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Communications in Theoretical Physics

Computing Ground State Solution of Bose–Einstein Condensates Trapped in One-Dimensional Harmonic Potential

Yuan Qing-Xin, Guo-Hui Ding

Published Nov 1, 2006• 2 citations
Publisher

Abstract

For Bose–Einstein condensates trapped in a harmonic potential well, we present numerical results from solving the time-dependent nonlinear Schrodinger equation based on the Crank–Nicolson method. With this method we are able to find the ground state wave function and energy by evolving the trial initial wave function in real and imaginary time spaces, respectively. In real time space, the results are in agreement with [Phys. Rev. A 51 (1995) 4704], but the trial wave function is restricted in a very small range. On the contrary, in imaginary time space, the trial wave function can be chosen widely, moreover, the results are stable.

DOI
10.1088/0253-6102/46/5/021

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For Bose–Einstein condensates trapped in a harmonic potential well, we present numerical results from solving the time-dependent nonlinear Schrodinger equation based on the Crank–Nicolson method. With this method we are able to find the ground state wave function and energy by evolving the trial initial wave function in real and imaginary time spaces, respectively. In real time space, the results are in agreement with [Phys. Rev. A 51 (1995) 4704], but the trial wave function is restricted in a very small range. On the contrary, in imaginary time space, the trial wave function can be chosen widely, moreover, the results are stable.
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 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.
为在一个泛音潜力套住很好的 Bose 爱因斯坦冷凝物，我们从解决时间依赖者的现在的数字结果非线性的 Schroedinger 方程基于的在 Crank-Nicolson 方法。与这个方法，我们能由发展试用发现扎根的州的波浪功能和精力在真实、想象的时间空格的起始的波浪功能分别地。在实时空间，结果与一致[Phys。加快。一 51 (1995 ) 4704 ] ，但是试用波浪功能在一个很小的范围被限制。在想象的时间空间，相反，试用波浪函数能广泛地被选择，而且，结果是稳定的。
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.