Conditions for order and chaos in the dynamics of a trapped Bose-Einstein condensate in coordinate and energy space

Abstract

We investigate numerically conditions for order and chaos in the dynamics of an interacting BoseEinstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size.

Effect of trapping geometry on the parametric resonances in a disordered Bose–Einstein condensate driven by an oscillating potential

“…The time-dependent Gross-Pitaevskii equation (TDGPE) [79][80][81][82][83][84][85][86][87][88][89][90][91][92][93] is an excellent tool to model the present system, which is currently facilitated by the numerous codes available for its solution [57,[94][95][96][97]. For the 1D case, the TDGPE is reduced from the three-dimensional (3D) to the 1D form by integrating out the contributions in the transverse direction (see e.g.…”

“…so that we refer to the signal energy as D kin . The kinetic energy is particularly considered because it is an important property of the condensate [99] as it has been used quite a number of times in the description of BEC dynamics [11,85,86,100,101]. However, other quantities such as the radial size and the potential energy reveal similar PRs at the same positions Ω as those of E kin in the results to be presented below and could also be used.…”

Application of the Lagrangian variational method to a one-dimensional Bose gas in a dimple trap

“…Note that the phase in (19) arises as a complicated function of x, α, and γ so that perhaps q a g ~+ x x 2 . In the future, the LVM can be used to analyze the same system under the effect of a moving DT, that would complement previous work [11,14,15]. If our Bose gas were stirred by a laser, then the total energy varies with time until the stirrer leaves the hard wall [11].…”

“…Analytical results based on this approach have been found to agree with experiment [1]. We therefore aim to explore the small-oscillation dynamics, and provide further support by the ground-state properties, of a weakly interacting one-dimensional (1D) Bose gas that is confined by a box potential [11,14,15] with a DT situated at the center. An external harmonic trap is added just to simulate the experimental conditions.…”

“…The importance of the DT has been outlined in previous work [15]. Nevertheless, some additional words about its usefulness are in order here.…”