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Physical Review A

Dynamics of Bose-Einstein condensates in a time-dependent trap

Vaduganathan Ramesh Kumar, R. Radha, Prasanta Kumar Panigrahi

Published Feb 8, 2008• 37 citations
Publisher

Abstract

In this paper, we generate the Lax pair for the one-dimensional Gross-Pitaevskii equation with time-dependent scattering length in the presence of a confining or expulsive harmonic time-dependent trap. We then exploit the Lax pair profitably to construct multisoliton solutions using gauge transformation from a trivial input solution. In particular, we have investigated the effect of both expulsive and confining traps on soliton interaction. Even though we find that the amplitude of the bright soliton relies upon the time-dependent scattering length and the external time-dependent trap with the velocity being dictated by the external trap alone, the observation of interdependence of the scattering length on the trap shows that the bright solitons not only can be compressed into a desirable width and amplitude but also can be remote controlled and driven anywhere in the plane by suitably maneuvering the external time-dependent trap alone.

DOI
10.1103/physreva.77.023611

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In this paper, we generate the Lax pair for the one-dimensional Gross-Pitaevskii equation with time-dependent scattering length in the presence of a confining or expulsive harmonic time-dependent trap. We then exploit the Lax pair profitably to construct multisoliton solutions using gauge transformation from a trivial input solution. In particular, we have investigated the effect of both expulsive and confining traps on soliton interaction. Even though we find that the amplitude of the bright soliton relies upon the time-dependent scattering length and the external time-dependent trap with the velocity being dictated by the external trap alone, the observation of interdependence of the scattering length on the trap shows that the bright solitons not only can be compressed into a desirable width and amplitude but also can be remote controlled and driven anywhere in the plane by suitably maneuvering the external time-dependent trap alone.
In this paper, we generate the Lax pair of the nonconservative Gross–Pitaevskii (GP) equation with time-dependent linear and harmonic oscillator potentials and construct a multisoliton solution using gauge transformation. We show how an interplay between the dispersion coefficient, scattering length and atomic feeding can suitably be exploited to remote control the dynamics of solitons, thereby generating favorable profiles of Bose–Einstein condensates (BECs), notable among them being the matter wave similaritons.
Due to their relevance to physics and technology, the Bose–Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross–Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz–Kaup–Newell–Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates.
We present a family of exact solutions of the one-dimensional nonlinear Schro dinger equation which describes the dynamics of a bright soliton in Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Our results show that, under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the absolute value of the atomic scattering length, which can provide an experimental tool for investigating the range of validity of the one-dimensional Gross-Pitaevskii equation. We also find that the number of atoms in the bright soliton keeps dynamic stability: a time-periodic atomic exchange is formed between the bright soliton and the background.
We present an analytical study on the dynamics of bright and dark solitons in Bose–Einstein condensates with time-varying atomic scattering length in a time-varying external parabolic potential. A set of exact soliton solutions of the one-dimensional Gross–Pitaevskii equation are obtained, including fundamental bright solitons, higher-order bright solitons, and dark solitons. The results show that the soliton's parameters (amplitude, width, and period) can be changed in a controllable manner by changing the scattering length and external potential. This may be helpful to design experiments.
We present three families of one-soliton solutions for (2+l)-dimensional Gross-Pitaevskii equation withboth time-dependent scattering length and gain or loss in a harmonic trap.Then we investigate the dynamics of thesesolitons in Bose-Einstein condensates (BECs) by some selected control functions.Our results show that the intensitiesof these solitons first increase rapidly to the condensation peak,then decay very slowly to the background;thus thelifetime of a bright soliton,a train of bright solitons and a dark soliton in BECs can be all greatly extended.Our resultsoffer a useful method for observing matter-wave solitons in BECs in future experiments.
We present three families of one-soliton solutions for (2+1)-dimensional Gross—Pitaevskii equation with both time-dependent scattering length and gain or loss in a harmonic trap. Then we investigate the dynamics of these solitons in Bose—Einstein condensates (BECs) by some selected control functions. Our results show that the intensities of these solitons first increase rapidly to the condensation peak, then decay very slowly to the background; thus the lifetime of a bright soliton, a train of bright solitons and a dark soliton in BECs can be all greatly extended. Our results offer a useful method for observing matter-wave solitons in BECs in future experiments.