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Journal of the Physical Society of Japan

Dynamics of a Wavefunction for the Attractive Nonlinear Schrödinger Equation under Isotropic Harmonic Confinement Potential

Hirofumi Morise, Miki Wadati

Published Dec 15, 2001• 2 citations
Publisher

Abstract

We study dynamics of a wavefunction for the nonlinear Schrödinger equation with attractive self-interaction under a harmonic potential. By use of the variance identity and a rigorous inequality, we obtain a sufficient condition for the collapse of the wavefunction. By applying this method, we investigate the dynamic stability of a trapped Bose–Einstein condensate and its dependence on the initial shape of the condensate. The result agrees qualitatively well with the results of the time-dependent variational method. We find that the anisotropy introduced in the initial state has a significant effect on the stabilization.

DOI
10.1143/jpsj.70.3529

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We study dynamics of a wavefunction for the nonlinear Schrödinger equation with attractive self-interaction under a harmonic potential. By use of the variance identity and a rigorous inequality, we obtain a sufficient condition for the collapse of the wavefunction. By applying this method, we investigate the dynamic stability of a trapped Bose–Einstein condensate and its dependence on the initial shape of the condensate. The result agrees qualitatively well with the results of the time-dependent variational method. We find that the anisotropy introduced in the initial state has a significant effect on the stabilization.
The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.
Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.