An optical lattice is the periodic potential that atoms experience via the ac-Stark shift when they are illuminated by counter-propagating laser beams that form a standing-wave pattern. Optical lattices have been widely used as a versatile platform in cooling, trapping, controlling atoms, and also for the study of a variety of problems in physics. The many-body states of ultracold atoms in optical lattices can be characterized by the quantum-correlations encoded in time-of-flight images. In this thesis, we mainly discuss the use of correlations function as a natural framework for characterizing quantum states in optical lattices.\nThe outline of the thesis is as follows. Chapter 1 gives a brief introduction to optical lattice potentials and ways for describing particles moving in a periodic potential. Chapter 2 explains the importance of correlations and discusses common methods to detect them in cold atom experiments.\nChapter 3 presents work done to study the many-body Schrödinger equation in a quasiperiodic potential and discusses its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We posed a possible visualization of such a connection in experimentally accessible many-body observables. These observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson, and band-insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil's staircases. Our numerical treatment is complemented with a perturbative analysis that provides insight on the underlying physics. The perturbative-theory approach is particularly useful in illuminating the distinction between the Anderson-insulator and the band-insulator phases in terms of paired sets of dimerized states.\nChapter 4 discusses several theoretical procedures developed to understand a recent experiment on macroscopic quantum self-trapping (ST) performed in a 2D optical lattice. Mean field and truncated-Wigner-approximation (TWA) calculations are performed trying to reproduce the experimental observations. The discrepancy between the theory and the experiment lead to the hypothesis of a new type of ST caused by strong correlations. We analyze toy models to support it.
An optical lattice is the periodic potential that atoms experience via the ac-Stark shift when they are illuminated by counter-propagating laser beams that form a standing-wave pattern. Optical lattices have been widely used as a versatile platform in cooling, trapping, controlling atoms, and also for the study of a variety of problems in physics. The many-body states of ultracold atoms in optical lattices can be characterized by the quantum-correlations encoded in time-of-flight images. In this thesis, we mainly discuss the use of correlations function as a natural framework for characterizing quantum states in optical lattices.\nThe outline of the thesis is as follows. Chapter 1 gives a brief introduction to optical lattice potentials and ways for describing particles moving in a periodic potential. Chapter 2 explains the importance of correlations and discusses common methods to detect them in cold atom experiments.\nChapter 3 presents work done to study the many-body Schrödinger equation in a quasiperiodic potential and discusses its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We posed a possible visualization of such a connection in experimentally accessible many-body observables. These observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson, and band-insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil's staircases. Our numerical treatment is complemented with a perturbative analysis that provides insight on the underlying physics. The perturbative-theory approach is particularly useful in illuminating the distinction between the Anderson-insulator and the band-insulator phases in terms of paired sets of dimerized states.\nChapter 4 discusses several theoretical procedures developed to understand a recent experiment on macroscopic quantum self-trapping (ST) performed in a 2D optical lattice. Mean field and truncated-Wigner-approximation (TWA) calculations are performed trying to reproduce the experimental observations. The discrepancy between the theory and the experiment lead to the hypothesis of a new type of ST caused by strong correlations. We analyze toy models to support it.
This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting system and going into the many-body physics that describes the strongly correlated Mott insulator regime. Even though this thesis is a theoretical work all the chapters are linked either with experiments already done or with ongoing experimental efforts. This thesis can be divided into four different parts. The first part comprises chapters 1 to 3. In these chapters, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials. The second part deals with the superfluid weakly interacting regime where standard mean field techniques can be applied. This is covered in chapters 4 and 5. Specifically, chapter 4 introduces the discrete nonlinear Schroedinger equation (DNLSE) and uses it to model some experiments. In chapter 5 I go one step further and include the small quantum fluctuations neglected in the DNLSE by studying quadratic approximations of the Bose-Hubbard Hamiltonian. Chapters 6 to 8 can be grouped as the third part of the thesis. In them I adopt an effective action formalism, the so called two particle irreducible effective action (2PI) together with the closed time path (CTP) formalism to study far-from-equilibrium dynamics. The many-body techniques discussed in these chapters systematically include higher order quantum corrections, not included in the quadratic approximations of the Hamiltonian, which we show are crucial for a correct description of the quantum dynamics outside the very weakly interacting regime. Finally, chapter 9 to 11 are devoted to study the Mott insulator phase. In these chapters using perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the system to Bragg spectroscopy, and propose a mechanism to correct for the residual quantum coherences inherent to the Mott insulator ground state. Even though small these are not ideal for the use of neutral atoms in optical lattice as a tool for quantum computation.
Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up.
Ultracold atoms in optical lattices provide an extremely clean and controllable setting to explore quantum many-body phases of matter. Here we give a brief review of the strong-correlation physics that has been realized using such ultracold atoms in optical lattices ranging from the realization of Hubbard models to studies of quantum magnetism and the detection of single atoms with lattice site resolution. All this has opened up fundamentally new opportunities for the investigation of quantum many-body systems. Introduction Over the past years, ultracold atoms in optical lattices have emerged as versatile new systems to explore the physics of quantum many-body systems. On the one hand, they can be helpful in gaining a better understanding of known phases of matter and their dynamical behavior; on the other hand, they allow one to realize completely novel quantum systems that have not been studied before in nature [1, 2, 3]. Commonly, the approach of exploring quantum many-body systems in such a way is referred to as “quantum simulations.” Examples of some of the first strongly interacting many-body phases that have been realized both in lattices and in the continuum include the quantum phase transition from a superfluid to a Mott insulator [4, 5, 6], fermionic Mott insulators [7, 8], the achievement of a Tonks- Girardeau gas [9, 10], and the realization of the Bose-Einsten condensate (BEC)– Bardeen-Cooper-Schrieffer (BCS) crossover (see also Chapter 12) in Fermi gas mixture [11] using Feshbach resonances [12]. In almost all of these experiments, detection was limited to time-of-flight imaging or more refined derived techniques that mainly characterized the momentum distribution of the quantum gas [2]. For several years, researchers in the field have therefore aspired to employ in situ single-particle detection methods for the analysis of ultracold quantum gases. Only recently has it become possible to implement such imaging techniques, marking a milestone for the characterization and manipulation of ultracold quantum gases [13, 14, 15, 16, 17]. In our discussion, we will focus on one of the most successful of these techniques based on high-resolution fluorescence imaging. Despite being a rather new technique, such quantum gas microscopy has already proven to be an enabling technology for probing and manipulating quantum many-body systems. For the first time, controllable and strongly interacting many-body systems, as realized with ultracold atoms, could be observed on a local scale [17, 16].
Ultracold atomic gases in optical lattices provide an unique framework to study quantum phenomena in strongly correlated systems. In addition to the precise control over all relevant parameters in the experiment, these experiments can be mapped to a fundamental model from solid-state physics. For the bosonic version of this model, the so-called Bose-Hubbard model, a phase transition from a superfluid to a Mott insulator was theoretically predicted and later experimentally observed in an ultracold gas of Rb-87 atoms in three-dimensional as well as in one-dimensional optical lattices. Apart from homogeneous optical lattices one can introduce more complex lattice topologies such as two-color superlattices which give rise to a rich phase diagram including more exotic phases like the Bose-glass. We employ various powerful many-body techniques like exact diagonalization in complete and truncated Hilbert spaces and the Density-Matrix Renormalization Group (DMRG) algorithm to study the phase diagrams of the one-dimensional Bose-Hubbard and the one-dimensional Bose-Fermi-Hubbard Hamiltonian. Most theoretical studies of these systems discuss the phase diagrams with respect to the generic parameters of the Hubbard model. These Hubbard parameters, however, depend non-trivially on the control parameters used in experiments. The focus of this work is on the ab-initio calculation of the phase diagram of ultracold Rb-87 in one-dimensional optical superlattices starting directly from the experimental setup. To this end, we first employ band-structure calculations to extract the Hubbard parameters from the experimental parameters. Then, we use state-of-the-art DMRG techniques to solve the many-body problem for realistic particle numbers and lattice sizes that occur in experiments. Our results show that by using the intensities of the two laser fields forming the two-color superlattice as control parameters while keeping all other experimental parameters fixed, it is possible to access all relevant quantum phases of the system. Furthermore, we found out that the critical values of the laser intensities for the different phase transitions depend strongly on a third parameter that has to be included for a realistic description of the experiment. This third parameter is the strength of a harmonic trapping potential which accounts for the Gaussian shape of the laser fields and an additional magnetic potential used to confine the atoms in the center of the trap.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
Ultracold atoms in optical lattices have proven to be a versatile model system in which strong correlation effects in a many-body quantum system can be studied [1]. The tunability and control over almost all underlying system parameters makes them rather unique and allows one to switch between different ground states of the underlying many-body quantum system. One of the most prominent effects in this respect is the quantum phase transition (QPT) from a superfluid (SF) to a Mott insulator (MI) that started the field of strong-correlation physics with ultracold gases. First theoretically proposed by Fisher et al. [2], the Bose-Hubbard model (BHM) was introduced into the field of ultracold atoms by Jaksch et al. [3], eventually being realized in subsequent experiments [4-7]. This article focuses on giving a brief summary of the main results achieved in the SF-MI transition with ultracold bosons in optical lattices. A brief introduction into the setup of optical lattices is followed by a discussion of key results such as the coherence properties, the density profiles, transport properties and SF characteristics across the QPT.