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Chaos An Interdisciplinary Journal of Nonlinear Science

Discrete nonlinear dynamics of weakly coupled Bose–Einstein condensates

Augusto Smerzi, Andrea Trombettoni

Published May 29, 2003• 30 citations
Publisher

Abstract

The dynamics of a Bose-Einstein condensate trapped in a periodic potential is governed by a discrete nonlinear equation. The interplay/competition between discreteness (introduced by the lattice) and nonlinearity (due to the interatomic interaction) manifests itself on nontrivial dynamical regimes which disappear in the continuum (translationally invariant) limit, and have been recently observed experimentally. We review some recent efforts on this highly interdisciplinary field, with the goal of stimulating interexchanges among the communities of condensed matter, quantum optics, and nonlinear physics.

DOI
10.1063/1.1577271

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The dynamics of a Bose-Einstein condensate trapped in a periodic potential is governed by a discrete nonlinear equation. The interplay/competition between discreteness (introduced by the lattice) and nonlinearity (due to the interatomic interaction) manifests itself on nontrivial dynamical regimes which disappear in the continuum (translationally invariant) limit, and have been recently observed experimentally. We review some recent efforts on this highly interdisciplinary field, with the goal of stimulating interexchanges among the communities of condensed matter, quantum optics, and nonlinear physics.
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We study the dynamical phase diagram of a dilute Bose-Einstein condensate (BEC) trapped in a periodic potential. The dynamics is governed by a discrete nonlinear Schrödinger equation: intrinsically localized excitations, including discrete solitons and breathers, can be created even if the BEC's interatomic potential is repulsive. Furthermore, we analyze the Anderson-Kasevich experiment [Science 282, 1686 (1998)], pointing out that mean field effects lead to a coherent destruction of the interwell Bloch oscillations.
Matter waves inside periodic potentials are well known from solid-state physics, where electrons interacting with a crystal lattice are considered. Atomic Bose-Einstein condensates inside light-induced periodic potentials (optical lattices) share many features with electrons in solids, but also with light waves in nonlinear materials and other nonlinear systems. Generally, atom-atom interactions in Bose-Einstein condensates lead to rich and interesting nonlinear effects. Furthermore, the experimental control over the parameters of the periodic potential and the condensate make it possible to enter regimes inaccessible in other systems. In this review, an introduction to the physics of ultracold bosonic atoms in optical lattices is given and an overview of the theoretical and experimental advances to date.
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.