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arXiv:1104.1521 · New Journal of Physics

Coherence and instability in a driven Bose–Einstein condensate: a fully dynamical number-conserving approach

T. P. Billam, S. A. Gardiner

Published Jan 20, 2012• 22 citations
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Abstract

We consider a Bose–Einstein condensate driven by periodic δ -kicks. In contrast to first-order descriptions, which predict rapid, unbounded growth of the noncondensate in resonant parameter regimes, the consistent treatment of condensate depletion in our fully time-dependent, second-order description acts to damp this growth, leading to oscillations in the (non)condensate population and the coherence of the system.

DOI
10.1088/1367-2630/14/1/013038

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We consider a Bose–Einstein condensate driven by periodic δ -kicks. In contrast to first-order descriptions, which predict rapid, unbounded growth of the noncondensate in resonant parameter regimes, the consistent treatment of condensate depletion in our fully time-dependent, second-order description acts to damp this growth, leading to oscillations in the (non)condensate population and the coherence of the system.
While the Gross-Pitaevskii equation is well established as the canonical dynamical description of atomic Bose-Einstein condensates (BECs) at zero temperature, describing the dynamics of BECs at finite temperatures remains a difficult theoretical problem, particularly when considering low-temperature, nonequilibrium systems in which depletion of the condensate occurs dynamically as a result of external driving. In this paper, we describe a fully time-dependent numerical implementation of a second-order, number-conserving description of finite-temperature BEC dynamics. This description consists of equations of motion describing the coupled dynamics of the condensate and noncondensate fractions in a self-consistent manner, and is ideally suited for the study of low-temperature, nonequilibrium, driven systems. The $\ensuremath{\delta}$-kicked-rotor BEC provides a prototypical example of such a system, and we demonstrate the efficacy of our numerical implementation by investigating its dynamics at finite temperature. We demonstrate that the qualitative features of the system dynamics at zero temperature are generally preserved at finite temperatures, and predict a quantitative finite-temperature shift of resonance frequencies which would be relevant for, and could be verified by, future experiments.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
Bose-Einstein condensates subject to short pulses (``kicks'') from standing waves of light represent a nonlinear analog of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (i.e., exponential proliferation of noncondensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics.
We develop the number-conserving approach, which has previously been used in a single-component Bose-Einstein condensed dilute atomic gas, to describe consistent coupled condensate and noncondensate number dynamics, to an nnn-component condensate. The resulting system of equations is comprised, for each component, of a generalized Gross-Pitaevskii equation coupled to modified Bogoliubov--de Gennes equations. Lower order approximations yield general formulations for multicomponent Gross-Pitaevskii equations, and systems of multicomponent Gross-Pitaevskii equations coupled to multicomponent modified number-conserving Bogoliubov--de Gennes equations. The analysis is left general, such that, in the nnn-component condensate, there may or may not be mutually coherent components. An expansion in powers of the ratio of noncondensate-to-condensate particle numbers for each coherent set is used to derive the self-consistent, second-order, dynamical equations of motion. The advantage of the analysis developed in this article is in its applications to dynamical instabilities that appear when two (or more) components are in conflict and where a significant noncondensed fraction of atoms is expected to appear.