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Jisuan wuli

Numerical Solution of Two-dimensional Gross-Pitaevskii Equation for Bose-Einstein Condensation in a Non-harmonic Trap

Zhao Jiangang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

A two-dimensional Gross-Pitaevskii equation describing Bose-Einstein condensation in an axial symmetry non-harmonic trap is solved.Converged wave functions,chemical potentials,and internal energies in Bose-Einstein condensation are obtained.Ground-state of Bose-Einstein condensation with atomic interaction is investigated.A comparison between harmonic and non-harmonic traps is reported.

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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.
We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates using the numerical solution of the Gross-Pitaevskii (GP) equation with both spherical and axial symmetries. We consider time-evolution problems initiated by suddenly changing the interatomic scattering length or harmonic trapping potential in a stationary condensate. These changes introduce oscillations in the condensate which are studied in detail. We use a time iterative split-step method for the solution of the time-dependent GP equation, where all nonlinear and linear non-derivative terms are treated separately from the time propagation with the kinetic energy terms. Even for an arbitrarily strong nonlinear term this leads to extremely accurate and stable results after millions of time iterations of the original equation.