In order to explore a quantum version of a discrete nonlinear Schrödinger equation (DNLS), we quantize one nonlinear Schrödinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrödinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.

The Bose-Hubbard model is used to study bosons in optical lattices. In this thesis we will use an extended Bose-Hubbard model to study a type of completely localized solutions, called compactons. The compactons are a special case of the much studied solitons. The soliton is a familiar concept in non-linear physics. It is a stable, localized wave-solution, found in a range of different systems; from DNA-molecules to optical fibers. The compacton is a soliton that is completely localized, i.e. strictly zero outside a given area. The dynamics of the (extended) Bose-Hubbard model is based on the tunneling of particles between the lattice sites. The ordinary Bose-Hubbard model only accounts for one-particle tunneling processes. We will consider a model that also takes some two-particle tunneling processes into account, basically by considering long-range effects of the particle interaction. The aim of this thesis is to find and study the quantum analog of the compactons found in an extended Discrete Non-Linear Schrödinger equation. We will study analytical solutions and try to find if and under which conditions specific compactons exist. Numerical calculations are made to study the properties of the compactons and to study how compacton solutions arise in the classical limit.

Ultracold atoms in optical lattices offer an important tool for studying dynamics in many-body interacting systems in a pristine environment. This thesis focuses on three theoretical works motivated by recent optical lattice experiments. In the first, we theoretically study the center of mass dynamics of states derived from the disordered Bose-Hubbard model in a trapping potential. We find that the edge states in the trap allow center of mass motion even with insulating states in the center. We identify short and long-time mechanisms for edge state transport in insulating phases. We also argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our zero temperature results offer important insights into mechanisms of transport of atoms in trapped optical lattices while putting bounds on center of mass dynamics expected at non-zero temperature. In the second work, we study the domain wall expansion dynamics of strongly interacting bosons in 2D optical lattices with disorder in a recent experiment {[}J.-y. Choi et al., Science 352, 1547 (2016)]. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system. The last work features our study of phase diagrams of the 2D Bose-Hubbard model in an optical lattice with synthetic spin-orbit coupling. We investigate the transitions between superfluids with different phase patterns, which may be detected by measuring the spin-dependent momentum distribution. / Ph. D. / Ultracold atoms in optical lattices, a periodic potential generated by laser beams, offer an important tool for quantum simulations in a pristine environment. Motivated by recent optical lattice experiments with the implementation of disorder and synthetic spin-orbit coupling, we utilize Gutzwiller mean-field theory (GMFT) to study the dynamics of disordered state in an optical lattice under the sudden shift of the harmonic trap, the domain wall expansion of strongly interacting bosons in 2D lattices with disorder, and spin-orbit-driven transitions in the Bose-Hubbard model. We argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our findings show that evidence for many-body localization claimed in experiments [J.-y. Choi et al., Science 352, 1547 (2016)] must lie in the differences between GMFT and experiments. We also find that strong spin-orbit coupling alone can generate superfluids with finite momentum and staggered phase patterns.

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.

Cette thèse étudie les phases classiques et quantiques des systèmes atomiques ou moléculaires de basse dimension en mettant un accent particulier sur le crossover dimensionnel de une à deux dimensions.La première partie de la thèse est consacrée à la description d'un système d'atomes froids interagissants avec un potentiel de contact. Plus précisément, nous étudions le dé-piégeage dynamique qui, suite à l'extinction rapide d'un réseau optique, s'opère dans un gaz composé de bosons impénétrables dans un guide d'onde atomique linéaire.  Nous employons une solution exacte, basée sur une  correspondance entre bosons en forte interaction et fermions sans interaction pour déduire l'évolution dynamique quantique exacte. Dans la limite thermodynamique, nous observons l'approche vers un état stationnaire hors équilibre, caractérisé par l'absence d'ordre hors diagonal à longue distance et une visibilité réduite de la distribution en impulsions. Des caractéristiques similaires sont observées dans un système de taille finie pour des temps  correspondant à la moitié du temps de récurrence, lors desquels nous observons que le système approche un état  quasi-stationnaire auquel le système s'approche avec une dépendance temporelle en loi de puissance.La deuxième partie de la thèse analyse l'effet des interactions dipolaires sur l'état fondamental du système. L'inclusion de l'interaction dipôle-dipôle donne lieu à de nouvelles phases quantiques du système unidimensionnel, mais peut également entraîner une instabilité transverse.Cette instabilité est tout d'abord analysée dans le régime classique. Nous considérons des particules classiques  avec interactions dipolaires, confinés sur un anneau par un potentiel harmonique radiale.  Les dipôles sont polarisés perpendiculairement au plan de confinement. En diminuant le confinement dans la direction radiale, les particules classique montrent une transition entre une chaîne simple et une chaîne double (en zigzag).  Nous montrons que cette transition est faiblement du premier ordre. Nous expliquons que la nature de cette transition est déterminée par le couplage entre les modes d'excitation transversaux et axiaux de la chaîne des dipôles.  Ce résultat est très différent du  comportement observé dans les systèmes Coulombiens, où la transition entre la chaîne linéaire et la chaîne en zigzag est continue et appartient à la classe d'universalité de la transition ferromagnétique. Nos résultats s'appliquent aux systèmes dipolaires classiques et aux atomes Rydberg, qui peuvent constituer un banc d'essai pour simuler le comportement critique des aimants couplés à des grilles.Dans le régime quantique, nous considérons un système des bosons dipolaires sur un réseaux optique, confinés par un potentiel harmonique anisotrope. Dans le régime favorisant l'instabilité d'une chaîne simple, nous démontrons que le système peut être  décrit  par un modèle de  Bose-Hubbard étendu à plusieurs modes couplés entre eux, dont les coefficients peuvent être déterminés en utilisant une théorie de basse énergie. La méthode d'intégrale de chemin Monte Carlo, la diagonalisation exacte et TEBD sont utilisés pour déterminer l'état fondamental de modèle de Bose-Hubbard étendu et démontrent que ce modèle capture la transition entre la chaîne linéaire et la chaîne en zigzag. / In this work, the classical and quantum phases of low-dimensional atomic or molecular systems is studied with a particular focus on the regime where a system goes over from a strictly one-dimensional to a two dimensional system.The first part of the thesis is dedicated to atoms interacting via contact interactions. In particular, we study the dynamical depinning following a sudden turn off of an optical lattice for a gas of impenetrable bosons in a tight atomic waveguide. We use an exact solution, which is based on an equivalence between strongly interacting bosons and noninteraction fermions, in order to derive the exact quantum dynamical evolution. At long times, in the thermodynamic limit, we observe the approach to a nonequilibrium steady state, characterized by the absence of quasi-long-range order and a reduced visibility in the momentum distribution. Similar features are found in a finite-size system at times corresponding to half the revival time, where we find that the system approaches a quasisteady state with a power-law behavior.In the second part, we study the effect of additional dipolar interactions on the ground state of the system. The inclusion of dipole-dipole interaction leads to new quantum phases of the one-dimensional system, but can also lead to a transverse instability.This instability is first analyzed in the classical regime. We study classical particles with dipolar interactions, that are confined on a chain by a harmonic potential. The dipoles are polarised perpendicular to the plane of confinement. Classical particles with repulsive power-law interactions undergo a transition from a single to a double chain (zigzag) by decreasing the confinement in the transverse direction. We theoretically characterize this transition when the particles are classical dipoles, polarized perpendicularly to the plane in which the motion occurs, and argue that this transition is of first order, even though weakly. The nature of the transition is determined by the coupling between transverse and axial modes of the chain and contrasts with the behavior found in Coulomb systems, where the linear-zigzag transition is continuous and belongs to the universality class of the ferromagnetic transition. Our results hold for classical dipolar systems and Rydberg atoms, which can offer a test bed for simulating the critical behavior of magnets with lattice coupling.In the quantum regime, we consider dipolar bosons in an optical lattice, tightly confined by an anisotropic harmonic potential. In the regime where a single chain becomes unstable, we show that the system can be mapped onto an extended multi-mode Bose-Hubbard model, where the coefficients can be determined by means of a low energy theory. A path integral Monte Carlo method, exact diagonalization and TEBD are used to determine the ground state of the extended Bose-Hubbard models. and show that the model captures the linear to zigzag transition.

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.

Orientador: Eduardo Miranda / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física "Gleb Wataghin" / Made available in DSpace on 2018-09-24T13:52:11Z (GMT). No. of bitstreams: 1 HerazoWarnes_JesusMaria_D.pdf: 4836710 bytes, checksum: 5b7290f1db20bc31b153f3e7202fff39 (MD5) Previous issue date: 2011 / Resumo: A questão da natureza das transições de fases de sistemas de redes de bósons tem se tornado cada vez mais urgente à vista da capacidade de carregamento de átomos ultrafrios em redes ópticas. Nesta tese, tentamos avançar este conhecimento através do estudo de 3 modelos básicos de redes de bósons interagentes. Inicialmente, determinamos o diagrama de fases c as propriedades físicas do modelo bosônico de impureza única de Anderson. Este modelo é interessante tanto em si mesmo quanto por causa de sua relação com outras abordagens teóricas tais como a teoria dinâmica de campo médio bosônica. Usamos como estratégia a inclusão de um pequeno campo externo acoplado ao parâmetro de ordem superfluido, que quebra a simetria global de calibre do modelo. Desta forma, foi possível estudar a transição de condensação de Base-Einstein através do critério de quebra espontânea de simetria global de calibre. Outras quantidades como a ocupação da impureza, o desvio padrão da ocupação e a susceptibilidade com respeito ao campo externo também foram calculadas, caracterizando a transição de fase do modelo. Alguns desses resultados foram comparados com aqueles já obtidos na literatura através do grupo de renormalização numérico. Encontramos bom acordo entre os dois métodos. O segundo estudo realizado nesta tese refere-se ao comportamento crítico do modelo de Bose-Hubbard desordenado através da chamada teoria de campo médio estocástica. O objeto central dessa teoria de campo médio é a distribuição de parâmetros de ordem P(?). Estudos numéricos estabelecem que perto da linha crítica que separa as fases superfluida e vidro de Base do modelo, essa distribuição exibe uma grande região com comportamento de lei de potência P(?) ~ ? ^-(1+ß_c), onde ß_c < 1. Usando esse comportamento como tentativa, obtivemos analiticamente tanto a fronteira de fases quanto o valor do expoente crítico da lei de potência ß_c , encontrando um razoável acordo com os resultados numéricos e avançando o entendimento da natureza da transição de fase específica ao modelo desordenado. Finalmente, o modelo de Bose-Hubbard desordenado para partículas de spin-1 foi estudado dentro da teoria de campo médio estocástica. As distribuições de probabilidade de várias quantidades físicas como o parâmetro de ordem superfluido, o desvio padrão da ocupação por sítio, a fração do condensado, o quadrado do operador de spin, bem como seus valores médios, foram determinados para as três fases do modelo, a saber, o superfluido polar, o isolante de Mott e o vidro de Bose. Uma completa caracterização das propriedades físicas dessas fases e das transições de fase entre elas foi estabelecida / Abstract: The question of the nature of phase transitions of systems of lattice bosons has become increasingly more pressing in view of the capability of loading ultracold atoms in opticallattices. In this thesis we try to advance this understanding through the study of 3 basic models of interacting lattice bosons. Initially, we determined the phase diagram and physical properties of the bosonic singleimpurity Anderson model. This model is interesting both in its own right and because of its relation to other theoretical approaches such as the bosonic dynamical field theory method. We used as strategy the inclusion of a small external field coupled to the superfluid order parameter, which breaks the global gauge symmetry of the model. Thus, it was possible to study the Base-Einstein condensation transition through the criterion of the onset of spontaneous broken global gauge symmetry. Other quantities such as the occupation of the impurity, the standard deviation of the occupation and the susceptibility with respect to the external! Field were calculated characterizing the phase transition in the model. Some of the results were compared with those already reported in the literature, obtained with tic numerical renormalization group. We found good agreement between the two methods. The second study carried out in this thesis concerned the critical behavior of the disordered Bose-Hubbard model within the so-called stochastic mean-field theory. The central object of this mean-field theory is the distribution of order parameters P(?). Numerical studies establish that near the critical line separating the superfluid and Bose glass phases of this model, this distribution shows a wide region of power-law behavior P(?) ~ ? ^-(1+ß_c), where ß_c < 1. Using this behavior as an Ansatz, we obtained analytically both the phase boundary and the value of the critical power-law exponent ß_c, finding a reasonably good agreement with the numerical results and thus shedding new light on the nature of this phase transition specific to disordered model. Finally, the disordered Bose-Hubbard model for spin-1 particles was studied within the stochastic mean-field theory. The probability distributions of various physical quantities, such as the superfluid order parameter, the standard deviation of the occupation per site, the condensate fraction, the square of the spin operator, as well as their average values, were determined for the three phases of the model, namely, the polar superfluid, the Mott insulating and the Bose glass phases. A complete characterization of the physical properties of these phases and the phase transitions between them was then established / Doutorado / Física da Matéria Condensada / Doutor em Ciências

No description available.

La réalisation de condensats de Bose-Einstein et de gaz de Fermi dégénérés ont déclenché d'énormes progrès dans les méthodes théoriques ainsi que dans la mise en place de nouvelles techniques expérimentales. Parmi celles-ci, de fascinantes possibilités viennent de l'implémentation de réseaux optiques : potentiels périodiques pour atomes neutres créés à travers l'interférence de rayons laser. Un gaz dégénéré dans un réseau optique peut être forcé dans des pièges fortement anisotropes, jusqu'à réduire la dimensionnalité du système physique. Du point de vue fondamental, le comportement des ondes de matière en dimensions réduites éclaircit les propriétés intrinsèques des interactions entre particules. En outre, ces systèmes à dimensionnalité réduite peuvent être manipulés afin de créer des simulateurs quantiques de la matière condensée, comme par exemple des réseaux à deux dimensions, dans un environnement pur et contrôlable. Motivés par les passionnantes perspectives de ce domaine, on a consacré cette Thèse à l'étude théorique de deux systèmes dans lesquels une onde de matière se propage en dimensions réduites. L'interaction dipôle-dipôle, à longue portée et anisotrope, affecte fortement le comportement des gaz quantiques. Les progrès expérimentaux dans ce domaine florissant permettront bientôt de piéger dans des réseaux optiques un gaz dégénéré de dipôles. Dans la première partie de cette thèse, on considère l'apparition d'une seule résonance dipolaire dans l'interaction entre deux particules pour différents systèmes quasi-unidimensionnels. On propose une approche à deux canaux qui décrit cette résonance dans un piège harmonique fortement allongé “en forme de cigare”, qui représente l'approximation d'un site d'un réseau optique quasi-unidimensionnel. A` ce stade, on développe un nouveau modèle étendu de Bose-Hubbard atome-dimère, qui est valable pour des bosons dipolaires dans un réseau optique quasi-unidimensionnel. On étudie donc le diagramme de phase du modèle pour T =0 par la diagonalisation exacte de systèmes de petite taille, en soulignant les effets de la résonance dipolaire sur la physique à plusieurs corps dans le réseau. Dans la seconde partie de la thèse, on propose un modèle pour réaliser des simulateurs quantiques de cristaux bidimensionnels avec des atomes froids, basé sur le piégeage indépendant de deux espèces atomiques. La première constitue une onde de matière bidimensionnelle qui interagit exclusivement avec les atomes de la seconde espèce, piégés aux nœuds d'un réseau optique bidimensionnel. En introduisant une approche théorique générale, on examine les propriétés de transport de l'onde de matière. On propose des exemples d'application pour réseaux soit de Bravais (carré, triangulaire), soit de non-Bravais (graphène, kagomé), en étudiant soit des systèmes périodiques idéaux, soit des systèmes de taille expérimentale et désordonnés. Les caractéristiques d'un réseau atomique artificiel dépendent de l'intensité de l'interaction entre les deux espèces, qu'on montre être largement réglable grâce à des résonances à dimensionnalité mixte de type 0D-2D. / The experimental achievement of Bose-Einstein condensation and Fermi degeneracy with ultracold gases boosted tremendous progresses both in theoretical methods and in the development of new experimental tools. Among them, intriguing possibilities have been opened by the implementation of optical lattices: periodic potentials for neutral atoms created by interfering laser beams. Degenerate gases in optical lattices can be forced in highly anisotropic traps, reducing the effective dimensionality of the system. From a fundamental point of view, the behavior of matter waves in reduced dimensions sheds light on the intimate properties of interparticle interactions. Furthermore, such reduced-dimensional systems can be engineered to quantum-simulate fascinating solid state systems, like bidimensional crystals, in a clean and controllable environment. Motivated by the exciting perspectives of this field, we devote this Thesis to the theoretical study of two systems where matter waves propagate in reduced dimensions.The long-range and anisotropic character of the dipole-dipole interaction critically affects the behavior of dipolar quantum gases. The continuous experimental progresses in this flourishing field might lead very soon to the creation of degenerate dipolar gases in optical potentials. In the first part of this Thesis, we investigate the emergence of a single dipolar-induced resonance in the two-body scattering process in quasi-one dimensional geometries. We develop a two-channel approach to describe such a resonance in a highly elongated cigar-shaped harmonic trap, which approximates the single site of a quasi-one- dimensional optical lattice. At this stage, we develop a novel atom-dimer extended Bose- Hubbard model for dipolar bosons in this quasi-one-dimensional optical lattice. Hence we investigate the T=0 phase diagram of the model by exact diagonalization of a small- sized system, highlighting the effects of the dipolar-induced resonance on the many-body behavior in the lattice.In the second part of the Thesis, we present a general scheme to realize cold-atom quantum simulators of bidimensional atomic crystals, based on the possibility to independently trap two different atomic species. The first one constitutes a two-dimensional matter wave which interacts only with the atoms of the second species, deeply trapped around the nodes of a two-dimensional optical lattice. By introducing a general analytic approach, we investigate the matter-wave transport properties. We propose some illustrative appli- cations to both Bravais (square, triangular) and non-Bravais (graphene, kagomé) lattices, studying both ideal periodic systems and experimental-sized, eventually disordered, ones. The features of the artificial atomic crystal critically depend on the two-body interspecies interaction strength, which is shown to be widely tunable via 0D-2D mixed-dimensional resonances.

Orientador: Marcos César de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-18T12:10:11Z (GMT). No. of bitstreams: 1 Farias_ReginaldodeJesusCosta_D.pdf: 4228413 bytes, checksum: 9e7ff892eed5441839ee091ddb080501 (MD5) Previous issue date: 2011 / Resumo: Partindo de uma derivação matemática do modelo de Bose-Hubbard, desenvolvemos os cálculo para o emaranhamento bi e multipartido, através da transição Isolante de Mott-superuida, entre os modos de uma rede óptica para as situações mais simples de dois, três e quatro átomos (N) depositados nestas com igual número de sítios M, ocasionando um fator de preenchimento ? = N=M unitário. Apresentamos uma investigação sobre o controle da dinâmica de um condensado de Bose-Einstein aprisionado em um poço duplo através da ação de um potencial externo dependente do tempo. Apresentamos também uma investigação preliminar de codificação e operações quânticas embasadas em colisões controladas entre átomos em múltiplos poços / Abstract: Starting from a mathematical derivation of the Bose-Hubbard Model (BHM) we analyze the developing of bipartite and multipartite entanglement through the Mott insulator-superuid quantum phase transition, among the modes of an optical lattice to the simplest situations of two, three and four atoms (N) deposited in such optical lattice with equal number of sites (M), where a filling factor ? = N=M = 1 per lattice site is considered. We present an investigation about the controlled dynamic of a Bose-Einstein condensate in a double well driven by an external time dependent potential. Beyond we present some preliminar notes on codi cations and quantum operations in cold controlled collisions among atoms in multiples wells / Doutorado / Física da Matéria Condensada / Doutor em Ciências