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A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates. (English) Zbl 1149.76039

Summary: A generalized-Laguerre-Hermite pseudospectral method is proposed for computing symmetric and central vortex states in Bose-Einstein condensates (BECs) in three dimensions with cylindrical symmetry. The new method is based on the properly scaled generalized-Laguerre-Hermite functions and a normalized gradient flow. It enjoys three important advantages: (i) it reduces a three-dimensional (3D) problem with cylindrical symmetry into an effective two-dimensional (2D) problem; (ii) it solves the problem in the whole space instead of in a truncated artificial computational domain, and (iii) it is spectrally accurate. Extensive numerical results for computing symmetric and central vortex states in BECs are presented for one-dimensional (1D) BEC, 2D BEC with radial symmetry and 3D BEC with cylindrical symmetry.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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[1] Bao, W.; Chai, M.H., A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems, Commun. comput. phys., 4, 135-160, (2008) · Zbl 1364.81015
[2] Bao, W.; Chern, I.-L.; Lim, F.Y., Efficient and spectrally accurate numerical methods for computing ground and first excited states in bose – einstein condensates, J. comput. phys., 219, 836-854, (2006) · Zbl 1330.82031
[3] Bao, W.; Du, Q., Computing the ground state solution of bose – einstein condensates by a normalized gradient flow, SIAM J. sci. comput., 25, 1674-1697, (2004) · Zbl 1061.82025
[4] Bao, W.; Ge, Y.; Jaksch, D.; Markowich, P.A.; Weishäupl, R.M., Convergence rate of dimension reduction in bose – einstein condensates, Comput. phys. commun., 177, 832-850, (2007) · Zbl 1196.82065
[5] Bao, W.; Lim, F.Y., Computing ground states of spin-1 bose – einstein condensates by the normalized gradient flow, SIAM J. sci. comput., 30, 1925-1948, (2008) · Zbl 1173.81028
[6] Bao, W.; Shen, J., A fourth-order time-splitting laguerre – hermite pseudo-spectral method for bose – einstein condensates, SIAM J. sci. comput., 26, 2010-2028, (2005) · Zbl 1084.35083
[7] Bao, W.; Wang, H., A mass and magnetization conservative and energy diminishing numerical method for computing ground state of spin-1 bose – einstein condensates, SIAM J. numer. anal., 45, 2177-2200, (2007) · Zbl 1149.82018
[8] Bao, W.; Wang, H.; Markowich, P.A., Ground, symmetric and central vortex states in rotating bose – einstein condensates, Commun. math. sci., 3, 57-88, (2005) · Zbl 1073.82004
[9] Castin, Y.; Hadzibabic, Z.; Stock, S.; Dalibard, J.; Stringari, S., Quantized vortices in the ideal Bose gas: a physical realization of random polynomials, Phys. rev. lett., 96, (2006), article 040405
[10] Chiofalo, M.L.; Succi, S.; Tosi, M.P., Ground state of trapped interacting bose – einstein condensates by an explicit imaginary-time algorithm, Phys. rev. E, 62, 7438-7444, (2000)
[11] Guo, B.-Y.; Shen, J., Laguerre – galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. math., 86, 635-654, (2000) · Zbl 0969.65094
[12] Guo, B.-Y.; Shen, J.; Xu, C.-L., Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation, Adv. comput. math., 19, 35-55, (2003) · Zbl 1032.33004
[13] Guo, B.-Y.; Zhang, X.-Y., A new generalized Laguerre spectral approximation and its applications, J. comput. appl. math., 181, 342-363, (2005) · Zbl 1072.65155
[14] Huepe, C.; Tuckerman, L.S.; Metens, S.; Brachet, M.E., Stability and decay rates of nonisotropic attractive bose – einstein condensates, Phys. rev. A, 68, (2003), article 023609 · Zbl 1060.76528
[15] Kapale, K.T.; Dowling, J.P., Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in bose – einstein condensates via optical angular momentum beams, Phys. rev. lett., 95, 173601, (2005)
[16] Khabibrakhmanov, I.K.; Summers, D., The use of generalized Laguerre polynomials in spectral methods for nonlinear differential equations, Comput. math. appl., 36, 65-70, (1998) · Zbl 0932.65091
[17] Klein, A.; Jaksch, D.; Zhang, Y.; Bao, W., Dynamics of vortices in weakly interacting bose – einstein condensates, Phys. rev. A, 76, (2007), article 043602
[18] Leanhardt, A.E.; Gorlitz, A.; Chikkatur, A.P.; Kielpinski, D.; Shin, Y.; Pritchard, D.E.; Ketterle, W., Imprinting vortices in a bose – einstein condensate using topological phases, Phys. rev. lett., 89, (2002), article 190403
[19] Lieb, E.H.; Seiringer, R.; Yugvason, J., Bosons in a trap: a rigorous derivation of the gross – pitaevskii energy functional, Phys. rev. A, 61, 3602, (2000)
[20] Ma, H.; Sun, W.; Tang, T., Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains, SIAM J. numer. anal., 43, 58-75, (2005) · Zbl 1087.65097
[21] Madison, K.W.; Chevy, F.; Wohlleben, W.; Dalibard, J., Vortex formation in a stirred bose – einstein condesate, Phys. rev. lett., 84, 806-809, (2000)
[22] Matthews, M.R.; Anderson, B.P.; Haljan, P.C.; Hall, D.S.; Wieman, C.E.; Cornell, E.A., Vortices in a bose – einstein condensate, Phys. rev. lett., 93, 2498, (1999)
[23] Muruganandam, P.; Adhikari, S.K., Bose – einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods, J. phys. B-at. mol. opt. phys., 36, 2501-2513, (2003)
[24] Pitaevskii, L.P.; Stringari, S., Bose – einstein condensation, (2003), Clarendon Press · Zbl 1110.82002
[25] Rokhsar, D.S., Vortex stability and persistent currents in trapped Bose-gas, Phys. rev. lett., 79, 2164-2167, (1997)
[26] Shen, J., Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. numer. anal., 38, 1113-1133, (2000) · Zbl 0979.65105
[27] G. Szegö, Orthogonal Polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence, RI, 1975.
[28] Tang, T., The Hermite spectral method for Gaussian-type functions, SIAM J. sci. comput., 14, 594-606, (1993) · Zbl 0782.65110
[29] Wu, J.-M.; Shen, L.-J., Spectral analysis of the first-order Hermite cubic spline collocation differentiation matrices, J. comput. math., 20, 551-560, (2002) · Zbl 1018.15014
[30] Xu, C.-L.; Guo, B.-Y., Laguerre pseudospectral method for nonlinear partial differential equations, J. comput. math., 20, 413-428, (2002) · Zbl 1005.65115
[31] Zhang, Y.; Bao, W.; Li, H., Dynamics of rotating two-component bose – einstein condensates and its efficient computation, Phys. D: nonlinear phenom., 234, 49-69, (2007) · Zbl 1130.82023
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