Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms

TLDR

The study evaluates how time‑dependent experiments can diagnose Bose‑Einstein condensation in a trap. Numerical solutions of the time‑dependent NLSE for a weakly interacting Bose‑Einstein condensate in a small harmonic trap at zero temperature were obtained, enabling examination of the macroscopic wave‑function evolution when the trap potential is varied on a timescale comparable to condensate dynamics. The method yields ground‑state condensate wave functions in 1D and spherically symmetric 3D traps that agree with time‑independent results, and demonstrates stable solutions for both positive and negative s‑wave scattering lengths, with negative‑scattering‑length solutions exhibiting soliton‑like behavior in 1D and 3D, though stability in 3D is limited to a modest nonlinearity range.

Abstract

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.