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Pseudo-Arclength Continuation Algorithms for Binary Rydberg-Dressed Bose-Einstein Condensates

Part of: Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  12 April 2016

Sirilak Sriburadet ,
Y.-S. Wang ,
C.-S. Chien  and
Y. Shih
Sirilak Sriburadet
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan
Y.-S. Wang
Affiliation:
Department of Computer Science and Information Engineering, Chien Hsin University of Science and Technology, Zhongli 320, Taiwan
C.-S. Chien*
Affiliation:
Department of Computer Science and Information Engineering, Chien Hsin University of Science and Technology, Zhongli 320, Taiwan
Y. Shih
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan
*
*Corresponding author. Email addresses:yuijungja@yahoo.com (S. Sriburadet), wang04.wang25@msa.hinet.net (Y.-S. Wang), cschien@uch.edu.tw (C.-S. Chien), Yintzer_Shih@nchu.edu.tw (Y. Shih)
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Abstract

We study pseudo-arclength continuation methods for both Rydberg-dressed Bose-Einstein condensates (BEC), and binary Rydberg-dressed BEC which are governed by the Gross-Pitaevskii equations (GPEs). A divide-and-conquer technique is proposed for rescaling the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the affect of the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.

Keywords

Ground state solutionsolution branchspectral collocation methoddivide-and-conquer

MSC classification

Secondary: 65N35: Spectral, collocation and related methods 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 1067 - 1093
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Cinti, F., Jain, P., Boninsegni, M., Micheli, A., Zoller, P. and Pupillo, G., Supersolid droplet crystal in a dipole-blockaded gas, Phys. Rev. Lett., 105 (2010), 135301.CrossRefGoogle Scholar
[2]Saccani, S., Moroni, S. and Boninsegni, M., Phase diagram of soft-core bosons in two dimensions, Phys. Rev. B, 83 (2011), 092506.Google Scholar
[3]Henkel, N., Nath, R. and Pohl, T., Three-dimensional roton excitations and supersolid formation in Rydberg-excited Bose-Einstein condensates, Phys. Rev. Lett., 104 (2010), 195302.Google Scholar
[4]Henkel, N., Cinti, F., Jain, P., Pupillo, G. and Pohl, T., Supersolid vortex crystals in Rydberg-dressed Bose-Einstein condensates, Phys. Rev. Lett., 108 (2012), 265301.Google Scholar
[5]Hsueh, C.-H., Lin, T.-C., Horng, T.-L. and Wu, W. C., Quantum crystals in a Rydberg-dressed Bose-Einstein condensate, Phys. Rev. A, 86 (2012), 013619.Google Scholar
[6]Muruganandam, P. and Adhikari, S. K., Bose-Einstein condensation dynamics in three dimension by the pseudospectral and finite-difference methods, J. Phys. B: At. Mol. Opt. Phys., 36 (2003), 2501–2513.CrossRefGoogle Scholar
[7]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
[8]Kasamatsu, K., Tsubota, M. and Ueda, M., Structure of vortex lattice in rotating two-component Bose-Einstein condensates, Physica B, 329-333 (2003), 23–24.Google Scholar
[9]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.CrossRefGoogle Scholar
[10]Kasamatsu, K. and Tsubota, M., Vortex sheet in rotating two-component Bose-Einstein condensates, Phys. Rev. A, 79 (2009), 023606.Google Scholar
[11]García-Ripoll, J. J. and Pérez-García, V. M., Optimizing Schrödinger functional using Sobolev gradients: Applications to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23 (2001), 1316–1334.Google Scholar
[12]Bao, W. andDu, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674–1697.Google Scholar
[13]Bao, W., Ground states and dynamics of multi-component Bose-Einstein condensates, SIAM J. Multiscale Model. Simul., 2 (2004), 210–236.Google Scholar
[14]Bao, W., Wang, H. and Markowich, P. A., Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3 (2005), 57–88.Google Scholar
[15]Bao, W., Chern, I.-L. and Lim, F. Y., Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates, J. Comput. Phys., 219 (2006), 836–854.Google Scholar
[16]Bao, W. and Lim, F. Y., Computing ground states of spin-1 Bose-Einstein condensates by the normalized gradient flow, SIAM J. Sci. Comput., 30 (2008), 1925–1948.CrossRefGoogle Scholar
[17]Danaila, I. and Kazemi, P., A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32 (2010), 2447–2467.Google Scholar
[18]Govaerts, W. J. F., Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM Publications, Philadelphia, 2000.Google Scholar
[19]Allgower, E. L. and Georg, K., Introduction to Numerical Continuation Methods, SIAM, Philadelphia, PA, 2003.Google Scholar
[20]Keller, H. B., Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, Berlin, 1987.Google Scholar
[21]Jepson, A. and Spence, A., Folds in solutions of two parameter systems and their calculation, Part I, SIAM J. Numer. Anal., 22 (1985), 347–368.Google Scholar
[22]Rheinboldt, W. C., Numerical Analysis of Parametrized Nonlinear Equations, Wiley, NY, 1986.Google Scholar
[23]Rheinboldt, W. C., On the computation of multidimensional solution manifolds of parametrized equations, Numer. Math., 53 (1988), 165–181.CrossRefGoogle Scholar
[24]Chang, S.-L., Chien, C.-S. and Jeng, B.-W., Tracing the solution surface with folds of a two-parameter system, Int. J. Bifurcation and Chaos, 15 (2005), 2689–2700.Google Scholar
[25]Chang, S.-L., Chien, C.-S. and Jeng, B.-W., Computing wave functions of nonlinear Schrödinger equations: a time-independent approach, J. Comput. Phys., 226 (2007), 104–130.Google Scholar
[26]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
[27]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
[28]Zezyulin, D. A., Alfimov, G. L., Konotop, V. V. and Pérez-García, V. M., Control of nonlinear modes by scattering-length management in Bose-Einstein condensates, Phys. Rev. A, 76 (2007), 013621.Google Scholar
[29]Zezyulin, D. A., Alfimov, G. L., Konotop, V. V. and Pérez-García, V. M., Stability of excited states of a Bose-Einstein condensates in an anharmonic trap, Phys. Rev. A, 78 (2008), 013606.Google Scholar
[30]Chen, H.-S., Chang, S.-L.and Chien, C.-S., Spectral collocation methods using sine functions for a rotating Bose-Einstein condensation in optical lattices, J. Comput. Phys., 231 (2012), 1553–1569.Google Scholar
[31]Chang, S.-L. and Chien, C.-S., Computing multiple peak solutions for Bose-Einstein condensates in optical lattices, Comput. Phys. Commun., 180 (2009), 926–947.Google Scholar
[32]Wang, Y.-S., Jeng, B.-W. and Chien, C.-S., A two-parameter continuation method for rotating two-component Bose-Einstein condensates in optical lattices, Commun. Comput. Phys., 13 (2013), 442–460.Google Scholar
[33]Hsueh, C.-H., Tsai, Y.-C., Wu, K.-S., Chang, M.-S. and Wu, W. C., Pseudospin orders in the supersolid phases in binary Rydberg-dressed Bose-Einstein condensates, Phys. Rev. A, 88 (2013), 043646.Google Scholar
[34]Jeng, B.-W., Chien, C.-S. and Chern, I.-L., Spectral collocation and a two-level continuation scheme for dipolar Bose-Einstein condensates, J. Comput. Phys., 256 (2014), 713–727.CrossRefGoogle Scholar