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CRC Press eBooks

Stationary Solutions of the Gross-Pitaevskii Equation for Neutral Atoms in Harmonic Trap

Wei Hua, Xue‐Shen Liu, Peizhu Ding

Published Apr 29, 2019• 1 citation
Publisher

Abstract

We solve the time-independent Gross-Pitaevskii equation that describes the dilute Bose-condensed atoms in harmonic trap at T =0 by symplectic shooting method (SSM). Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bounded state eigenvalues as well as the corresponding wave functions are presented.

DOI
10.1201/9780429081385-53

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Acta Physica Sinica•Jan 1, 2006•1 citation
We solve the time-independent Gross-Pitaevskii equation that describes the dilute Bose-condensed atoms in harmonic trap at T =0 by symplectic shooting method (SSM). Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bounded state eigenvalues as well as the corresponding wave functions are presented.
The ground state wavefunctions of dilute Bose-condensed atoms in a harmonic trap at T=0 are evaluated by a symplectic shooting method.Stability of the wavefunctions is tested,and a stable wavefunction is used as the initial input of the time-dependent Gross-Pitaevskii equation. The dynamic property of stable wavefunctions is numerically examined in two phase spaces when the harmonic potential is altered suddenly.The figures in the two phase spaces are regular even after a long time of interations.For negative nonlinear coefficients,two eigenvalues related to the same negative nonlinear coefficient are calculated,and the stability of the corresponding two wavefunctions is tested.
We try to find the analytical solutions to the time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in the simulation of Bose—Einstein condensates trapped in a harmonic potential. Both the homotopy analysis method and the Galerkin spectral method are applied. We investigate the one-dimensional case and obtain the approximate analytical solutions successfully. Comparison between the analytical solutions and the numerical solutions has been made. The results indicate that they agree very well with each other when the atomic interaction is not too strong.
It is well known that bosonic particles with attractive interaction in a uniform gas do not form a condensate. Here we investigate a dilute Bose gas and study stationary solutions of the Gross–Pitaevskii equation with attractive interaction. We have also used a higher order stabilising term in the presence of a harmonic confining potential. We show that there are three possible types of stationary solutions corresponding to stable, metastable and unstable phases. These results are discussed in relation to a 7Li condensate.
Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.