Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T19:49:05.888Z Has data issue: false hasContentIssue false
  • English
  • Français

A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

Published online by Cambridge University Press:  03 June 2015

Y.-S. Wang ,
B.-W. Jeng  and
C.-S. Chien
Y.-S. Wang*
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan
B.-W. Jeng*
Affiliation:
Department of Mathematics Education, National Taichung University of Education, Taichung 403, Taiwan
C.-S. Chien*
Affiliation:
Department of Computer Science and Information Engineering, Ching Yun University, Jungli 320, Taiwan
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected], [email protected]
Get access

Abstract

We study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.

Keywords

65N3535P3035Q55Spectral collocation methodsecond kind Chebyshev polynomialsperiodic potentialbifurcation
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 442 - 460
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Madison, K. W., Chevy, F., Wohlleben, W. and Dalibard, J., Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000), 806–809.CrossRefGoogle Scholar
[2]Abo-Shaeer, J. R., Raman, C., Vogels, J. M. and Ketterle, W., Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), 476–479.CrossRefGoogle ScholarPubMed
[3]Haljan, P. C., Coddington, I., Engels, P. and Cornell, E. A., Driving Bose-Einstein-condensate vorticity with a rotating normal cloud, Phys. Rev. Lett., 87 (2001), 210403.CrossRefGoogle ScholarPubMed
[4]Hodby, E., Hechenblaikner, G., Hopkins, S. A., Marago, O. M. and Foot, C. J., Vortex nucleation in Bose-Einstein condensates in an oblate, purely magnetic potential, Phys. Rev. Lett., 88 (2002), 010405.Google Scholar
[5]Mueller, E. J. and Ho, T.-L., Two-component Bose-Einstein condensates with a large number of vortices, Phys. Rev. Lett., 88 (2002), 180403.CrossRefGoogle ScholarPubMed
[6]Kasamatsu, K., Tsubota, M. and Ueda, M., Vortex phase diagram in rotating two-component Bose-Einstein condensates, Phys. Rev. Lett., 91 (2003), 150406.Google Scholar
[7]Kasamatsu, K., Tsubota, M. and Ueda, M., Structure of vortex lattices in rotating two-component Bose-Einstein condenstates, Phys. B, 329-333 (2003), 23–24.Google Scholar
[8]Kasamatsu, K., Tsubota, M. and Ueda, M., Vortex states of two-component Bose-Einstein condensates with and without internal Josephson coupling, J. Low Temp. Phys., 134 (2004), 719–724.Google Scholar
[9]Kasamatsu, K. and Tsubota, M., Vortex sheet in rotating two-component Bose-Einstein condensates, Phys. Rev. A, 79 (2009), 023606.Google Scholar
[10]Zhang, Y., Bao, W. and Li, H., Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Phys. D, 234 (2007), 49–69.Google Scholar
[11]Wang, H., Numerical simulations on stationary states for rotating two-component Bose-Einstein condensates, J. Sci. Comput., 38 (2009), 149–163.Google Scholar
[12]Mason, J. C. and Handscomb, D. C., Chebyshev Polynomials, Chapman & Hall/CRC, New York, 2003.Google Scholar
[13]Shen, J., Tang, T. and Wang, L.-L., Spectral Method: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, New York, 2011.Google Scholar
[14]Chang, S.-L. and Chien, C.-S., Adaptive continuation algorithms for computing energy levels of rotating Bose-Einstein condensates, Comput. Phys. Commun., 177 (2007), 707–719.Google Scholar
[15]Li, Z.-C., Chen, S.-Y., Chien, C.-S. and Chen, H.-S., A spectral collocation method for a rotating Bose-Einstein condensation in optical lattices, Comput. Phys. Commun., 182 (2011), 1215–1234.Google Scholar
[16]Wang, Y.-S. and Chien, C.-S., A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross-Pitaevskii equation, J. Comput. Appl. Math., 235 (2011), 2740–2757.Google Scholar
[17]Chen, H.-S. and Chien, C.-S., Multilevel spectral-Galerkin and continuation methods for nonlinear Schrodinger equations, SIAM J. Multiscale Model. Simul., 8 (2009), 370–392.CrossRefGoogle Scholar
[18]Chien, C.-S., Chang, S.-L. and Wu, B., Two-stage continuation algorithms for Bloch waves of Bose-Einstein condensates in optical lattices, Comput. Phys. Commun., 181 (2010), 1727–1737.Google Scholar
[19]Chien, C.-S., Chang, S.-L. and Mei, Z., Tracing the buckling of a rectangular plate with the block GMRES method, J. Comput. Appl. Math., 136 (2001), 199–218.Google Scholar
[20]Kuo, Y.-C., Lin, W.-W., Shieh, S.-F. and Wang, W., A minimal energy tracking continuation method for coupled nonlinear Schrodinger equations, J. Comput. Phys., 228 (2009), 7941–7956.Google Scholar
[21]Chang, S.-M., Kuo, Y.-C., Lin, W.-W. and Shieh, S.-F., A continuation BSOR-Lanczos-Galerkin method for positive bound states of a multi-component Bose-Einstein condensate, J. Comput. Phys., 210 (2005), 439–458.Google Scholar
[22]Chen, J.-H., Chern, I.-L. and Wang, W., Exploring ground states and excited states of spin-1 Bose-Einstein condensates by continuation methods, J. Comput. Phys., 230 (2011), 2222–2236.Google Scholar
[23]Alfimov, G. L. and Zezyulin, D. A., Nonlinear modes for the Gross-Pitaevskii equation-a demonstrative computation approach, Nonlinearity, 20 (2007), 2075–2092.Google Scholar
[24]Zezyulin, D. A., Alfimov, G. L., Konotop, V. V. and M, V.Pérez-Garcia, Control of nonlinear modes by scattering-length management in Bose-Einstein condensates, Phys. Rev. A., 76 (2007), 013621.Google Scholar
[25]Zezyulin, D. A., Alfimov, G. L., Konotop, V. V. and Péréz-Garcia, V. M., Stability of excited states of a Bose-Einstein condensates in an anharmonic trap, Phys. Rev. A., 78 (2008), 013606.Google Scholar
[26]Chang, S.-L., Chien, C.-S. and Jeng, B.-W., Computing wave functions of nonlinear Schrodinger equations: A time-independent approach, J. Comput. Phys., 226 (2007), 104130.Google Scholar
[27]Golub, G. H. and Van Loan, C. F., Matrix Computations, 3rd ed., The Johns Hopkins Univ. Press, Baltimore, 1996.Google Scholar
[28]Allgower, E. L. and Georg, K., Introduction to Numerical Continuation Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.Google Scholar
[29]Keller, H. B., Lectures on Numerical Methods in Bifurcation Problems, Springer, Berlin, 1987.Google Scholar
[30]Chan, T. F., Deflation techniques and block-elimination algorithm for solving Bordered singular system, SIAM J. Sci. Statist. Comput., 5 (1984), 121–134.Google Scholar
[31]Budden, P. J. and Norbury, J., A nonlinear elliptic eigenvalue problem, J. Inst. Math. Appl., 24 (1979), 9–33.Google Scholar
[32]Chien, C.-S. and Jeng, B.-W., A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput., 27 (2006), 1287–1304.Google Scholar