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arXiv:cond-mat/0406545 · Physical Review A

Nonlinear atom-optical δ -kicked harmonic oscillator using a Bose-Einstein condensate

G. J. Duffy, Angela S. Mellish, Katharine J. Challis, A. C. Wilson

Published Oct 11, 2004• 54 citations
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Abstract

We experimentally investigate the atom-optical $\ensuremath{\delta}$-kicked harmonic oscillator for the case of nonlinearity due to collisional interactions present in a Bose-Einstein condensate. A Bose condensate of rubidium atoms tightly confined in a static harmonic magnetic trap is exposed to a one-dimensional optical standing-wave potential that is pulsed on periodically. We focus on the quantum antiresonance case for which the classical periodic behavior is simple and well understood. We show that after a small number of kicks the dynamics are dominated by dephasing of matter wave interference due to the finite width of the condensate's initial momentum distribution. In addition, we demonstrate that the nonlinear mean-field interaction in a typical harmonically confined Bose condensate is not sufficient to give rise to chaotic behavior.

DOI
10.1103/physreva.70.041602

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We experimentally investigate the atom-optical \ensuremathδ\ensuremath{\delta}\ensuremathδ-kicked harmonic oscillator for the case of nonlinearity due to collisional interactions present in a Bose-Einstein condensate. A Bose condensate of rubidium atoms tightly confined in a static harmonic magnetic trap is exposed to a one-dimensional optical standing-wave potential that is pulsed on periodically. We focus on the quantum antiresonance case for which the classical periodic behavior is simple and well understood. We show that after a small number of kicks the dynamics are dominated by dephasing of matter wave interference due to the finite width of the condensate's initial momentum distribution. In addition, we demonstrate that the nonlinear mean-field interaction in a typical harmonically confined Bose condensate is not sufficient to give rise to chaotic behavior.
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We report measurements of the early-time momentum diffusion for the atom-optical delta-kicked rotor. In this experiment a Bose-Einstein condensate provides a source of ultracold atoms with an ultranarrow initial momentum distribution, which is then subjected to periodic pulses (or "kicks") using an intense far-detuned optical standing wave. We characterize the effect of varying the effective Planck's constant for the system, while keeping all other parameters fixed. The observed behavior includes both quantum resonances (ballistic energy growth) and antiresonances (re-establishment of the initial state). Our experimental results are compared with theoretical predictions.
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We consider the case of a cubic nonlinear Schr\"odinger equation with an additional chaotic potential, in the sense that such a potential produces chaotic dynamics in classical mechanics. We derive and describe an appropriate semiclassical limit to such a nonlinear Schr\"odinger equation, using a semiclassical interpretation of the Wigner function, and relate this to the hydrodynamic limit of the Gross-Pitaevskii equation used in the context of Bose-Einstein condensation. We investigate a specific example of a Gross-Pitaevskii equation with such a chaotic potential, the one-dimensional \ensuremath{\delta}-kicked harmonic oscillator, and its semiclassical limit, discovering in the process an interesting interference effect, where increasing the strength of the repulsive nonlinearity promotes localization of the wave function. We explore the feasibility of an experimental realization of such a system in a Bose-Einstein condensate experiment, giving a concrete proposal of how to implement such a configuration, and considering the problem of condensate depletion.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor, where the nonlinearity stems from the mean-field interactions between the condensed atoms. For weak interactions, 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. In the stable regime, the system remains predominantly in the two lowest energy states and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and discuss the route to instability. We numerically explore a parameter regime for the occurrence of instability and reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.