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Journal of the Optical Society of America B

Recurrence for motion of solitons of the Bose–Einstein condensate in a dynamic trap

Nikolay N. Rosanov, N. V. Vysotina

Published Apr 8, 2015• 11 citations
Publisher

Abstract

We consider, analytically and numerically, the dynamics of an atomic Bose–Einstein condensate soliton bouncing in a magnetic or optical trap between two ideal barriers oscillating harmonically with coinciding frequency and opposite phases. There are zones of parametric instability of the soliton’s central position in the trap for certain ranges of the frequency of the barriers’ oscillations. We demonstrate the Fermi–Pasta–Ulam recurrence in the Fermi–Ulam scheme for solitons: the soliton’s deviations from the trap’s center grow and decay quasi-periodically in time. The results allow one to excite and investigate various structures of matter waves in traps and to manipulate their localization and dynamics.

DOI
10.1364/josab.32.000b20

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Optics and Spectroscopy•Jan 1, 2016•2 citations
We consider, analytically and numerically, the dynamics of an atomic Bose–Einstein condensate soliton bouncing in a magnetic or optical trap between two ideal barriers oscillating harmonically with coinciding frequency and opposite phases. There are zones of parametric instability of the soliton’s central position in the trap for certain ranges of the frequency of the barriers’ oscillations. We demonstrate the Fermi–Pasta–Ulam recurrence in the Fermi–Ulam scheme for solitons: the soliton’s deviations from the trap’s center grow and decay quasi-periodically in time. The results allow one to excite and investigate various structures of matter waves in traps and to manipulate their localization and dynamics.
We analyze the motion of solitons in a self-attractive Bose-Einstein condensate, loaded into a quasi-one-dimensional parabolic potential trap, which is subjected to time-periodic modulation with an amplitude epsilon and frequency Omega. First, we apply the variational approximation, which gives rise to decoupled equations of motion for the center-of-mass coordinate of the soliton, xi (t), and its width a (t). The equation for xi (t) is the ordinary Mathieu equation (ME) (it is an exact equation that does not depend on the adopted ansatz), the equation for a (t) being a nonlinear generalization of the ME. Both equations give rise to the same map of instability zones in the (epsilon,Omega) plane, generated by the parametric resonances (PRs), if the instability is defined as the onset of growth of the amplitude of the parametrically driven oscillations. In this sense, the double PR is predicted. Direct simulations of the underlying Gross-Pitaevskii equation give rise to a qualitatively similar but quantitatively different stability map for oscillations of the soliton's width a (t). In the direct simulations, we identify the soliton dynamics as unstable if the instability (again, realized as indefinite growth of the amplitude of oscillations) can be detected during a time comparable with, or smaller than, the lifetime of the condensate (therefore accessible to experimental detection). Two-soliton configurations are also investigated. It is concluded that multiple collisions between solitons are elastic, and they do not affect the instability borders.
Dynamics of solitons in a one-dimensional trap with motionless and moving walls is studied by the example of bright solitons of an atomic Bose-Einstein condensate. Newton-type equations of the soliton's center motion are proposed and analyzed, and regimes of periodic, almost-periodic, and deterministically chaotic dynamics are presented. The predictions based on these approximate equations are in excellent agreement with that based on the Gross-Pitaevskii equation in the case of sufficiently narrow and slow solitons.
Motion of a soliton of Bose–Einstein condensate of atoms captured by a trap with rapidly oscillating walls has been studied. This motion can be described using both the Gross–Pitaevskii equation for a condensate wave function and an approximate equation in the form of the Newton equation for the soliton center coordinate. An analytical approach for solving the Newton equation has been developed. This approach is based on the multiscale method where the solution is sought for in the form of small-parameter expansion. This parameter is a ratio of the frequency of intrinsic slow soliton oscillations around the equilibrium position to the frequency of fast oscillations of the trap walls. In the first part of the study, an approach based on two time scales is described and the case of extremely fast wall oscillations is investigated. The calculation performed within the zero approximation shows a very good coincidence with the numerical solution of the Newton equation with respect to all parameters. A good agreement with the numerical solutions of the Gross–Pitaevskii equation is also demonstrated for calculations of the parameters such as oscillation frequency and shift of the soliton equilibrium position under the action of the wall motion. In the second part, the role of corrections to the obtained solution is analyzed for a decreasing wall-oscillation frequency and the range of applicability of the used analytical approach is discussed.
We consider the evolution of an interacting dilute Bose-Einstein condensate transferred from a harmonic trap to a perfect box of finite size. We study numerically the influence of the nonlinearities in the system on its dynamics. The parallelism between this problem and the Fermi-Pasta-Ulam problem in the context of matter waves is established. Criteria for the appearance of statistical behavior are discussed. We show that by increasing the nonlinearity we pass from a regime of collapses and revivals for the wave function to a regime of collapses and revivals of regular motion.
This work is a continuation of our study [1], in which a two-scale analytical approach to the investigation of a soliton oscillon in a trap with rapidly oscillating walls has been developed. In terms of this approach, the solution to the equation of motion of the soliton center is sought as a series expansion in powers of a small parameter, which is a ratio of the intrinsic frequency of slow soliton oscillations to the frequency of fast trap wall oscillations. In [1], we have examined the case ε ≪ 1, in which, to describe the motion of the soliton, it is sufficient to restrict the consideration to the zero approximation of the sought solution. However, when the frequency of wall oscillations begins to decrease, while the parameter begins to increase, it is necessary to take into account corrections to the zero approximation. In this work, we have calculated corrections of the first and second orders in to this approximation. We have shown that, with an increase in, the role played by the corrections related to fast oscillations of the trap walls increases, which results in a complex shape of the envelope of oscillations of the soliton center. It follows from our calculations that, if the difference between the amplitudes of wall oscillations is not too large, the analytical solution of the equation of motion of the soliton center will coincide very well with the numerical solution. However, with an increase in this difference, as well as with a decrease in the wall oscillation frequency, the discrepancy between the numerical and analytical solutions generally begins to increase. Regimes of irregular oscillations of the soliton center arise. With a decrease in the frequency of wall oscillations, the instability boundary shows a tendency toward a smaller difference between the wall oscillation amplitudes. In general, this leads to enlargement of the range of irregular regimes. However, at the same time, stability windows can arise in this range in which the analytical and numerical solutions correlate rather well with each other. Our comparative analysis of the analytical and numerical solutions has allowed us not only to study their properties in detail, but also to draw conclusions on the limits of applicability of the analytical approach.