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arXiv:0901.3868 · Physical Review A

Regular and chaotic Bose-Einstein condensate in an accelerated Wannier-Stark lattice

Wenhua Hai, Gengbiao Lu, Honghua Zhong

Published May 7, 2009• 9 citations
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Abstract

We investigate a Bose-Einstein condensate (BEC) held in a quasi-one-dimensional Wannier-Stark lattice which is a combination of linear potential with an accelerating optical lattice. It is demonstrated that the system can be reduced to a periodically driven Gross-Pitaevskii one, in which we find the exact analytical solution and the regular and chaotic numerical solutions with accelerated atomic flow densities. The higher instabilities in the transport of BEC are predicted for the positive interatomic interaction. The results suggest an experimental scheme to generate and control the accelerated regular and chaotic matter waves.

DOI
10.1103/physreva.79.053610

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We investigate a Bose-Einstein condensate (BEC) held in a quasi-one-dimensional Wannier-Stark lattice which is a combination of linear potential with an accelerating optical lattice. It is demonstrated that the system can be reduced to a periodically driven Gross-Pitaevskii one, in which we find the exact analytical solution and the regular and chaotic numerical solutions with accelerated atomic flow densities. The higher instabilities in the transport of BEC are predicted for the positive interatomic interaction. The results suggest an experimental scheme to generate and control the accelerated regular and chaotic matter waves.
We investigate the phase effects of a periodically driven Bose–Einstein condensate (BEC) held in a spatially two-dimensional (2D) harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross–Pitaevskii equation is found, which depicts the shock wave with chaotic or periodic amplitude and phase. The atomic densities are illustrated numerically, and the circularly symmetric distributions in the separable phase and the axially symmetric BEC clusters in the inseparable phase are shown. It is demonstrated that the periodic driving may lead to chaos for both phases, which plays a role in avoiding the escape of the solution and restraining the BEC collapse and blast. The results suggest a method for controlling the directed transports of the 2D BEC.
It is demonstrated that the well-known Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose-Einstein condensate (BEC) driven by the time-periodic harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is constructed, which describes the matter shock waves with chaotic or periodic amplitudes and phases. When the periodic driving is switched off and the number of condensed atoms is conserved, we obtained the exact stationary states and nonstationary states. The former contains the stable ``nonpropagated'' shock wave, and in the latter the shock wave alternately collapses and grows for the harmonic trapping or propagates with exponentially increased shock-front speed for the antitrapping. It is revealed that the existence of chaos plays a role for suppressing the blast of matter wave. The results suggest a method for preparing the exponentially accelerated BEC shock waves or the stable stationary states.
Experiments on atomic Bose-Einstein condensates inside quasi-one-dimensional optical lattices and related developments in the realization of atom lasers are currently at the frontiers in atomic physics. We give a short review of theoretical progress in evaluating the coherent transport of matter in such configurations, which has been based on an adaptation of the Wannier function representation of quasi-particle states in periodic potentials and on the use of advanced numerical methods for the solution of the time-dependent Gross-Pitaevskii equation. We present various methods by which the band structure of the elementary excitations of a periodic condensate may be probed and describe in terms of Bloch oscillations the coherent emission of matter pulses from a condensate inside an optical lattice under the force of gravity. A harmonic force can be applied to a condensate inside a magnetic trap by a rapid displacement of the center of the trap and a transition of the condensate from superfluid to dissipative behavior is driven by superposing an optical lattice. Finally, in the superfluid regime the combination of a constant force and a harmonic force in the presence of an optical lattice drives Josephson-type oscillations of the condensate, leading to observable resonances and multimode behavior.
Bose-Einstein condensates of alkali atoms inside an optical lattice provide a unique opportunity to investigate such issues as coherent transport of matter, Josephson-type phenomena, and dissipation of superfluidity under highly controllable experimental conditions. We give a short review of theoretical progress in evaluating the collective excitations of a periodic condensate and the coherent transport of matter in different configurations, which has been based on an adaptation of the Wannier function representation of quasi-particle states in periodic potentials and on the use of advanced numerical methods for the solution of the Gross-Pitaevskii equation. We present various methods by which the band structure of the elementary excitations may be probed and show that in the superfluid regime the combination of a constant force and a harmonic force in the presence of the optical lattice drives Josephson-type oscillations of the condensate, leading to observable resonances and multimode behaviour. Finally, we discuss decoherence of the condensate and dissipation via sound wave emission above a local velocity threshold.
The dynamics of a bright matter wave soliton in a quasi one-dimensional Bose–Einstein condensate (BEC) with a periodically rapidly varying time trap is considered. The governing equation is based on averaging the fast modulations of the Gross–Pitaevskii (GP) equation. This equation has the form of a GP equation with an effective potential of a more complicated structure than an unperturbed trap. In the case of an inverted (expulsive) quadratic trap corresponding to an unstable GP equation, the effective potential can be stable. For the bounded space trap potential it is showed that bifurcation exists, i.e. the single-well potential bifurcates to the triple-well effective potential. The stabilization of a BEC cloud on-site state in the temporary modulated optical lattice is found. This phenomenon is analogous to the Kapitza stabilization of an inverted pendulum. The analytical predictions of the averaged GP equation are confirmed by numerical simulations of the full GP equation with rapid perturbations.
We present the results of experimental work on the dynamics of a Bose-Einstein condensate (BEC) loaded in a 1D moving optical lattice. The system is studied both in the regime of low atomic density (expanding BEC) and high atomic density (trapped BEC). In the low-density regime, the stable dynamics of the system is well-described in terms of Bloch waves. In the presence of nonlinearities, as in a high-density sample, the Bloch waves are not stable solutions of the Gross-Pitaevskii equation (GPE) and dynamical instability takes place.