Neste trabalho de mestrado é estudada a estabilidade de vórtices em condensados de Bose-Einstein com interação atrativa entre os átomos através da solução numérica da equação de Gross-Pitaevskii. Inicialmente são reproduzidos resultados da literatura, nos quais são estudados vórtices em condensados bidimensionais atrativos com potencial interatômico homogêneo em todo o condensado. A estabilidade de tais sistemas é inferida através da solução numérica das equações de Bogoliubov-de Gennes e da evolução temporal dos vórtices. Demonstra-se que esses vórtices são estáveis, até um certo número crítico de átomos, apenas para valores de vorticidade S=1. Em seguida foi proposto um modelo no qual a interação entre os átomos é espacialmente modulada. Neste caso é possível demonstrar que vórtices com valores de vorticidade de até S=6, pelo menos, são estáveis. Finalmente é estudada a estabilidade de vórtices em condensados tridimensionais atrativos, novamente com potencial interatômico homogêneo em todo o condensado. Assim como no caso bidimensional mostra-se que tais vórtices são estáveis para valores de vorticidade de S=1. Espera-se em breve estudar a estabilidade de vórtices em condesados tridimensionais com potencial de interação espacialmente modulado. / In this work we study the stability of vortices in attractive Bose-Einstein condensates by solving numerically the Gross-Pitaevskii equation. Initially we reproduce some results from the literature, in which vortices in two-dimensional attractive Bose-Einstein condensates with homogeneous interatomic potential are studied. The stability of these systems is determined by solving numerically the Bogoliubov-de Gennes equations and by studying the time evolution of these vortices. We demonstrate that these vortices are stable, up to a certain critical number of atoms, just for the value of vorticity S=1. After we propose a model in which the interatomic interaction are spatially modulated. In this case it is possible to verify that vortices with values of vorticity up to S=6 , at least, are stable. Finally, we study the stability of vortices in three-dimensional attractive condensates, again with a homogeneous interatomic potential. As in the two-dimensional case, we show that vortices in these systems are stable to values of vorticity S=1. The next step in this work is study the stability of vortices in three-dimensional condensates with spatially modulated interatomic interaction.

This thesis is divided in two parts. The first one is devoted to the dynamics of polariton condensates, with particular attention to their pattern-forming capabilities. In many configurations of physical interest, the dynamics of polariton condensates can be modelled by means of a non-linear PDE which is strictly related to the Gross-Pitaevskii and the complex Ginzburg-Landau equations. Numerical simulations of this equation are used to investigate the robustness of the rotating vortex lattice which is predicted to spontaneously form in a non-equilibrium trapped condensate. An idea for a polariton-based gyroscope is then presented. The device relies on peculiar properties of non-equilibrium condensates - the possibility of controlling the vortex emission mechanism and the use of pumping strength as a control parameter - and improves on existing proposals for superfluid-based gyroscopes. Finally, the important rôle played by quantum pressure in the recently observed transition from a phase-locked but freely flowing condensate to a spatially trapped one is discussed. The second part of this thesis presents work done in the context of the geometrical models of matter framework, which aims to describe particles in terms of 4-dimensional manifolds. Conserved quantum numbers of particles are encoded in the topology of the manifold, while dynamical quantities are to be described in terms of its geometry. Two infinite families of manifolds, namely ALF gravitational instantons of types A_k and D_k, are investigated as possible models for multi-particle systems. On the basis of their topological and geometrical properties it is concluded that A_k can model a system of k+1 electrons, and D_k a system of a proton and k-1 electrons. Energy functionals which successfully reproduce the Coulomb interaction energy, and in one case also the rest masses, of these particle systems are then constructed in terms of the area and Gaussian curvature of preferred representatives of middle dimension homology. Finally, an idea for constructing multi-particle models by gluing single-particle ones is discussed.

The presence of quantized vortices and a high level of control over trap geometries and other system parameters make dilute-gas Bose-Einstein condensates (BECs) a natural environment for studies of vortex dynamics and quantum turbulence in superfluids, primary interests of the BEC group at the University of Arizona. Such research may lead to deeper understanding of the nature of quantum fluid dynamics and far-from-equilbrium phenomena.Despite the importance of quantized vortex dynamics in the fields of superfluidity, superconductivity and quantum turbulence, direct imaging of vortices in trapped BECs remains a significant technical challenge. This is primarily due to the small size of the vortex core in a trapped gas, which is typically a few hundred nanometers in diameter. In this dissertation I present the design and construction of a new ^87Rb BEC apparatus with the goal of studying vortex dynamics in trapped BECs. The heart of the apparatus is a compact vacuum chamber with a custom, all-glass science cell designed to accommodate the use of commercial high-numerical-aperture microscope objectives for in situ imaging of vortices.The designs for the new system are, in part, based on prior work in our group on in situ imaging of vortices. Here I review aspects of our prior work and discuss some of the successes and limitations that are relevant to the new apparatus. The bulk of the thesis is used to described the major subsystems of the new apparatus which include the vacuum chamber, the laser systems, the magnetic transfer system and the final magnetic trap for the atoms. Finally, I demonstrate the creation of a BEC of ~2x10^6 ^87Rb atoms in our new system and show that the BEC can be transferred into a weak, spherical, magnetic trap with a well defined magnetic field axis that may be useful for future vortex imaging studies.

In this work, dipolar Bose-Einstein condensates in multiwell potentialsplaced to form dierent geometrical structures are studied theoretically inorder to determine how the ground state population of the particles in thepotential wells changes depending on the relative strength of the particlesdipole moment. In the analytical limit (neglecting intersite tunneling), asymmetry-breaking change in the number of wells that are populated byparticles is observed for all studied systems for a certain value of the rela-tive strength of the particles dipole moment. The numerical calculationsfor nonzero intersite tunneling show a non-degenerate bifurcation whichis not seen in the analytical limit.

Utilizamos o modelo de Bose-Hubbard para estudar as estabilidades dinâmica e termodinâmica dos condensados numa rede óptica periódica circular. O nosso principal objetivo foi investigar a existência de condensados metaestáveis no sistema. Deduzimos e resolvemos a equação de Gross-Pitaevskii e, a partir da análise das soluções, foi possível mostrar que o sistema se condensa em estados com momento modular bem definido. Esses estados formam uma base que diagonaliza o termo que descreve o tunelamento atômico no hamiltoniano de Bose-Hubbard. No contexto da teoria de Bogoliubov deduzimos para cada condensado, o hamiltoniano efetivo cuja diagonalização determina o espectro das excitações coletivas do sistema. Identificamos corretamente o modo de energia zero, conseqüência da violação da conservação do número de átomos, e verificamos que este possui momento modular igual ao do condensado. No estudo da estabilidade vimos que todos os condensados com momento modular nos 2º e 3º quadrantes são termodinamicamente instáveis e as respectivas condições de estabilidade dinâmica dependem dos parâmetros de controle do sistema. Por outro lado os condensados com momento modular nos 1º e 4º quadrantes são todos dinamicamente estáveis enquanto que, nesse caso, é a estabilidade termodinâmica que depende dos parâmetros de controle do sistema. Nessa análise verificamos que o condensado com momento modular zero, que corresponde ao mínimo global da energia, é sempre estável. Determinamos exatamente o intervalo nos parâmetros de controle a partir do qual podemos encontrar condensados metaestáveis no sistema. Examinamos como a competição entre as intensidades dos termos de tunelamento e repulsão local afeta a estabilidade dos condensados. Essa competição define dois regimes distintos: Rabi, onde a coerência entre estados localizados nos sítios é mantida, e Fock, onde não há mais essa coerência e a aplicabilidade da aproximação de Bogoliubov é questionável. / We use the Bose-Hubbard model to study the dynamical and thermodynamical stabilities of condensates in a circular periodic optical lattice. Our main goal was to investigate the existence of metastable condensates in the system. We derive and solve the Gross-Pitaevskii equation, and from the analysis of the solutions it was possible to show that the system condenses in states with well-defined modular momentum. These states constitute a basis that diagonalizes the term of the Bose-Hubbard Hamiltonian which describes the dynamics of atomic tunneling. In the framework of Bogoliubov theory we determine, for each condensate, the effective Hamiltonian whose diagonalization give us the collective excitation spectrum of the system. We show that the mode associated to a zero eigenvalue, which is a consequence of the violation of atoms number conservation, has the same modular momentum of the condensate. The condensates with modular momentum in the 2nd and 3rd quadrants are all thermodynamically unstable whereas the dynamical stability depends on the control parameters. On the other hand, the condensates with modular momentum in the 1st and 4th quadrants are all dynamically stable whereas the thermodynamical stability depends on the control parameters. Our analysis shows that the condensate with modular momentum zero, which corresponds to a global minimum of energy, is always stable independently of the control parameters. We determine, exactly, the range on the control parameters where it is possible to detect metastability in the system. We have studied how the competition between the intensities of the tunneling and local interaction terms affects the stability of the condensates. This competition defines two distinct regimes: Rabi, where the coherence between states localized in the sites is achieved, and Fock, where this coherence is not achieved and the validity of Bogoliubov approximation is questionable.

Utilizamos o modelo de Bose-Hubbard para estudar as estabilidades dinâmica e termodinâmica dos condensados numa rede óptica periódica circular. O nosso principal objetivo foi investigar a existência de condensados metaestáveis no sistema. Deduzimos e resolvemos a equação de Gross-Pitaevskii e, a partir da análise das soluções, foi possível mostrar que o sistema se condensa em estados com momento modular bem definido. Esses estados formam uma base que diagonaliza o termo que descreve o tunelamento atômico no hamiltoniano de Bose-Hubbard. No contexto da teoria de Bogoliubov deduzimos para cada condensado, o hamiltoniano efetivo cuja diagonalização determina o espectro das excitações coletivas do sistema. Identificamos corretamente o modo de energia zero, conseqüência da violação da conservação do número de átomos, e verificamos que este possui momento modular igual ao do condensado. No estudo da estabilidade vimos que todos os condensados com momento modular nos 2º e 3º quadrantes são termodinamicamente instáveis e as respectivas condições de estabilidade dinâmica dependem dos parâmetros de controle do sistema. Por outro lado os condensados com momento modular nos 1º e 4º quadrantes são todos dinamicamente estáveis enquanto que, nesse caso, é a estabilidade termodinâmica que depende dos parâmetros de controle do sistema. Nessa análise verificamos que o condensado com momento modular zero, que corresponde ao mínimo global da energia, é sempre estável. Determinamos exatamente o intervalo nos parâmetros de controle a partir do qual podemos encontrar condensados metaestáveis no sistema. Examinamos como a competição entre as intensidades dos termos de tunelamento e repulsão local afeta a estabilidade dos condensados. Essa competição define dois regimes distintos: Rabi, onde a coerência entre estados localizados nos sítios é mantida, e Fock, onde não há mais essa coerência e a aplicabilidade da aproximação de Bogoliubov é questionável. / We use the Bose-Hubbard model to study the dynamical and thermodynamical stabilities of condensates in a circular periodic optical lattice. Our main goal was to investigate the existence of metastable condensates in the system. We derive and solve the Gross-Pitaevskii equation, and from the analysis of the solutions it was possible to show that the system condenses in states with well-defined modular momentum. These states constitute a basis that diagonalizes the term of the Bose-Hubbard Hamiltonian which describes the dynamics of atomic tunneling. In the framework of Bogoliubov theory we determine, for each condensate, the effective Hamiltonian whose diagonalization give us the collective excitation spectrum of the system. We show that the mode associated to a zero eigenvalue, which is a consequence of the violation of atoms number conservation, has the same modular momentum of the condensate. The condensates with modular momentum in the 2nd and 3rd quadrants are all thermodynamically unstable whereas the dynamical stability depends on the control parameters. On the other hand, the condensates with modular momentum in the 1st and 4th quadrants are all dynamically stable whereas the thermodynamical stability depends on the control parameters. Our analysis shows that the condensate with modular momentum zero, which corresponds to a global minimum of energy, is always stable independently of the control parameters. We determine, exactly, the range on the control parameters where it is possible to detect metastability in the system. We have studied how the competition between the intensities of the tunneling and local interaction terms affects the stability of the condensates. This competition defines two distinct regimes: Rabi, where the coherence between states localized in the sites is achieved, and Fock, where this coherence is not achieved and the validity of Bogoliubov approximation is questionable.

No description available.

Motivated by the sustained interest in Bose Einstein condensates and the recent progress in the understanding of topological phases in condensed matter systems, we study quantum condensates and possible topological phases of bosons in coupled light-matter excitations, so-called polaritons. These bosonic quasi-particles emerge if electronic excitations (excitons) couple strongly to photons. In the first part of this thesis a polariton Bose Einstein condensate in the presence of disorder is investigated. In contrast to the constituents of a conventional condensate, such as cold atoms, polaritons have a finite life time. Then, the losses have to be compensated by continued pumping, and a non-thermal steady state can build up. We discuss how static disorder affects this non-equilibrium condensate, and analyze the stability of the superfluid state against disorder. We find that disorder destroys the quasi-long range order of the condensate wave function, and that the polariton condensate is not a superfluid in the thermodynamic limit, even for weak disorder, although superfluid behavior would persist in small systems. Furthermore, we analyze the far field emission pattern of a polariton condensate in a disorder environment in order to compare directly with experiments. In the second part of this thesis features of polaritons in a two-dimensional quantum spin Hall cavity with time reversal symmetry are discussed. We propose a topological invariant which has a nontrivial value if the quantum spin Hall insulator is topologically nontrivial. Furthermore, we analyze emerging polaritonic edge states, discuss their relation to the underlying electronic structure, and develop an effective edge state model for polaritons.

Dans les microcavités semiconductrices, les polaritons excitoniques sont obtenus à partir du couplage fort entre l'exciton et le photon. Le régime de laser à polaritons à température ambiante, première étape vers le condensat de Bose-Einstein (BEC), a été atteint dans des microcavités ZnO et nous avons dans cette thèse étudié les propriétés des condensats de polaritons. Nous avons réalisé la spectroscopie linéaire et déterminé les propriétés spatiales de nouvelles microcavités ZnO sur substrat Si structuré en mesa, de haut facteur de qualité Q. Plusieurs géométries de génération de condensats de polariton ont été mises en oeuvre et comparées. Nous avons également mesuré, au travers d'expériences d'imagerie 2D en champ proche et en champ lointain, et modélisé, en résolvant l'équation de Gross-Pitaevskii, la propagation des condensats. Nous avons ainsi pu décrire les phénomènes mis en jeu dans la propagation des condensats à 80 et 300K pour une excitation fortement focalisée par rapport à une excitation étendue à 2D. Ces travaux posent les bases de dispositifs polaritoniques à 300K dans lesquels les condensats seront façonnés et contrôlés. / In semiconductors microcavities, exciton-polaritons arise from the strong coupling between excitons and photons. The polariton laser at room temperature, which is the first step to Bose-Einstein condensation (BEC), has been achieved in ZnO microcavities and the study of polariton condensates is the main issue of this work. We have studied the linear spectroscopy and measured the spatial properties of new high-Q ZnO microcavities grown on a patterned Si-substrate. Many generation geometries have been set up and compared to control the shape of polariton condensates. We have also measured and simulated polariton condensates propagation, using respectively 2D imaging experiments in near-field and far-field and by resolving the Gross-Pitaevskii equation. Then we were able to describe the variety of phenomena involved in the condensates propagation at 80 and 300 K for a tightly focused excitation compared to a wide 2D excitation. Those experiments pave the way for the development of polariton devices operating at 300 K in which polariton condensates can be patterned and controlled.

Exciton-polaritons are the fundamental excitations arising from the strong coupling of quantum well excitons and cavity photons in semiconductor microcavities. They are compound bosons for which stimulated scattering and macroscopic occupation of single quantum states can occur at sufficiently high densities. One way of creating such polariton condensates is with nonresonant optical pumping. Doing so creates a large density of free- carriers and excitons that strongly interact and blueshift the polariton energy levels. Using spatially patterned nonresonant fields, the polariton potential landscape can be tailored and optically trapped condensates can be created. This thesis shows that the spin properties of polariton condensates are strongly modified by such trapping. Under linearly polarised pumping, helicity can spontaneously develop at a critical occupation, breaking the parity symmetry. This formation of spin-up/spin-down condensates is explained within a Gross-Pitaevskii model which accurately reproduces the influence of electric fields and condensate density. Under elliptically polarised pumping, two phenomena are observed: the formation of condensates with the opposite handedness to the pump and hysteresis of both occupation and spin with respect to pump power. The spatial dependence of these effects highlights the limitations of commonly used models of polariton condensation. Finally, the suitability of patterned optical fields for the creation of polariton lattices is explored. For small chains of condensates, controllable coupling between adjacent spins is demonstrated, with the formation of antiferromagnetic and ferromagnetic domains. The extent of these domains is strongly affected by sample nonuniformity, fundamentally limiting the scalability of these lattices.