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Existence, dynamics and mobility of Quantum Compactons in an extended Bose-Hubbard model

Peter Jason, Magnus Johansson

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Lattice Compactons, discrete breathers with compact support, were found for a discrete nonlinear Schrodinger (DNLS) equation extended with nearest neighbour intersite nonlinearities [1], a model originally studied with waveguide arrays in mind. These compactons were shown to exhibit very good mobility if the parameters are tuned close to the compactons stability boundary. The DNLS can also be used to model the behaviour of Bose-Einstein condensates in optical lattices, and the remarkable control over the experiments in this field of research has made it possible to study the quantum mechanics of strongly correlated atoms.We will define the concept of a Quantum Lattice Compacton [2] and discuss the existence and dynamics, with special emphasis on mobility [3], of these in an extended Bose-Hubbard model corresponding to above-mentioned extended DNLS equation in the quantum mechanical limit. The compactons is given 'a kick' by means of a phase-gradient and it is shown that the size of this phase is crucial for the mobility of the compactons. For small phase-gradients, corresponding to a slow coherent motion in the classical model, the time-scales of the quantum tunnelings become of the same order as the time-scale of the translational motion and the classical mobility is destroyed by quantum fluctuations. For large phase-gradients, corresponding to rapid classical motion, the classical and quantum time-scales separate so that a mobile, localized coherent quantum state can be translated many sites in the lattice already for small particle numbers of the order of 10 [3].Acknowledgements: This project has been financed by the Swedish Research Council.References[1] M. Oster, M. Johansson, and A. Eriksson 2003 Phys. Rev. E 67 056606[2] P. Jason and M. Johansson 2012 Phys. Rev. A 85 011603(R)[3] P. Jason and M. Johansson 2013 Phys. Rev. A 88 033605

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Lattice Compactons, discrete breathers with compact support, were found for a discrete nonlinear Schrodinger (DNLS) equation extended with nearest neighbour intersite nonlinearities [1], a model originally studied with waveguide arrays in mind. These compactons were shown to exhibit very good mobility if the parameters are tuned close to the compactons stability boundary. The DNLS can also be used to model the behaviour of Bose-Einstein condensates in optical lattices, and the remarkable control over the experiments in this field of research has made it possible to study the quantum mechanics of strongly correlated atoms.We will define the concept of a Quantum Lattice Compacton [2] and discuss the existence and dynamics, with special emphasis on mobility [3], of these in an extended Bose-Hubbard model corresponding to above-mentioned extended DNLS equation in the quantum mechanical limit. The compactons is given 'a kick' by means of a phase-gradient and it is shown that the size of this phase is crucial for the mobility of the compactons. For small phase-gradients, corresponding to a slow coherent motion in the classical model, the time-scales of the quantum tunnelings become of the same order as the time-scale of the translational motion and the classical mobility is destroyed by quantum fluctuations. For large phase-gradients, corresponding to rapid classical motion, the classical and quantum time-scales separate so that a mobile, localized coherent quantum state can be translated many sites in the lattice already for small particle numbers of the order of 10 [3].Acknowledgements: This project has been financed by the Swedish Research Council.References[1] M. Oster, M. Johansson, and A. Eriksson 2003 Phys. Rev. E 67 056606[2] P. Jason and M. Johansson 2012 Phys. Rev. A 85 011603(R)[3] P. Jason and M. Johansson 2013 Phys. Rev. A 88 033605
We study the dynamical properties, with special emphasis on mobility, of quantum lattice compactons (QLCs) in a one-dimensional Bose-Hubbard model extended with pair-correlated hopping. These are quantum counterparts of classical lattice compactons (localized solutions with exact zero amplitude outside a given region) of an extended discrete nonlinear Schr\"odinger equation, which can be derived in the classical limit from the extended Bose-Hubbard model. While an exact one-site QLC eigenstate corresponds to a classical one-site compacton, the compact support of classical several-site compactons is destroyed by quantum fluctuations. We show that it is possible to reproduce the stability exchange regions of the one-site and two-site localized solutions in the classical model with properly chosen quantum states. Quantum dynamical simulations are performed for two different types of initial conditions: ``localized ground states'' which are localized wave packets built from superpositions of compactonlike eigenstates, and SU(4) coherent states corresponding to classical two-site compactons. Clear signatures of the mobility of classical lattice compactons are seen, but this crucially depends on the magnitude of the applied phase gradient. For small phase gradients, which classically correspond to a slow coherent motion, the quantum time scale is of the same order as the time scale of the translational motion, and the classical mobility is therefore destroyed by quantum fluctuations. For a large phase instead, corresponding to fast classical motion, the time scales separate so that a mobile, localized, coherent quantum state can be translated many sites for particle numbers already of the order of 10.
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.
Abstract The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.
Discrete breathers, originally introduced in the context of biopolymers and coupled nonlinear oscillators, are also localized modes of excitation of Bose–Einstein condensates (BEC) in periodic potentials such as those generated by counter-propagating laser beams in an optical lattice. Static and dynamical properties of breather states are analysed in the discrete nonlinear Schrödinger equation that is derived in the limit of deep potential wells, tight-binding and the superfluid regime of the condensate. Static and mobile breathers can be formed by progressive re-shaping of initial Gaussian wave-packets or by transporting atomic density towards dissipative boundaries of the lattice. Static breathers generated via boundary dissipations are determined via a transfer-matrix approach and discussed in the two analytic limits of highly localized and very broad profiles. Mobile breathers that move across the lattice are well approximated by modified analytical expressions derived from integrable models with two independent parameters: the core-phase gradient and the peak amplitude. Finally, possible experimental realizations of discrete breathers in BEC in optical lattices are discussed in the presence of residual harmonic trapping and in interferometry configurations suitable to investigate discrete breathers' interactions.
We show that a Bose-Hubbard model extended with pair-correlated hopping has exact eigenstates, quantum lattice compactons, with complete single-site localization. These appear at parameter values where the one-particle tunneling is exactly canceled by nonlocal pair correlations, and correspond in a classical limit to compact solutions of an extended discrete nonlinear Schr\"odinger model. Classical compactons at other parameter values, as well as multisite compactons, generically get delocalized by quantum effects, but strong localization appears asymptotically for increasing particle number.
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.