Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.

This thesis presents theoretical investigations of triplet superfluidity (triplet superconductivity) in quasi-one-dimensional organic conductors and ultra-cold Fermi gases. Triplet superfluidity is different from its s-wave singlet counterpart since the order parameter is a complex vector and the interaction between fermions is in general anisotropic. Because of these distinctions, triplet superfluids have different physical properties in comparison to the s-wave case. The author discusses in this thesis the interplay between triplet superconductivity and spin density waves in quasi-one-dimensional organic conductors, and proposes a coexistence region of the two orders. Within the coexistence region, the interaction between the two order parameters acquires a vector structure, and induces an anomalous magnetic field effect. Furthermore, the author analyzes the matter-wave interference between two p-wave Fermi condensates, and proposes a polarization effect. For a single harmonically trapped p-wave Fermi condensate, the author also shows that the expansion upon release from the trap can be anisotropic, which reflects the anisotropy of the p-wave interaction.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].

The work presented in this thesis revolves around spectral inequalities and their applications in quantum mechanics. In Paper A, the ground state energy of an atom confined to two dimensions is analyzed in the limit when the charge of the nucleus Z becomes very large. The main result is a two-term asymptotic expansion of the ground state energy in terms of Z. Paper B deals with Hardy inequalities for the kinetic energy of a particle in the presence of an external magnetic field. If the magnetic field has a non-trivial radial component, we show that Hardy’s classical lower bound can be improved by an extra term depending on the magnetic field. In Paper C we study interacting Bose gases and prove Lieb-Thirring type estimates for several types of interaction potentials, such as the hard-sphere interaction in three dimensions, the hard-disk interaction in two dimensions as well as homogeneous potentials. / <p>QC 20140520</p>

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

In this thesis, we present the generation and studies of a 87Rb Bose-Einstein condensate (BEC) perturbed by an oscillatory excitation. The atoms are trapped in a harmonic magnetic trap where, after an evaporative cooling process, we produce the BEC. In order to study the effect caused by oscillatory excitations, a quadrupole magnetic field time oscillatory is superimposed to the trapping potential. Through this perturbation, collective modes were observed. The dipole mode is excited even for low excitation amplitudes. However, a minimum excitation energy is needed to excite the condensate quadrupole mode. Observing the excited cloud in TOF expansion, we note that for excitation amplitude in which the quadrupole mode is excited, the cloud expands without invert its aspect ratio. By looking these clouds, after long time-of-flight, it was possible to see vortices and, sometimes, a turbulent state in the condensed cloud. We calculated the momentum distribution of the perturbed BECs and a power law behavior, like the law to Kolmogorov turbulence, was observed. Furthermore, we show that using the method that we have developed to calculate the momentum distribution, the distribution curve (including the power law exponent) exhibits a dependence on the quadrupole mode oscillation of the cloud. The randomness distribution of peaks and depletions in density distribution image of an expanded turbulent BEC, remind us to the intensity profile of a speckle light beam. The analogy between matter-wave speckle and light speckle is justified by showing the similarities in the spatial propagation (or time expansion) of the waves. In addition, the second order correlation function is evaluated and the same dependence with distance was observed for the both waves. This creates the possibility to understand the properties of quantum matter in a disordered state. The propagation of a three-dimensional speckle field (as the matter-wave speckle described here) creates an opportunity to investigate the speckle phenomenon existing in dimensions higher than 2D (the case of light speckle). / Nesta tese, descrevemos a produção e os estudos de condensados de Bose-Einstein, em átomos de 87Rb, perturbados através de excitações oscilatórias. Os átomos aprisionados são aprisionados em uma armadilha magnética harmônica onde produzimos o condensado de Bose-Einstein após o processo de resfriamento evaporativo. Com o objetivo de estudar o efeito de excitações oscilatórias, um campo magnético quadrupolar temporalmente oscilanteé superposto ao campo de aprisionamento. Através dessa perturbação, podemos observar a excitação de modos coletivos no condensado. Mesmo para baixas amplitudes de excitação, o modo dipolar é facilmente excitado. Porém, observamos que para excitar o modo quadrupolar no condensado é necessária uma energia mínima. Através da expansão em tempo de voo da nuvem excitada, identificamos que, para amplitude de excitação na quail o modo quadrupolar é excitado, a nuvem expande sem inverter o aspect ratio. Analisando essas nuvens por longos tempos de voo, foi possível observar alguns vórtices e, às vezes, um estado turbulento na nuvem condensada. Calculamos a distribuição de momento dessas nuvens e notamos que ela exibe um comportamento de lei de potência, parecido com a lei de Kolmogorov para turbulência. Além disso, mostramos que pelo nosso método que desenvolvemos para calcular a distribuição de momento, a forma da curva dessa distribuição (inclusive o expoente da lei de potência) exibe uma dependência com o modo quadrupolar de oscilação da nuvem. A distribuição desordenada de picos e depleções, na imagem da distribuição de densidade do condensado turbulento expandido, assemelha-se ao perfil de intensidade de um feixe de luz com speckle. A analogia entre speckle de onda de matéria e de luz é fundamentada através das semelhanças entre a propagação (ou expansão) dessas duas ondas. Além disso, a função de correlação de segunda ordem foi calculada e a mesma dependência com a distância foi observada para as duas ondas. Isto cria a possibilidade de entender melhor as propriedades da matéria quântica em um estado de desordem. A propagação de um campo de speckle tridimensional (como é o caso do speckle de onda de matéria aqui descrito) cria uma oportunidade de investigar o fenômeno de speckle em dimensões maiores que 2D (o caso do speckle de luz).

Orientador: Silvio Antonio Sachetto Vitiello / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-26T08:38:32Z (GMT). No. of bitstreams: 1 Pedroso_VitorZampronio_M.pdf: 1351334 bytes, checksum: 9c35568e525b0eca7e7eb168eb22baa4 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho estudamos a condensação Bose-Einstein em sistemas formados por átomos de 4He. Para tanto, introduzimos uma nova função de onda sombra que permite o cálculo da fração de condensado do sistema a T = 0 K diretamente das configurações geradas no cálculo da energia variacional. A função proposta é construída através da integração sobre um conjunto de variáveis auxiliares que interagem com todos os átomos do sistema. Esta função é translacionalmente invariante mesmo na fase sólida e satisfaz a estatística de Bose-Einstein. Utilizando essa nova função de onda concluímos que aproximadamente 4% dos átomos estão no estado de menor energia e o sistema apresenta ordem de longo alcance fora da diagonal. Resultados de energia variacional e função de distribuição radial de pares também são apresentados. As integrais de configuração espaciais foram realizadas utilizando o método de Monte Carlo / Abstract: We study the Bose-Einstein condensation in systems formed by 4He atoms. To this end, we introduce a new shadow wave function that enables the calculation of the condensate fraction of the system at T = 0 K directly from the configurations generated in the calculation of the variational energy. The proposed function is constructed by an integration over a set of auxiliary variables that interact with all the atoms of the system. This function is translationally invariant even in the solid phase and satisfies the Bose- Einstein statistics. Using this new wave function we conclude that approximately 4% of the atoms are in the lowest energy state and the system displays off-diagonal long ranged order. Results from variational energy and pair distribution function are also presented. The configuration integrations were performed using the Monte Carlo method / Mestrado / Física / Mestre em Física

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.

Cette thèse est dédiée à l'interprétation théorique des expériences sur les gaz froids des excitons indirects dans des nanostructures semi-conductrices. La théorie proposée explique la formation de l'état des excitons macroscopiquement ordonnés ("MOES") et des taches lumineuses localisées dans les images de photoluminescence des excitons. Dans la première partie je montrerai que la séparation macroscopique de charge induite par laser mène à l'apparition d'un champ électrique situé dans le plan de la structure. A cause de ce champ les états quantiques 1s et 2p de l'exciton se croisent et son moment dipolaire s'incline. Par conséquent, l'exciton va se localiser à la frontière entre les deux domaines d'une charge différente, où le champ électrique est le plus fort. Ensuite, j'étudierai un gaz d'excitons mis dans de tels pièges bidimensionnels en négligeant sa structure de spin. J'analyserai la possibilité de la condensation de Bose-Einstein dans le système considéré en utilisant les méthodes puissantes de la théorie à N-corps développées pour des gaz atomiques. En me basant sur le Hamiltonien pour un segment du cercle bidimensionnel ("2D cigar"), je montrerai que la dispersion cohérente des excitons mène à l'autolocalisation accompagnée par une modulation périodique de la densité. L'idée principale de la théorie est, ensuite, de modéliser cet état périodique par une chaîne de condensats piégés (Le Modèle de Chaîne). Un tel modèle permettra de dire que le système peut exhiber la transition de phase de second ordre pour certaines valeurs du paramètre qui caractérise les interactions. La valeur critique de ce paramètre peut être trouvée en analysant le comportement des fluctuations de phase à la température nulle. Le nombre de condensats dans le régime où les interactions sont fortes est déterminé par la balance entre les contributions de l'énergie cinétique est l'entropie dans l'énergie libre du système. Le Modèle de Chaîne permettra aussi de révéler l'invariance d''échelle et l'universalité du phénomène. J'obtiendrai l'expression pour la température unique de la transition de phase dans le système excitonique et discuterai l'effet de désordre. Je finirai par une discussion du rôle des interactions à N-corps et des effets de spin dans la condensation de Bose-Einstein des excitons. Je proposerai un modèle de gaz idéal pour décrire les textures de polarisation linéaire observées autour de chaque tache lumineuse et chaque fragment de MOES. Selon ce modèle, le domaine central incohérent de tous ces objets est composé d'une glace excitonique quantique. / The present Thesis is devoted to theoretical interpretation of intriguing observations made recently in cold gases of indirect excitons in semiconductor quantum wells. The proposed theory provides simple intuitive explanation for the basic phenomenology of the macroscopically ordered exciton state (MOES) and the localized bright spots (LBS) in the exciton photoluminescense pattern. The Thesis is organized as follows.First, we provide an important insight into the formation process of the external ring and LBS. We show that the macroscopic charge separation induced by the photoexcitation results in appearance of an in-plane electric field in the vicinity of the boundary. The field hybridizes 1s and 2p quantum states of an indirect exciton, effectively tilting its dipole moment. Thus polarized exciton seeks for the regions with higher in-plane electric field and, hence, becomes localized at the ring-shaped boundary.As a next step, we consider a gas of spinless dipolar bosons put in such two-dimensional (2D) traps. We analyze the possibility for occurence of Bose-Einstein condensation (BEC) in the system under consideration by means of the powerful many-body theoretical methods developed for ultracold atomic gases. Starting from the Hamiltonian for a segment of the ring (2D cigar) we show, howthe coherent scattering of excitons can result in autolocalization accompanied by a buildup of the diagonal long-range order. The crucial point of the theory then consists in replacement of the periodic coherent state by a chain of harmonically trapped condensates (Chain Model). We argue, that for sufficiently strong contact interaction between the excitons the system can exhibit the true second order phase transistion at finite temperature. The critical value of the interaction parameter can be found by analyzing the behaviour of the quantum phase fluctuations at zero temperature. The number of condensates at the ring in the strongly interacting regime is defined by the balance between the kinetic energy and the entropy terms in the free energy of the system.Futhermore, the use of the Chain Model of the MOES allows one to reveal scale invariance and universality of the pnenomenon. We obtain the expression for the unique critical temperature of the second order phase transition in the exciton system and discuss the effect of disorder.Finally, we comment on the role of many-body interactions and spin degrees of freedom in excitonic BEC. We suggest that each bead (or, equivalently, LBS) has the internal structure: it consists of a solid core (Quantum Exciton Iceberg) surrounded by a coherent exciton fluid. We develop an ideal gas model for the coherent four-component exciton fluid which allows one to explain the measured linear polarization patterns.