用SSCN法数值研究BEC的宏观量子性质
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摘要
玻色—爱因斯坦凝聚（Bose-Einstein Condensates,简称BEC）在稀薄原子气体中实现以来，很快就成为现代物理学研究热点之一。而谐振势和光晶格中BEC宏观性质的研究又是BEC研究领域的两个重要内容。许多围绕它们而展开的基础性理论和实验研究工作已经取得了令世人瞩目的成果。理论上，研究BEC的动力学性质，主要是通过求解基于平均场理论的Gross-Pitaevskii equation（G-P方程）来获得。由于G-P方程中含有非线性项，因此很难得到波函数的解析表达式，往往通过简化和近似解G-P方程；或者直接进行数值求解。在众多数值算法中，Split-step Crank-Nicholson method（SSCN法）是近年来出现的一种高效且无条件稳定的算法，尤其对于方程中非线性系数很大或所需观察的演化时间很长的情况，优势更为明显。因此，SSCN方法非常适用于BEC问题的数值研究。
本文运用SSCN方法，首先数值研究了谐振势阱中BEC的一些宏观性质。在保留和撤掉外部谐振势两种情况下，研究了单磁阱中BEC的密度分布函数随时间演化规律，探讨了非线性系数C对凝聚体密度分布的影响。然后，我们将研究拓展到双势阱中两组份BEC在撤掉外势后，两组份BEC相干行为随时间的演化；以及在不撤掉外势时，两组份BEC在外势作用下出现周期性的振荡和叠加现象。当两组份BEC相遇叠加时，满足波的叠加原理。最后，我们以光晶格中的BEC为研究模型，运用SSCN算法求解由光晶格势与谐振势构成的组合势阱中BEC的G-P方程，数值模拟了一维以及二维光晶格中BEC的相干现象随时间的演化规律，对比研究了非线性系数C=1和C=5时的BEC演化特性。作为本课题的延伸，我们还探讨了光晶格中带涡旋BEC基态波函数，运用SSCN方法研究其随时间演化的宏观量子性质将是本课题今后进一步要深入开展的工作。
Bose-Einstein Condensation (BEC) has become a hot spot of physics research after its realization in dilute atomic gas quickly. The macroscopic properties study of BEC in harmonic or optical lattice trap are two important aspects, and it has achieved remarkable success between theoretical and experimental work. In theory the dynamic properties of BEC are mainly by solving the mean-field Gross-Pitaevskii equation (G-P equation). It is difficult to acquire the precise solution of G-P equation because of the nonlinear term. So approximate or direct numerical methods are used usually. The Split-step Crank-Nicholson method (SSCN method) is a very efficient and stable algorithm among lots of numerical methods. It has obvious advantages especially when the nonlinear coefficient C is large or the iterative time is long. So the SSCN method is very suitable for numerical researching of BEC.
At first, the macroscopic properties of BEC in harmonic trap is studied numerically in this paper using the SSCN method. One obtain the density distribution function of the evolution over time for a BEC when the magnetic trap is switched off or, is still hold on. The effect between the nonlinear coefficient C and the density distribution is discussed. Then we development the study into two BEC in double well. The interference evolution with time of two BEC when the trap is switched off are simulated numerically. When the trap is still hold on, the density distribution shows the periodic oscillation and overlap. The overlap is consistent with the superposition principle of wave. At last, we take the optical lattices as a research model, the G-P equation in the combined potential which is comprised of optical lattices and harmonic potential is solved by the SSCN method, the time evolution of BEC in 1D and 2D optical lattices are simulated numerically, and the evolution properties with C=1and C=5 is contrastive studied. The ground state function of BEC with vortex in optical lattice is also discussed as a extension study of our topic. And it is the important work for us to study its macroscopic properties of time evolution by the SSCN method.
本文运用SSCN方法，首先数值研究了谐振势阱中BEC的一些宏观性质。在保留和撤掉外部谐振势两种情况下，研究了单磁阱中BEC的密度分布函数随时间演化规律，探讨了非线性系数C对凝聚体密度分布的影响。然后，我们将研究拓展到双势阱中两组份BEC在撤掉外势后，两组份BEC相干行为随时间的演化；以及在不撤掉外势时，两组份BEC在外势作用下出现周期性的振荡和叠加现象。当两组份BEC相遇叠加时，满足波的叠加原理。最后，我们以光晶格中的BEC为研究模型，运用SSCN算法求解由光晶格势与谐振势构成的组合势阱中BEC的G-P方程，数值模拟了一维以及二维光晶格中BEC的相干现象随时间的演化规律，对比研究了非线性系数C=1和C=5时的BEC演化特性。作为本课题的延伸，我们还探讨了光晶格中带涡旋BEC基态波函数，运用SSCN方法研究其随时间演化的宏观量子性质将是本课题今后进一步要深入开展的工作。
Bose-Einstein Condensation (BEC) has become a hot spot of physics research after its realization in dilute atomic gas quickly. The macroscopic properties study of BEC in harmonic or optical lattice trap are two important aspects, and it has achieved remarkable success between theoretical and experimental work. In theory the dynamic properties of BEC are mainly by solving the mean-field Gross-Pitaevskii equation (G-P equation). It is difficult to acquire the precise solution of G-P equation because of the nonlinear term. So approximate or direct numerical methods are used usually. The Split-step Crank-Nicholson method (SSCN method) is a very efficient and stable algorithm among lots of numerical methods. It has obvious advantages especially when the nonlinear coefficient C is large or the iterative time is long. So the SSCN method is very suitable for numerical researching of BEC.
At first, the macroscopic properties of BEC in harmonic trap is studied numerically in this paper using the SSCN method. One obtain the density distribution function of the evolution over time for a BEC when the magnetic trap is switched off or, is still hold on. The effect between the nonlinear coefficient C and the density distribution is discussed. Then we development the study into two BEC in double well. The interference evolution with time of two BEC when the trap is switched off are simulated numerically. When the trap is still hold on, the density distribution shows the periodic oscillation and overlap. The overlap is consistent with the superposition principle of wave. At last, we take the optical lattices as a research model, the G-P equation in the combined potential which is comprised of optical lattices and harmonic potential is solved by the SSCN method, the time evolution of BEC in 1D and 2D optical lattices are simulated numerically, and the evolution properties with C=1and C=5 is contrastive studied. The ground state function of BEC with vortex in optical lattice is also discussed as a extension study of our topic. And it is the important work for us to study its macroscopic properties of time evolution by the SSCN method.
引文
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[5] Liu W M, Fan W B, Zheng W M, Liang J Q and Chui S T Quantum tunneling of Bose-Einstein condensates in optical lattices under gravity [J]. Phys. Rev. Lett., 2002, 88(17): 170408
[6] Xiong H W, Liu S J, Huang G X and Xu Z J. Evolution of a coherent array of Bose-Einstein condensates in a magnetic trap [J]. Phys B: At. Mol. Opt. Phys., 2002, 35(23): 4863-4873
[7] Liu S J, Xiong H W, Xu Z J and Huang G X. Interference patterns of Bose-condensed gases in a two-dimensional optical lattice [J]. Phys.B:At. Mol. Opt. Phys., 2003, 36 (10): 2083-2092
[8] Xu Zhijun, Wang Dongmei, Li Zhen. Interference of Bose-condensed gas in a 1D optical lattice [J]. Acta Physica Sinica, 2007, 56(6): 3076-3082
[9] Xu Z J, Zhang D M. Evolution of Matter Wave Interference of Bose-Condensed Gas in a 1D Optical Lattice [J]. Chin. Phys. Lett., 2007, 24(9): 2493-2496
[10] Chen H J, Xue J K. The ground state solutions of two-component Bose-Einstein condensates in Bessel optical lattices [J]. Acta Physica Sinica, 2008, 57(7): 3962-3968
[11] Bao W, Tang W. Ground state solution of trapped interacting Bose-Einstein condensate by directly minizing the energy functional [J]. J. Comput. Phys., to appear
[12] Edwards M, Burnett K. Numerical solution of the nonlinear Schr?dinger equation for small samples of trapped neutral atoms [J]. Phys. Rev. A, 1995, 51(2): 1382-1386
[13] Adhikari S K. Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms [J]. Phys. Lett. A, 2000, 265(1-2): 91-96
[14] Adhikari S K. Numerical study of the spherically symmetric Gross-Pitaevskii equation in two space dimensions [J]. Phy. Rev. E, 2000, 62(2): 2937-2944
[15] Chiofalo M L, Succi S, Tosi M P. Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm [J]. Phys. Rev. E, 2000, 62(5): 7438-7444
[16] Schneider B I, Feder D L. Numerical approach to the ground and excited states of a Bose-Einstein condensated gas confined in a completely anisotropic trap [J]. Phys. Rev. A, 1999, 59(3): 2232-2242
[17] Dodd R J. Approximate solutions of the nonlinear Schr?dinger equation for ground and excited states of Bose-Einstein condensates [J]. J. Res. Natl. Inst. Stan., 1996, 101(4): 545-552
[18] Cerimele M M, Chilfalo M L, Pistella F, Succi S, Tosi M P. Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates [J]. Phys. Rev. E, 2000, 62(1): 1382-1389
[19] Cerimele M M, Pistella F, Succi S. Particle-inspired for the Gross-Pitaevskii equation: an application to Bose-Einstein condensation [J]. Comput. Phys. Commun., 2000, 129(1-3): 82-90
[20] Ruprecht P A, Holland M J, Burrett K, Edwards M. Time-dependent solution of the nonlinear Schr?dinger equation for Bosed-condensed trapped neutral atoms [J]. Phys. Rev. A, 1995, 51(6): 4704-4711
[21] Ruprecht P A, Holland M J, Burrett K and Edwards M. Time-Dependent solution of the nonlinear Schr?dinger equation for Bose-condensed trapped neutral atoms [J]. Phys. Rev. A, 1995, 51(6): 4704-4711
[22] Cerimele M M, Chiofalo M L, Pistella F, Succi S, Tosi M P. Numerical solution ofthe Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose-Einstein condesates [J]. Phys. Rev. E, 2000, 62(1): 1382-1389
[23] Edwards M and Brunett K. Numerical solution of the nonlinear Schr?dinger equation for small samples of trapped neutral atoms [J]. Phy. Rev. A, 1995, 51(2): 1382-1386
[24] Cgiofalo M L, Succi S and Tosi M P. Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm [J]. Phys. Rev. E, 2000, 62(5): 7438-7444
[25] Grynberg. G, Lounis. B, Verkerk. P, Courtois. J Y and Salomon.C. Quantized Motion of Cold Cesium Atoms in 2-Dimensional and 3-Dimensional Optical Potentials [J]. Phys. Rev. Lett., 1993, 70(15): 2249-2252
[26] Liu W M, Wu B and Niu Q. Nonlinear effects in interference of Bose-Einstein condesates [J]. Phys. Rev. Lett., 2000, 84(11): 2294-2297
[27] Wu Y and Yang X X. Analytical results for energy spectrum and eigenstates of a Bose-Einstein condensate in a Mott insulator state [J]. Phys. Rev. A, 2003, 68(1): 013608
[28] Huang G X, Makarov V A and Velarde M G. Two-dimensional solitons in Bose-Einstein condensates with a disk-shaped trap [J]. Phys. Rev. A, 2003, 67(2): 023604
[29] Xiong H W, Liu S J, Huang G X and Xu Z X. Canonical statistics of trapped ideal and interacting Bose gases [J]. Phys. Rev. A, 2002, 65(3): 033609
[30] Xu Z J, Cheng C, Yang H S, Wu Q and Xiong H W, et al. The groud-state wave function and evolution of the interference pattern for a Bose-condensed gas in 3D optical lattices [J] Acta Phys. Sin., 53(9): 2835-2842
[31] Adhikari S K. Matter-wave interference, Josephson oscillation and its disruption in a Bose-Einstein condensate on an optical lattice [J]. Nucl. Phys. A, 2004, 737: 289-293
[32] Pedri P, Pitaevskii L, Stringari S. Expansion of a coherent array of Bose-Einstein condensates [J]. Phys. Rev. Lett., 2001, 87(22): 220401-220404
[33] Jaksch D, Bruder C, Cirac J I, Gardiner C W and Zoller P. Cold bosonic atoms in optical lattices [J]. Phys. Rev. Lett.,1998, 81(15): 3108-3111
[34] Greiner M, Mandel O, Esslinger T, H?nsch T W and Bloch I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms [J]. Nature, 2002, 415(6867): 39-44
[35] Greiner M, Mandel O, Hansch T W and Bloch I. Collapse and revival of the matter wave field of a Bose-Einstein condensate [J]. Nature, 2002, 419(6902): 51-54
[36] Peil S, Porto J V, Tolra B L, Obrecht J M, King B E, et al. Patterned loading of a Bose-Einstein condensate into an optical lattice [J]. Phys. Rev. A, 67(5): 051603-051606
[37] Adhikari S K, Muruganandam P. Mean-field model for the interference of matter waves from a three-dimensional optical trap [J]. Phys. Lett. A, 2003, 310(2-3): 229-235
[38] Ai Khawaja U, Stoof H T C, Hulet R G, et al. Bright soliton trains of trapped Bose-Einstein condensates [J]. Phys. Rev. Lett., 2002, 89(20): 200404
[39] Adhikari S K, Muruganandam P. Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation [J]. Phys. B, 2002, 35(12): 2831-2843
[40]郝柏林.中性原子的玻色—爱因斯坦凝聚[J].物理学进展, 1997, 17(3): 223-232
[41] Bronski J C, Carr L D, Deconinck B, Kutz J N. Bose-Einstein condensates in standing waves: The cubic nonlinear Schrodinger equation with a periodic potential [J]. P. R. Lett., 2001, 86(8): 1402-1405
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