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Chinese Physics B

Chaos in a Bose—Einstein condensate

Zhixia Wang, Zhengguo Ni, Fuzhong Cong, Xue‐Shen Liu, Lei Chen

Published Nov 1, 2010• 12 citations
Publisher

Abstract

It is demonstrated that Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose—Einstein condensate driven by 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.

DOI
10.1088/1674-1056/19/11/113205

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It is demonstrated that Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose—Einstein condensate driven by 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.
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.
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.
We investigate a one-dimensional open Bose-Einstein condensate with attractive interaction, by considering the effect of feeding from nonequilibrium thermal cloud and applying the time-periodic inverted-harmonic potential. Using the direct perturbation method and the exact shock wave solution of the stationary Gross—Pitaevskii equation, we obtain the chaotic perturbed solution and the Melnikov chaotic regions. Based on the analytical and the numerical methods, the influence of the feeding strength on the chaotic motion is revealed. It is shown that the chaotic regions could be enlarged by reducing the feeding strength and the increase of feeding strength plays a role in suppressing chaos. In the case of "nonpropagated" shock wave with fixed boundary, the number of condensed atoms increases faster as the feeding strength increases. However, for the free boundary the metastable shock wave with fixed front density oscillates its front position and atomic number aperiodically, and their amplitudes decay with the increase of the feeding strength.
We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave-function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance zzz and the phase difference $\ensuremath{\phi}$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\ensuremath{\xi}$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z, \ensuremath{\phi}$, and $\ensuremath{\xi}$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.
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.