservativos pode produzir situações de instabilidade resultando no fenômeno conhecido como caos espaço temporal. Por outro lado, verificamos também quais combinações de parâmetros leva a soluções de equilíbrio, tipo solitônica. Nessa pesquisa, utilizando esse modelo de campo médio com uma parametrização conhecida, estudamos as propriedades de tais sistemas para alguns valores dos parâmetros não-conservativos, por meio de métodos numéricos e variacionais. / We investigate the stationary and dynamical solutions of the Gross-Pitaevskii equation extended for atomic systems with confining potential in the presence of nonconservative terms associated to atomic dissipation and feeding, in order to describe Bose-Einstein Condensates. We considered the cases of negative (attractive interaction) and positive (repulsive interaction) two-body lenght. We verified how the variation of the parameters associated to those nonconservative terms could produce instabilities resulting in occurrence of spacetemporal chaos. In other hand, we looked for parameters combinations that give us stable solitonic-like solutions. In this research, by using the mean-field approach with a particular parameterization, we studied the properties of these systems for some values of the nonconservative parameters, by means of numerical and variational methods.

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2004-10Bitstream added on 2014-06-13T19:06:28Z : No. of bitstreams: 1 brtka_m_dr_ift.pdf: 1102252 bytes, checksum: 1a7b66f943a95b6ff94475b02062691c (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos as oscilações não-lineares de um condensado de Bose-Einstein em três dimensões numa armadilha com simetria esférica e com comprimento de espalhamento periódico no tempo. Usamos o método variacional dependente do tempo e deduzimos a equação para a largura do condensado. A partir desta equação, analisaremos as características das ressonâncias não-lineares e o efeito de bi-estabilidade em oscilações da largura do condensado. Previsões teóricas são confirmadas pelas simulações numéricas da equação de Gross-Pitaevskii. / Nonlinear oscillations of a 3D radial symmetric Bose-Einstein condensate (BEC) under periodic variation in time of the atomic scattering length have been studied. The time-dependent variational approach is used for analysis of the characteristics of nonlinear resonances in the oscillations of the condensate. The bistability in oscillations of the BEC width is investigated. Predictions of the theory are confirmed by numerical simulations of the full Gross-Pitaevskii equation.

Resumo: Estudamos as oscilações não-lineares de um condensado de Bose-Einstein em três dimensões numa armadilha com simetria esférica e com comprimento de espalhamento periódico no tempo. Usamos o método variacional dependente do tempo e deduzimos a equação para a largura do condensado. A partir desta equação, analisaremos as características das ressonâncias não-lineares e o efeito de bi-estabilidade em oscilações da largura do condensado. Previsões teóricas são confirmadas pelas simulações numéricas da equação de Gross-Pitaevskii. / Abstract: Nonlinear oscillations of a 3D radial symmetric Bose-Einstein condensate (BEC) under periodic variation in time of the atomic scattering length have been studied. The time-dependent variational approach is used for analysis of the characteristics of nonlinear resonances in the oscillations of the condensate. The bistability in oscillations of the BEC width is investigated. Predictions of the theory are confirmed by numerical simulations of the full Gross-Pitaevskii equation. / Orientador: Roberto A. Kraenkel / Coorientador: Arnaldo Gammal / Banca: Felipe Barbedo Rizzato / Banca: Lauro Tomio / Banca: Emerson José Veloso dos Passos / Doutor

No presente trabalho foi estudada a dinâmica de um sistema de muitas partículas no regime de temperaturas ultra-baixas. Realizamos um estudo dinâmico de sistemas condensados bidimensionais em uma rede óptica não-linear em uma direção e também na presença de uma armadilha harmônica assimétrica. Investigamos alguns aspectos sobre a estabilização e propagação de sólitons em condensados de Bose-Einstein. O colapso da função de onda é evitado pela não-linearidade periódica dissipativa, no caso de um meio com campo de fundo positivo (com sistemas atômicos atrativos). A variação adiabática do comprimento de espalhamento de fundo leva a existência de sólitons de onda de matéria metaestáveis. Um sóliton dissipativo pode existir no meio atrativo bidimensional (2D) com uma não-linearidade periódica unidimensional (1D), quando um mecanismo de alimentação atômica é utilizado. Um sóliton estável pode existir no caso de condensados repulsivos, em um campo de fundo negativo, com uma armadilha harmônica em uma direção e uma rede óptica não-linear na outra direção. Os resultados inteiramente numéricos, para a equação de Gross-Pitaevskii 2D, confirmam as simulações da abordagem variacional. / In this work the dynamics of a system of many particles in a ultra-low temperature regime was studied. We performed a dynamic study of two-dimensional condensate systems into a nonlinear optical lattice in one direction and also in the presence of an asymmetrical harmonic trap. We investigated some aspects of the stabilization and spread of solitons in a Bose-Einstein condensate. In the case of positive background field media (with attractive atomic systems), the collapse of the wave-packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in an attractive bidimensional (2D) media with unidimensional (1D) periodic nonlinearity. In the case of repulsive condensates, with a negative background field, a stable soliton may exist when we have an harmonic trap in one direction and a nonlinear optical lattice in the other. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.

Ce travail est une contribution à l'étude des équations de Schrödinger non-linéaires (NLS) en dimension un d'espace. De telles équations interviennent notamment comme modèles dans plusieurs domaines de la physique mathématique, tels l'optique non-linéaire, la superfluidité, la supraconductivité et la condensation de Bose-Einstein.Cette thèse contient trois thèmes connexes inclus dans les chapitres 2, 3 et 4. Dans la première partie (chapitre 2), on s'intéresse à la construction des solutions en multi-solitons de l'équation de Gross-Pitaevskii (NLS défocalisante avec non-linéarité cubique), comme une superposition approximative des ondes progressives (solitons). Cette partie contient également une description détaillée des interactions entre les solitons. Ces résultats sont obtenus en exploitant l'intégrabilité de l'équation de Gross-Pitaevskii et son système de Marchenko associé.La deuxième partie (chapitre 4) clarifie les relations entre la formulation classique et la formulation dite hydrodynamique de l'équation de Gross-Pitaevskii. Cette dernière a un sens lorsque la solution ne s'annule jamais dans le domaine spatial. La dernière partie (chapitre 3) est consacrée à l'étude du problème de Cauchy d'une famille d'équations aux dérivées partielles quasi-linéaires qui généralise l'équation du flot par courbure binormal d'une courbe dans l'espace euclidien de dimension trois. Cette dernière est liée formellement à NLS par la transformation de Hasimoto. Dans notre généralisation, la vitesse d'un point de la courbe est toujours dirigée dans la direction du vecteur binormal, mais son amplitude peut dépendre de l'abscisse curviligne ainsi de la position dans l'espace. Notre approche pour prouver l'existence est le suivant: schéma semi-discret (discret en espace et continu en temps), obtention de bornes sur les problèmes discrets et argument par compacité. Un théorème de comparaison entraîne l'unicité. / This work is a contribution to the study of nonlinear Schrödinger equations (NLS) in the one-dimensional space. Such equations arise in many physical fields, including nonlinear optics and Bose-Einstein condensation. The thesis contains three connected themes included in chapters 2, 3 and 4. The first part (chapter 2) constructs multi-soliton solutions of the Gross-Pitaevskii (or defocussing NLS) equation, as an approximate superposition of traveling waves (solitons). This part contains also a detailed description of the interactions between solitons. These results are obtained by exploiting the integrability of the the Gross-Pitaevskii equation and its associated Marchenko system. The second part (chapter 4) clarifies the relations between the classical formulation and the so-called hydrodynamical formulation that only has a meaning when the solution does not vanish anywhere in the spatial domain The last part (chapter 3) of this thesis concerns existence and uniqueness results for a family of quasi-linear partial differential equations that generalize the equation of the binormal curvature flow for a curve in the three-dimensional space. The latter equation is in connection to the focussing cubic NLS by Hasimoto transformation. In our generalization, the velocity of a point on the curve is still directed along the binormal vector (so that in particular the length of the curve is preserved) but the magnitude of the speed is allowed to depend both on the curvilinear parameter and on the position in space. Existence is proven using spatial discretization together with some a priori bounds on the approximate solutions. Uniqueness follows from a comparison theorem.

A prescription is given to obtain some exact results for certain external potentials �� (r) of the time-independent Gross-Pitaevskii and Schrodinger equations. The study motivation is the ability to program �� (r) experimentally in cold atom Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given �� (r), we ask which �� (r) will have a given pdf (probability density function) �� (r). Several examples in 1 dimension (1D), 2 dimensions (2D), and 3 dimensions (3D) are presented for well-known pdfs in the position space. Exact potentials with zero, one and two walls are obtained and explained in detail. Apart from position space, the method is also applicable to obtain exact solutions for the Time-independent Schr¨odinger equation (TISE) and Gross-Pitaevskii equation (GPeq) for pdfs in momentum space. For this, we derived the potentials which are generated from the pdfs of the hydrogen atom in the real space as well as in the momentum space. However, the method was also extended for the time-dependent case. The prescription is also applicable to solve time-dependent pdfs. The aim is to find the ��(r, ��) which generates the pdf ��(r, ��). As a special case, we tested our method by studying the well known case for the Gaussian wave packet in 1D with zero potential ��(��, ��) = 0.

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.