Title: Modelling of Ultracold Gases in Multidimensional Optical Lattices Author: Miroslav Urbanek Department: Department of Chemical Physics and Optics Supervisor: doc. Ing. Pavel Soldán, Dr. Abstract: Optical lattices are experimental devices that use laser light to confine ultracold neutral atoms to periodic spatial structures. A system of bosonic atoms in an optical lattice can be described by the Bose-Hubbard model. Although there exist powerful analytic and numerical methods to study this model in one dimension, their extensions to multiple dimensions have not been as successful yet. I present an original numerical method based on tree tensor networks to simulate time evolution in multidimensional lattice systems with a focus on the two-dimensional Bose-Hubbard model. The method is used to investigate phenomena accessible in current experiments. In particular, I have studied phase collapse and revivals, boson expansion, and many-body localization in two-dimensional optical lattices. The outcome of this work is TEBDOL - a program for modelling one-dimensional and two-dimensional lattice systems. Keywords: Bose-Hubbard model, multidimensional system, optical lattice, tensor network

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Durant cette thèse, je me suis intéressé à deux thématiques générales qui peuvent être explorées dans des systèmes d'atomes froids : d'une part, la dynamique hors-équilibre d'un système quantique isolé, et d'autre part l'influence du désordre sur un système fortement corrélé à basse température. Dans un premier temps, nous avons développé une méthode de champ moyen, qui permet de résoudre la dynamique unitaire dans un modèle à géométrie particulière, le réseau complètement connecté. Cette approche permet d'établir une correspondance entre la dynamique unitaire du système quantique et des équations du mouvement classique. Nous avons mis à profit cette méthode pour étudier le phénomène de transition dynamique qui se signale, dans des modèles de champ moyen, par une singularité des observables aux temps longs, en fonction des paramètres initiaux ou finaux de la trempe. Nous avons montré l'existence d'une transition dynamique quantique dans les modèle de Bose-Hubbard, d'Ising en champ transverse et le modèle de Jaynes-Cummings. Ces résultats confirment l'existence d'un lien fort entre la présence d'une transition de phase quantique et d'une transition dynamique.Dans un second temps, nous avons étudié un modèle de théorie des champs relativiste avec symétrie O(N) afin de comprendre l'influence des fluctuations sur ces singularités. À l'ordre dominant en grand N, nous avons montré que la transition dynamique s'apparente à un phénomène critique. En effet, à la transition dynamique, les fonctions de corrélations suivent une loi d'échelle à temps égaux et à temps arbitraires. Il existe également une longueur caractéristique qui diverge à l'approche du point de transition. D'autre part, il apparaît que le point fixe admet une interprétation en terme de particules sans masse se propageant librement. Enfin, nous avons montré que la dynamique asymptotique au niveau du point fixe s'apparente à celle d'une trempe d'un état symétrique dans la phase de symétrie brisée. Le troisième volet de cette thèse apporte des éléments nouveaux pour la compréhension du diagramme des phases du modèle de Bose-Hubbard en présence de désordre. Pour ce faire,nous avons utilisé et étendu la méthode de la cavité quantique en champ moyen de Ioffe et Mézard, qui doit être utilisée avec la méthode des répliques. De cette manière, il est possible d'obtenir des résultats analytiques pour les exposants des lois de probabilité de la susceptibilité.Nos résultats indiquent que dans les différents régimes de la transition de phase de superfluide vers isolant, les lois d'échelle conventionnelles sont tantôt applicables, tantôt remplacées par une loi d'activation. Enfin, les exposants critiques varient continûment à la transition conventionnelle.

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

Cold bosonic atoms trapped in optical lattices formed by standing wave interference patterns of multiple laser beams constitute excellent emulators of models of strongly correlated quantum systems of bosons. In this thesis, we develop and deploy strong-coupling expansion (i.e., an expansion in terms of the ratio of the inter-site hopping amplitude of the bosons to the strength of their interactions) techniques for studying the properties of three different instances of such systems. In the first instance, we have used strong coupling expansion techniques to calculate the density pro le for bosonic atoms trapped in an optical lattice with an overall harmonic trap at finite temperatures and large on site interaction in the presence of super fluid regions. Our results match well with quantum Monte Carlo simulations at finite temperature. We present calculations for the entropy per particle as a function of temperature which can be used to calibrate the temperature in experiments. Our calculations for the scaled density in the vacuum-to-super fluid transition agree well with the experimental data for appropriate temperatures. We also discuss issues connected with the demonstration of universal quantum critical scaling in the experiments. Experimental realizations of “atomtronic" Josephson junctions have recently been created in annular traps in relative rotation with respect to potential barriers that generate the weak links. If these devices are additionally subjected to optical lattice potentials, then they can incorporate strong-coupling Mott physics within the design, which can modify the behaviour and can allow for interesting new configurations of system generated barriers and of super fluid ow patterns. we have examined theoretically the behavior of a Bose super fluid in an optical lattice in the presence of an annular trap and a barrier across the annular region which acts as a Josephson junction. As the fluid is rotated relative to the barrier, it generates circulating super-currents until, at larger speeds of rotation, it develops phase slips which are typically accompanied by vortices. We use a finite temperature strong-coupling expansion about the mean- held solution of the Bose Hubbard model to calculate various properties of the device. In addition, we discuss some of the rich behavior that can result when there are Mott regions within the system. Rubidium-Cesium dipolar molecule formation through Feshbach resonance is an area of great current interest, for, the dipolar molecules, once formed, interact via v long range dipolar forces, leading to possibilities of novel phases. Experimentalists currently make such systems mostly using trial and error, and the resulting efficiencies for molecule formation tend to be low. With a goal to assist cold-atom experimentalists to achieve higher e ciencies of molecule formation, we have estimated the trap parameters for Rb and Cs atoms in a 3D optical lattice required to create single occupancy per site Mott phase for both the species in the same regions of the trap. We thus identify the ne tuning of the external magnetic held near Rb-Cs Feshbach resonance required to achieve highest probability for creating single Rb-Cs Feshbach molecules in the system. We have used the Falicov-Kimball model to describe the relevant system and strong-coupling expansions about the mean- held solution to calculate the density pro les for both species and efficiency for molecule formation, determined by overlapping regions of single occupancy for both Rb and Cs, up to second order in the expansion. We also calculate the entropy per particle which serves as an estimation of the temperature in the experimental system

<p dir="ltr">Complexity has played an increasingly important role in recent years. In this dissertation, we study some notions of complexity in systems that exhibit critical behaviour. Our results show that complexity as it is generally understood in holographic and lattice models of criticality can have several ambiguities. But despite these ambiguities, there are some features that are universally true. On the phase diagram of the system, it is the critical point which has the most complex ground state. States of physical systems with a large complexity tend to be hard to simulate using quantum circuits. Near the critical point, there is a part of complexity which is non-analytic and scales universally, i.e, the scaling is independent of the microscopic details of the Hamiltonian but depends only on the dimensionality of the system, and of the deforming operator. The coefficient of this term is unambiguous, i.e, it is not affected by the various changes in the definition of complexity which plague all the analytic terms near the critical point. We show this in lattice, field-theoretic and holographic calculations. These results were first presented in our earlier studies.</p>

This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. This thesis can be divided into five different parts. In Chapter 1, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials and a short overview of the experiments on ultracold atoms in an optical lattice. In Chapter 2 we develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this poten¬tial, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator(MI), density-wave(DW), and supersolid (SS) phases in the plane of the chemical potential and on-site repulsion ; we present phase diagrams for representative values of , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI ,DW ,and SSphases. We explore the implications of our study for experiments on cold-atom dipolar con¬densates in optical lattices in a confining potential. In Chapter3 we present an extensive study of Mottinsulator( MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with har¬monic traps. For this we develop an inhomogeneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees quantitatively with quan¬tum Monte Carlo(QMC) simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional(3D) systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also generalize our inhomogeneous mean-field theory to study BH models with har¬monic traps and(a) two species of bosons or(b) spin-1bosons. With two species of bosons we obtain rich phase diagrams with a variety of SF and MI phases and as¬sociated shells, when we include a quadratic confining potential. For the spin-1BH model we show, in a representative case, that the system can display alternating shells of polar SF and MI phases; and we make interesting predictions for experi¬ments in such systems. . In Chapter 4 we carry out an extensive study of the phase diagrams of the ex-tended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane ¬insulator(HI) phases. We show that the critical exponents and central charge for the HI-DW,MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) uni¬versality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different bound¬ary conditions lead to different phase diagrams.. In Chapter 5 we obtain the excitation spectra of the following three generalized of Bose-Hubbard(BH) models:(1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we show how to obtain excitation spectra by using the random phase approximation (RPA). We compare the results of our work with earlier studies of related models and discuss implications for experiments.

Durant cette thèse, je me suis intéressé à deux thématiques générales qui peuvent être explorées dans des systèmes d’atomes froids : d’une part, la dynamique hors-équilibre d’un système quantique isolé, et d’autre part l’influence du désordre sur un système fortement corrélé à basse température. Dans un premier temps, nous avons développé une méthode de champ moyen, qui permet de résoudre la dynamique unitaire dans un modèle à géométrie particulière, le réseau complètement connecté. Cette approche permet d’établir une correspondance entre la dynamique unitaire du système quantique et des équations du mouvement classique. Nous avons mis à profit cette méthode pour étudier le phénomène de transition dynamique qui se signale, dans des modèles de champ moyen, par une singularité des observables aux temps longs, en fonction des paramètres initiaux ou finaux de la trempe. Nous avons montré l’existence d’une transition dynamique quantique dans les modèle de Bose-Hubbard, d’Ising en champ transverse et le modèle de Jaynes-Cummings. Ces résultats confirment l’existence d’un lien fort entre la présence d’une transition de phase quantique et d’une transition dynamique.Dans un second temps, nous avons étudié un modèle de théorie des champs relativiste avec symétrie O(N) afin de comprendre l’influence des fluctuations sur ces singularités. À l’ordre dominant en grand N, nous avons montré que la transition dynamique s’apparente à un phénomène critique. En effet, à la transition dynamique, les fonctions de corrélations suivent une loi d’échelle à temps égaux et à temps arbitraires. Il existe également une longueur caractéristique qui diverge à l’approche du point de transition. D’autre part, il apparaît que le point fixe admet une interprétation en terme de particules sans masse se propageant librement. Enfin, nous avons montré que la dynamique asymptotique au niveau du point fixe s’apparente à celle d’une trempe d’un état symétrique dans la phase de symétrie brisée. Le troisième volet de cette thèse apporte des éléments nouveaux pour la compréhension du diagramme des phases du modèle de Bose-Hubbard en présence de désordre. Pour ce faire,nous avons utilisé et étendu la méthode de la cavité quantique en champ moyen de Ioffe et Mézard, qui doit être utilisée avec la méthode des répliques. De cette manière, il est possible d’obtenir des résultats analytiques pour les exposants des lois de probabilité de la susceptibilité.Nos résultats indiquent que dans les différents régimes de la transition de phase de superfluide vers isolant, les lois d’échelle conventionnelles sont tantôt applicables, tantôt remplacées par une loi d’activation. Enfin, les exposants critiques varient continûment à la transition conventionnelle. / The fast progress of cold atoms experiments in the last decade has allowed to explore new aspects of strongly correlated systems. This thesis deals with two such general themes: the out of equilibrium dynamics of closed quantum systems, and the impact of disorder on strongly correlated bosons at zero temperature. Among the different questions about out of equilibrium dynamics, the phenomenon of dynamical transition is still lacking a complete understanding. The transition is typically signalled, in mean-field, by a singular behaviour of observables as a function of the parameters of the quench. In this thesis, a mean field method is developed to give evidence of a strong link between the quantum phase transition at zero temperature and the dynamical transition. We then study using field theory techniques a relativistic O($N$) model, and show that the dynamical transition bears similarities with a critical phenomenon. In this context, the dynamical transition also appears to be formally related to the dynamics of symmetry breaking. The second part of this thesis is about the disordered Bose-Hubbard model and the nature of its phase transitions. We use and extend the cavity mean field method, introduced by Ioffe and Mezard to obtain analytical results from the quantum cavity method and the replica trick. We find that the conventional transition, with power law scaling, is changed into an activated scaling in some regions of the phase diagram. Furthermore, the critical exponents are continuously varying along the conventional transition. These intriguing properties call for further investigations using different methods.

Les systèmes fortement corrélés, pouvant adopter des phases surprenantes de la matière, émergent dans le domaine des atomes ultra-froids ou dans celui de l’électrodynamique quantique en cavité (CQED). Ceux-ci sont au centre d’intenses travaux expérimentaux et théoriques. Dans cette thèse, nous présentons une étude de deux modèles de bosons avec deux ou zéro états internes. Ceux-ci peuvent se déplacer sur un réseau, et sont localement couplés avec des spins 1/2. Notre intérêt réside dans la détermination du diagramme de phase de l’état fondamental de ces systèmes ainsi que de l’étude des propriétés de phase et des transitions entre ces dernières. Nous avons utilisé deux outils : une approximation de champ moyen et des simulations de Monte-Carlo quantique, qui fournit des résultats numériquement exacts. Le premier modèle, appelé modèle de Kondo bosonique sur réseau, s’inscrit dans le contexte des atomes ultra-froids sur réseau. Nous trouvons que sa physique est proche de celle du modèle de Bose-Hubbard, présentant des phases de Mott et superfluide. Le couplage local renforce le caractère isolant et on observe l’émergence de phases magnétiques au travers de couplage direct ou indirect entre bosons et/ou spins. Les effets thermiques, inhérents à tout dispositif expériemental, sont aussi étudiés. Le second modèle s’inscrit dans le domaine de la CQED sur réseau, décrit un régime de couplage ultra-fort entre des photons et des atomes, et est appelé modèle de Rabi sur réseau. Le diagramme de phase présente juste deux phases : une phase cohérente dans laquelle les spins locaux s’ordonnent ferromagnétiquement ainsi qu’une phase incohérente compressible paramagnétique / Strongly correlated systems, where new surprising phases of matter may appear both in the context of ultra-cold atoms and cavity quantum electrodynamics, are the focus of intense experimental and theoritical activity. In this thesis we present a study of two models of bosons with two or zero internal states, that is to say spin-1/2 or spin-0 bosons. These particles can move around a lattice, and they are locally coupled to immobile spins 1/2. Our interest was to determine the ground state phase diagram, study phase properties and quantum phase transitions. We used two methods: an approximate one using a mean field approach and the other using quantum Monte-Carlo simulations, which provides numerically exact results. The first model, namely the bosonic Kondo lattice model, is in the context of ultra-cold atoms in optical lattices. We found that its physics is close to that of the Bose-Hubbard model, exhibiting Mott and superfluid phases. The local coupling strengthens the insulating behaviour of the system and magnetism emerges through indirect or direct coupling between bosons. Thermal effects, inherent in experiments, are also studied. The second model, which is in the context of light-matter interaction, describes a situation of an ultra-strong coupling between spin-0 bosons (photons) and local spins 1/2 (two levels atoms) and is known as the Rabi lattice model. The phase diagram generally consists of only two phases: a coherent phase and a compressible incoherent one. The locals

Through our whole work we study the XY model with all its entirety, a particular spin model where spins are confined in a plane. We try to bring out a good understanding of this model with all different types of phases and phase transition, it undergoes in critical situations. We conceive of these external conditions from very different physical models like High Tc Superconductor, Ultracold atoms in optical lattice which are in focus of recent research. Firstly we model high Tc Superconductors with very simple 2D XY model to get an idea about the diamagnetic response exhibited by these materials when kept in a external magnetic field. This modeling is reasonable because most of the physics of cuprate High Tc Superconductors are governed by their 2D copper oxide planes which insists us to consider 2D models. Later we shifted to a more realistic 3D anisotropic XY model , as the coupling between cuprates plane may have a considerable role in devising physics of those materials. We particularly focus on the 2D to 3D crossover effect on magnetisation showed by these models, with keeping an eye on how all these can be relate to the experimentally acquired magnetisation profile of High Tc Supercondutors. On the second project we investigate on the phase diagram of a fully frustrated 2-leg ladder Bose Hubbard model. After mapping it properly to a classical model, a bi-layer Fully Frustrated XY model on square lattice, we found that the frustration leads to the emergence of a new phase "Chiral Mott insulator(CMI)" sandwiched between "Chiral Superfluid(CSF)" and "regular Mott insulator(MI)" phase. We divide the whole report into four parts. The first chapter is basically contain-ing introductory part comprising the motivation. In the second chapter we discuss various types of phases and phase transitions of the 2D & 3D XY models. We try to address their critical behaviors. In the third chapter and onwards we consider our model in external magnetic field and observe magnetisation in these systems. Here we specially focus on 2D to 3D crossover effect on magtisation measurement. Lastly in the fourth chapter we bring out a correspondence of XY model with the 2 leg ladder fully frustrated Bose Hubbard Model. There we report the emergence of a new phase, Chiral Mott Insulator(CMI) due to frustration in system.