华东师范大学学报（自然科学版） >

2024 , Vol. 2024 >Issue 3: 84 - 90

DOI: https://doi.org/10.3969/j.issn.1000-5641.2024.03.009

原子、分子和光物理

一个高效求解含时Gross-Pitaevskii方程的算法

  • 舒丽莎 ,
  • 董光炯
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  • 华东师范大学 精密光谱科学与技术国家重点实验室, 上海　200241

收稿日期: 2023-05-09

  网络出版日期: 2024-05-25

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An efficient algorithm for solving time-dependent Gross-Pitaevskii equation

  • Lisha SHU ,
  • Guangjiong DONG
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  • State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai　200241, China

Received date: 2023-05-09

  Online published: 2024-05-25

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摘要

Gross-Pitaevskii方程广泛应用于玻色-爱因斯坦凝聚体(Bose-Einstein condensate, BEC)的动力学研究, 然而这个方程通常很难解析求解. 因此发展相应的高精度数值求解方法非常重要. 发展了结合算符劈裂法、Crank-Nicolson算法和四阶精度Numerov算法的高效求解Gross-Pitaevskii方程的新数值计算方法. 通过数值计算可以表明, 与传统的四阶精度的五点差分法相比, 所提出的算法具有高效和消耗内存小的优点.

关键词： Gross-Pitaevskii方程; 算符劈裂法; Crank-Nicolson算法; Numerov算法

本文引用格式

舒丽莎 , 董光炯 . 一个高效求解含时Gross-Pitaevskii方程的算法[J]. 华东师范大学学报（自然科学版）, 2024 , 2024(3) : 84 -90 . DOI: 10.3969/j.issn.1000-5641.2024.03.009

Abstract

The Gross-Pitaevskii equation is widely applied in Bose-Einstein condensate research, yet is rarely analytically determined; thus, it is important to develop a numerical method with high precision to resolve this. Accordingly, a numerical method was developed in this work, considering the splitting step method, Crank-Nicolson algorithm, and Numerov algorithm with four-order accuracy. The corresponding test shows that compared with the finite difference method using five points, the proposed algorithm is more efficient and costs less memory.

Key words： Gross-Pitaevskii equation; splitting step method; Crank-Nicolson algorithm; Numerov algorithm

参考文献

1 ZHANG D W, ZHAO Y X, LIU R B, et al.. Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice. Physical Review A, 2016, 93 (4): 043617.
2 GROSS C, BLOCH I.. Quantum simulations with ultracold atoms in optical lattices. Science, 2017, 357 (6355): 995- 1001.
3 AARONSON S.. Quantum randomness. American Scientist, 2014, 102 (4): 266.
4 HERRERO-COLLANTES M, GARCIA-ESCARTIN J C.. Quantum random number generators. Reviews of Modern Physics, 2017, 89 (1): 015004.
5 PETERS A, CHUNG K Y, CHU S.. Measurement of gravitational acceleration by dropping atoms. Nature, 1999, 400 (6747): 849- 852.
6 ROSI G, SORRENTINO F, CACCIAPUOTI L, et al.. Precision measurement of the Newtonian gravitational constant using cold atoms. Nature, 2014, 510 (7506): 518- 521.
7 COMPARAT D, FIORETTI A, STERN G, et al.. Optimized production of large Bose-Einstein condensates. Physical Review A, 2006, 73 (4): 043410.
8 BOGOLIUBOV N N.. On the theory of superfluidity. Engl Transl J Phys (USSR), 1947, 11 (1): 23- 32.
9 LIU Y M, BAO C G.. Analytical solutions of the coupled Gross–Pitaevskii equations for the three-species Bose–Einstein condensates. Journal of Physics A, 2017, 50 (27): 275301.
10 GAO Y J, MAYFIELD J, LUO S T. Numerical solutions of the time-dependent Schr?dinger equation with position-dependent effective mass[J]. Numerical Methods for Partial Differential Equations, 2023, 39(4): 3222-3245.
11 CAPLAN R M, CARRETERO-GONZáLEZ R.. Numerical stability of explicit Runge–Kutta finite-difference schemes for the nonlinear Schr?dinger equation. Applied Numerical Mathematics, 2013, 71, 24- 40.
12 AGRAWAL G P. Nonlinear Fiber Optics [M]. New York: The Institute of Optics University of Rochester, 2013: 47-51.
13 VAN DIJK W, TOYAMA F M.. Accurate numerical solutions of the time-dependent Schr?dinger equation. Physical Review E, 2007, 75 (3): 036707.
14 MOYER C A.. Numerov extension of transparent boundary conditions for the Schr?dinger equation in one dimension. American Journal of Physics, 2004, 72 (3): 351- 358.
15 ABDULLAEV F K, CAPUTO J G, KRAENKEL R A, et al.. Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length. Physical Review A, 2003, 67 (1): 013605.
16 GROSS E P.. Hydrodynamics of a superfluid condensate. Journal of Mathematical Physics, 1963, 4 (2): 195- 207.
17 PITAEVSKII L P.. Vortex lines in an imperfect Bose gas. Zh Eksper Teor Fiz, 1961, 40 (2): 646- 651.
18 GOLDBERG A, SCHEY H M, SCHWARTZ J L.. Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena. American Journal of Physics, 1967, 35 (3): 177- 186.
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