Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Annalen der Physik

The Bose‐Hubbard model: from Josephson junction arrays to optical lattices

Christoph Bruder, Rosario Fazio, Gerd Schön

Published Aug 18, 2005• 14 citations
Publisher

Abstract

Abstract Dedicated to Bernhard Mühlschlegel on the occasion of his 80th birthday The Bose‐Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We review some of its properties with emphasis on the implications on quantum phase transitions of Josephson junction arrays and quantum dynamics of topological excitations as well as the properties of ultra‐cold atoms in optical lattices.

DOI
10.1002/andp.200551709-1005

Related papers

Cold bosons in optical lattices

Vyacheslav I. Yukalov

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.

arXiv:0901.0636•Laser Physics•Jan 1, 2009•208 citations

Zoo of Quantum Phases and Excitations of Cold Bosonic Atoms in Optical Lattices

Ofir E. Alon, Alexej I. Streltsov, L. S. Cederbaum

Quantum phases and phase transitions of weakly to strongly interacting bosonic atoms in deep to shallow optical lattices are described by a single multiorbital mean-field approach in real space. For weakly interacting bosons in one dimension, the critical value of the superfluid to Mott insulator (MI) transition found is in excellent agreement with many-body treatments of the Bose-Hubbard model. For strongly interacting bosons, (i) additional MI phases appear, for which two (or more) atoms residing in each site undergo a Tonks-Girardeau-like transition and localize, and (ii) on-site excitation becomes the excitation lowest in energy. Experimental implications are discussed.

arXiv:cond-mat/0412734•Physical Review Letters•Jul 15, 2005•85 citations

The Bose-Hubbard model: from Josephson junction arrays to optical lattices

Christoph Bruder, Rosario Fazio, Gerd Schön

Annalen der Physik•Aug 18, 2005•51 citations

Dynamics of Bose-Einstein Condensates in an Optical Lattice - Some Surprising Results!

Sayan Choudhury

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.

•Jan 1, 2011•0 citations

Recent progress on quantum simulations of non-standard Bose–Hubbard models

Titas Chanda, Luca Barbiero, Maciej Lewenstein +2 more

In recent years, the systems comprising of bosonic atoms confined to optical lattices at ultra-cold temperatures have demonstrated tremendous potential to unveil novel quantum mechanical effects appearing in lattice boson models with various kinds of interactions. In this progress report, we aim to provide an exposition to recent advancements in quantum simulations of such systems, modeled by different 'non-standard' Bose-Hubbard models, focusing primarily on long-range systems with dipole-dipole or cavity-mediated interactions. Through a carefully curated selection of topics, which includes the emergence of quantum criticality beyond Landau paradigm, bond-order wave insulators, the role of interaction-induced tunneling, the influence of transverse confinement on observed phases, or the effect of cavity-mediated all-to-all interactions, we report both theoretical and experimental developments from the last few years. Additionally, we discuss the real-time evolution of systems with long-range interactions, where sufficiently strong interactions render the dynamics non-ergodic. And finally to cap our discussions off, we survey recent experimental achievements in this rapidly evolving field, underscoring its interdisciplinary significance and potential for groundbreaking discoveries.

arXiv:2405.07775•Reports on Progress in Physics•Mar 21, 2025•23 citations

Strong Correlation Effects with Ultracold Bosonic Atoms in Optical Lattices

Immanuel Bloch

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.

CRC Press eBooks•Nov 2, 2010•0 citations
Abstract Dedicated to Bernhard Mühlschlegel on the occasion of his 80th birthday The Bose‐Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We review some of its properties with emphasis on the implications on quantum phase transitions of Josephson junction arrays and quantum dynamics of topological excitations as well as the properties of ultra‐cold atoms in optical lattices.
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.
Quantum phases and phase transitions of weakly to strongly interacting bosonic atoms in deep to shallow optical lattices are described by a single multiorbital mean-field approach in real space. For weakly interacting bosons in one dimension, the critical value of the superfluid to Mott insulator (MI) transition found is in excellent agreement with many-body treatments of the Bose-Hubbard model. For strongly interacting bosons, (i) additional MI phases appear, for which two (or more) atoms residing in each site undergo a Tonks-Girardeau-like transition and localize, and (ii) on-site excitation becomes the excitation lowest in energy. Experimental implications are discussed.
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 recent years, the systems comprising of bosonic atoms confined to optical lattices at ultra-cold temperatures have demonstrated tremendous potential to unveil novel quantum mechanical effects appearing in lattice boson models with various kinds of interactions. In this progress report, we aim to provide an exposition to recent advancements in quantum simulations of such systems, modeled by different 'non-standard' Bose-Hubbard models, focusing primarily on long-range systems with dipole-dipole or cavity-mediated interactions. Through a carefully curated selection of topics, which includes the emergence of quantum criticality beyond Landau paradigm, bond-order wave insulators, the role of interaction-induced tunneling, the influence of transverse confinement on observed phases, or the effect of cavity-mediated all-to-all interactions, we report both theoretical and experimental developments from the last few years. Additionally, we discuss the real-time evolution of systems with long-range interactions, where sufficiently strong interactions render the dynamics non-ergodic. And finally to cap our discussions off, we survey recent experimental achievements in this rapidly evolving field, underscoring its interdisciplinary significance and potential for groundbreaking discoveries.
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.