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Bose-Hubbard model of cold atoms to higher orders

Laura Rinaldi

Published Sep 25, 2020• 0 citations
OpenAlex

Abstract

The aim of this work is to study the dynamics of cold atoms trapped in a one-dimensional periodic lattice generated by an optical potential. Such a system is described by a standard model in condensed matter, namely the so-called Bose-Hubbard model. Arrays of cold neutral atoms or ions are the basic elements of some proposals of quantum computers, which motivates the interest of understanding their dynamics from a mathematical point of view. The main targets of this Master Thesis are the following. First, starting from a classical Hamiltonian of the Frenkel-Kontorova type and thanks to Perturbation Theory, to derive the discrete Bougolyubov-Gross-Pitaevskii (dBGP) equation. Second, starting from first principle formulation, to provide a systematic, quantum mechanical, perturbative derivation of the Bose-Hubbard (BH) model to orders higher than the first one in the small parameter.

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In Chapter 1 we set up the physical and mathematical framework for the description of interacting bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian (BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field (classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview of the relevant literature and of the various physical systems that are captured by the BHH including, among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends with an outline of the remaining thesis. In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly, done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the system can show some rather interesting non-linear behavior. This includes features such as solitons and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical explanations are given for both the experiments. The damping of center-of-mass of the system is seen as a result of modulational instability. We give a different interpretation to this opening up lines for further investigations. We outline these at the end of the chapter. In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some calculations from the literature showing how the dissipation can be thought of as arising from quantum phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating stable and unstable regions of current which have been have obtained from mean-field calculations via the Gutzwiller ansatz. In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under renormalisation group in both cases and the supercurrent is revived. In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of phase fluctuations, which are encoded by quantum phase slips.

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Basic properties of cold Bose atoms in optical lattices are reviewed. The main principles of correct self-consistent description of arbitrary systems with Bose-Einstein condensate are formulated. Theoretical methods for describing regular periodic lattices are presented. A special attention is paid to the discussion of Bose-atom properties in the frame of the boson Hubbard model. Optical lattices with arbitrary strong disorder, induced by random potentials, are treated. Possible applications of cold atoms in optical lattices are discussed, with an emphasis of their usefulness for quantum information processing and quantum computing. An important feature of the present review article, distinguishing it from other review works, is that theoretical fundamentals here are not just mentioned in brief, but are thoroughly explained. This makes it easy for the reader to follow the principal points without the immediate necessity of resorting to numerous publications in the field.

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The aim of this work is to study the dynamics of cold atoms trapped in a one-dimensional periodic lattice generated by an optical potential. Such a system is described by a standard model in condensed matter, namely the so-called Bose-Hubbard model. Arrays of cold neutral atoms or ions are the basic elements of some proposals of quantum computers, which motivates the interest of understanding their dynamics from a mathematical point of view. The main targets of this Master Thesis are the following. First, starting from a classical Hamiltonian of the Frenkel-Kontorova type and thanks to Perturbation Theory, to derive the discrete Bougolyubov-Gross-Pitaevskii (dBGP) equation. Second, starting from first principle formulation, to provide a systematic, quantum mechanical, perturbative derivation of the Bose-Hubbard (BH) model to orders higher than the first one in the small parameter.
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.
In Chapter 1 we set up the physical and mathematical framework for the description of interacting bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian (BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field (classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview of the relevant literature and of the various physical systems that are captured by the BHH including, among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends with an outline of the remaining thesis. In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly, done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the system can show some rather interesting non-linear behavior. This includes features such as solitons and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical explanations are given for both the experiments. The damping of center-of-mass of the system is seen as a result of modulational instability. We give a different interpretation to this opening up lines for further investigations. We outline these at the end of the chapter. In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some calculations from the literature showing how the dissipation can be thought of as arising from quantum phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating stable and unstable regions of current which have been have obtained from mean-field calculations via the Gutzwiller ansatz. In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under renormalisation group in both cases and the supercurrent is revived. In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of phase fluctuations, which are encoded by quantum phase slips.
In this thesis, the physics of trapped, interacting Bose-Einstein condensates is analyzed by solving the many-body Schrödinger equation. Particular emphasis is put on coherence, fragmentation and reduced density matrices. First, the ground state of a trapped Bose-Einstein condensate and its correlation functions are obtained. Then the dynamics of a bosonic Josephson junction is investigated by solving the time-dependent many-body Schrödinger equation numerically exactly. These are the first exact results in literature in this context. It is shown that the standard approximations of the field, Gross-Pitaevskii theory and the Bose-Hubbard model fail at weak interaction strength and within their range of expected validity. For stronger interactions the dynamics becomes strongly correlated and a new equilibration phenomenon is discovered. By comparison with exact results it is shown that a symmetry of the Bose-Hubbard model between attractive and repulsive interactions must be considered an artefact of the model. A conceptual innovation of this thesis are time-dependent Wannier functions. Equations of motion for time-dependent Wannier functions are derived from the variational principle. By comparison with exact results it is shown that lattice models can be greatly improved at little computational cost by letting the Wannier functions of a lattice model become time-dependent.
In this thesis, we study interacting bosons on lattices consisting of a few sites. The main focus is on their response to external driving fields, their transport and decay properties. We address fundamental questions of quantum classical correspondence and investigate the implications of interactions in the transport properties of (ultra-)cold atom devices. Quantum mechanically, we describe bosons on lattices using a parametric Bose-Hubbard Hamiltonian while the corresponding classical limit is described by the discrete nonlinear Schrödinger equation.In the first part of this thesis we focus on the chaotic regime of the Bose-Hubbard Hamiltonian. We investigate the parametric change of eigenstates as the coupling strength between neighboring sites is varied, using perturbative and semiclassical methods. We then turn to the quantum evolution of driven interacting bosonic systems and study the resulting quantum dynamics and the irreversibility (fidelity) of the quantum motion.In the second part we initiate the study of quantum pumping/stirring in Bose-Einstein condensates with the aim to identify optimal pumping cycles and propose such a device in order to probe the interatomic interactions. Finally, in the last part we focus on the interplay of intrinsic dynamics with coupling to the continuum and investigate the structure of the resonance widths of a Bose-Hubbard dimer which is coupled to the continuum at one of the sites using an effective non-Hermitian Hamiltonian formalism.
We discuss applications of the theory of quantum chaos to one of the paradigm models of many-body quantum physics — the Bose–Hubbard (BH) model, which describes, in particular, interacting ultracold Bose atoms in an optical lattice. After preliminary, pure quantum analysis of the system we introduce the classical counterpart of the BH model and the governing semiclassical equations of motion. We analyze these equations for the problem of Bloch oscillations (BOs) of cold atoms where a number of experimental results are available. The paper is written for nonexperts and can be viewed as an introduction to the field.
Basic properties of cold Bose atoms in optical lattices are reviewed. The main principles of correct self-consistent description of arbitrary systems with Bose-Einstein condensate are formulated. Theoretical methods for describing regular periodic lattices are presented. A special attention is paid to the discussion of Bose-atom properties in the frame of the boson Hubbard model. Optical lattices with arbitrary strong disorder, induced by random potentials, are treated. Possible applications of cold atoms in optical lattices are discussed, with an emphasis of their usefulness for quantum information processing and quantum computing. An important feature of the present review article, distinguishing it from other review works, is that theoretical fundamentals here are not just mentioned in brief, but are thoroughly explained. This makes it easy for the reader to follow the principal points without the immediate necessity of resorting to numerous publications in the field.