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We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter α. For weak interactions, the shift in critical temperature is expected to be linear in α with constant of linearity c. Using Markov chain Monte Carlo methods and finite-size scaling, we find c = 0.618 ± 0.086. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations. The plan of this paper is as follows. We begin with a non-technical discussion of the historical context of the project, along with a mention of alternative approaches. Relevant previous works are cited, thus annotating the bibliography. The random-cycle approach to the BEC problem requires a model of spatial permutations. This model it is of its own probabilistic interest; it is developed mathematically, without reference to the Bose gas. Our Markov-chain Monte Carlo algorithms for sampling from the random-cycle distribution - the swap-only, swap-and-reverse, band-update, and worm algorithms - are presented, compared, and contrasted. Finite-size scaling techniques are used to obtain information about infinite-volume quantities from finite-volume computational data.

Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.

Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.

The properties of an atomic Bose-Einstein condensate in a double-well potential are investigated through a two-mode analysis. An analytic solution for the semiclassical tunnelling and self-trapping dynamics is compared with numerical simulations of the quantum dynamics, which exhibit collapses and revivals for a closed system. A continuous non-destructive measurement technique to monitor the Josephson tunnelling oscillations is presented, in which the condensate in one well dispersively shifts the phase of a coherent probe beam in proportion to atom-number. The evolution of the resulting homodyne photocurrent and Bloch Q distributions shows that oscillations develop even when the initial state possesses phase symmetry. The conditional dynamics of the condensate which result from measurement back-action also appear in certain semiclassical formulations. The homodyne measurement technique is incorporated into a proposed weak-force detector. A maximally entangled initial state, which is the ground state for a double condensate with strong attractive atomic interactions, enables a high-precision measurement. The dynamics of quantum many-body multimode systems of interacting bosons are simulated using phase-space methods. The use of the Wigner technique predicts novel noise effects in fibre solitons. The positive-P representation is used to simulate the formation of mesoscopic Bose-Einstein condensates via evaporative cooling in three dimensional atom traps. The results indicate highly non-classical behaviour near the critical point, and provide evidence for the spontaneous formation of vortices. Comparisons with corresponding mean-field calculations reveal large differences between the semiclassical and fully quantum results. Finally, the possibility of future progress with alternative phase-space methods is considered.

Thesis (Ph.D.) - University of St Andrews, January 2008.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Daniel J. Heinzen. Vita. Includes bibliographical references. Also available from UMI.

Les polaritons de microcavité sont des états hybrides lumière-matière à caractère bosonique.Dans les dernières décennies, un grand intérêt a été accordé à leur phase de condensation de Bose-Einstein.Nous avons développé dans ce travail des outils théoriques et numériques pour comprendre et interpréter la dynamique spatiale et temporelle de la formation des condensats de polaritons.Nous avons proposé une approche numérique pour la résolution complète des équations couplées du modèle Gross-Pitaevskii généralisé à deux dimensions en coordonnées cartésiennes.Nous avons cherché à comprendre les aspects du seuil de condensation sous différentes configurations spatiales et temporelles d'excitation optique non résonante.Nous avons en particulier proposé une nouvelle approche pour définir le seuil.Enfin, pour une condensation sous exciation focalisée, dans une microcavité ZnO, nous avons pu accéder à, et comprendre, quelques propriétés vues dans les expériences. / Microcavity polaritons are hybrid light-material states of a bosonic nature.In the last decades, an enormous interest has been paid to their Bose-Einstein condensation phase.We develop, in this work the theoretical and numerical tools to understand and interpret the spatial and temporal dynamics of the formation of condensates of polaritons.We propose a numerical approach for the comprehensive resolution of the generalized Gross-Pitaevskii model in two-dimensions in Cartesian coordinates.We sought to understand the aspects of the condensation threshold under different spatial and temporal configurations of non-resonant optical excitation.In particular, we propose a new approach to define the threshold.Finally, for a condensation under a focal exciation in a ZnO microcavity, we were able to access, and understand, some of the experimentally observed properties.

A turbulência clássica é um fenômeno de natureza caótica, mas de difícil estudo por ser constituída pela fusão e superposição de vórtices aleatórios, dificultando sua descrição matemática. A turbulência quântica (TQ), embora também caótica, é composta por vórtices quantizados, que favorecem o controle experimental e sua definição teórica. Embora a evidência experimental da TQ tenha sido obtida em sistemas de He líquido, sua caracterização em condensados de Bose-Einstein (BEC) ainda não foi totalmente realizada. Neste trabalho, estudamos a distribuição de momento em BECs expandidos em tempo de voo, nos regimes convencional e turbulento. Para a produção experimental da amostra quanticamente degenerada, utilizamos a técnica do resfriamento evaporativo em átomos de 87Rb, previamente resfriados em uma armadilha puramente magnética do tipo QUIC. A turbulência quântica foi produzida no sistema através de um par de bobinas de excitação capaz de produzir uma perturbação oscilatória na nuvem previamente condensada. O diagnóstico da amostra aprisionada é feito por imagem de absorção durante expansão livre da nuvem. Durante a expansão, tanto a nuvem condensada quanto a turbulenta, alcançaram um valor assintótico no aspect ratio, indicando uma evolução isotrópica. A partir deste resultado, elaboramos um método teórico capaz de determinar a projeção isotrópica da distribuição de momento, baseado na imagem produzida experimentalmente. Através de argumentos de simetria e de uma transformada integral, recuperamos a densidade de momento tridimensional da projeção, para então determinar o espectro de energia cinética da nuvem, observando uma lei de escala para um estreito intervalo de momento. A lei de escala já foi prevista teoricamente para sistemas quânticos e medida para o He superfluido, mas pela primeira vez foi evidenciada em um BEC. Desta forma, os resultados corroboram a existência da turbulência quântica em uma amostra quanticamente degenerada, introduzindo os BECs como candidatos alternativos ao He líquido superfluido no estudo deste fenômeno. / Classical turbulence is a chaotic phenomenon that requires labored work, because of its merging and overlapping of random vortices nature, which hinders its mathematical description. Quantum turbulence (QT), although chaotic, is comprised of quantized vortices that favor the experimental control and its theoretical definition. Although experimental evidence of QT has been proved in liquid helium systems, its characterization in Bose-Einstein condensates (BEC) has not been fully accomplished. In this work, we studied the momentum distribution of expanding turbulent and non-turbulent BEC. For experimental achievement of the quantum degenerated sample, we used evaporative cooling in rubidium atoms, previously cooled in a QUIC trap. Quantum turbulence was produced through a pair of excitation coils capable of producing an oscillatory perturbation in the cloud previously condensed. The diagnosis of the trapped sample is done by absorption image during free expansion of the cloud. During the expansion, both clouds achieved a asymptotic value of the aspect ratio, indicating an isotropic evolution. From this result, we have developed a theoretical method able to determine the projection of the isotropic distribution of momentum, based on the image produced experimentally. Through symmetry arguments and an integral transformation, we recovered the tridimensional momentum distribution of the projection and then determined the kinetic energy spectrum of the cloud, observing a scaling power law for a narrow range of momenta. The scaling law has been theoretically predicted for quantum systems and has been proved to liquid helium superfluid, but, in this work, was for the first time evidenced in a BEC. Thus, the results support the existence of quantum turbulence in our quantum degenerated sample, introducing the BECs as potential candidates besides liquid helium superfluid for the study of this phenomenon.

Neste trabalho é estudado o comportamento termodinâmico de um gás unidimensional de bosons sob a ação de uma impureza delta atrativa. O sistema apresenta o fenômeno da condensação de Bose- Einstein e a causa da transição é atribuida ao estado ligado introduzido pela impureza no espectro de partícula livre. A fase condensada é composta pelas partículas capturadas pela impureza, formando uma gota de partículas bem localizadas no espaço. Isto dá à condensação de Bose-Einstein apresentada pelo sistema a aparência da conhecida transição líquido-vapor. A ordem de transição é analisada pela equação de Clausius-Clayperon e interpretada como de primeira ordem. Deste modo, a semelhança entre a condensação de Bose-Einstein neste sistema ma e a transição líquido-vapor é reforçada.O cálculo do calor específico a comprimento constante, mostra a existência de uma descontinuidade finita na temperatura de transição. / The thermodynamic behavior of the one-dimensional Bose gas-attractive delta impurity system is studied in this work. The system is shown to undergo the Bose-Einstein condensation and the cause of the phase transition is attributed to the bound state introduced by the impurity in the free particle energy spectrum. The condensed phase is composed by particles captured by the impurity, forming a drop of particles well localized in space. This gives to the Bose-Einstein condensation in this system the appearance of the ordinary vapor-liquid phase transition. The order of the phase transition is analized with the aid of the Clausius-Clayperon equation wich allowed us to conclude that the transition is a first order one. This reinforce the interpretation of a vapor-liquid transition.The evaluation of the heat capacity at constant length shows the existence of a finite discontinuity at the transition temperature.