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Physical Review A

Stability of solutions of the nonlinear Schrödinger equation for trapped Bose-condensed atoms with negative scattering lengths

Kimitaka Watanabe, T. Mukai, Takaaki Mukai

Published May 1, 1997• 9 citations
Publisher

Abstract

We analyze the time evolution of solutions of nonlinear Schr\"odinger equations that describe a condensate composed of atoms with negative scattering lengths in a harmonic potential trap. It is theoretically demonstrated that if an initial condensate has negative energies due to negative scattering lengths, then the solutions diverge in a finite time that is determined by the condensate's energy, its initial phase, and the trap parameter.

DOI
10.1103/physreva.55.3639

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We analyze the time evolution of solutions of nonlinear Schr"odinger equations that describe a condensate composed of atoms with negative scattering lengths in a harmonic potential trap. It is theoretically demonstrated that if an initial condensate has negative energies due to negative scattering lengths, then the solutions diverge in a finite time that is determined by the condensate's energy, its initial phase, and the trap parameter.
We present numerical results from solving the time-dependent nonlinear Schr\"odinger equation (NLSE) that describes an inhomogeneous, weakly interacting Bose-Einstein condensate in a small harmonic trap potential at zero temperature. With this method we are able to find solutions for the NLSE for ground state condensate wave functions in one dimension or in three dimensions with spherical symmetry. These solutions corroborate previous ground state results obtained from the solution of the time-independent NLSE. Furthrmore, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering lengths in one-dimensional (1D) and 3D systems for a fixed number of atoms. In both the 1D and 3D cases, these negative scattering length solutions have solitonlike properties. In 3D, however, these solutions are only stable for a modest range of nonlinearities. We analyze the prospects for diagnosing Bose-Einstein condensation in a trap using several experiments that exploit the time-dependent behavior of the condensate.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.
We reinvestigate the dynamics of the grow and collapse of Bose-Einstein condensates in a system of trapped ultracold atoms with negative scattering lengths, and found a new behavior in the long time scale evolution: the number of atoms can go far beyond the static stability limit. The condensed state is described by the solution of the time-dependent nonlinear Schr\"odinger equation, in a model that includes atomic feeding and three-body dissipation. Our results for the model show that, by changing the feeding parameter and when a substantial depletion of the ground-state exists, a chaotic behavior is found. We consider a criterion proposed by Deissler and Kaneko [Phys. Lett. A 119, 397 (1987)] to diagnose spatiotemporal chaos.
We consider the one-dimensional nonlinear Schrödinger equations that describe the dynamics of a Bose–Einstein Condensates with time-dependent scattering length in a complex potential. Our results show that as long as the integrable relation is satisfied, exact solutions of the one-dimensional nonlinear Schrödinger equation can be found in a general closed form, and interactions between two solitons are modulated in a complex potential. We find that the changes of the scattering length and trapping potential can be effectively used to control the interaction between two bright soliton.