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Ground States of Two-component Bose-Einstein Condensates with an Internal Atomic Josephson Junction

  • Weizhu Bao (a1) and Yongyong Cai (a2)
Abstract
Abstract

In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.

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Corresponding author
Corresponding author. Email: bao@math.nus.edu.sg
Corresponding author. Email: caiyongyong@nus.edu.sg
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[1]Anderson M. H., Ensher J. R., Matthewa M. R., Wieman C. E. and Cornell E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), pp. 198201.
[2]Bao W., Ground states and dynamics of multicomponent Bose-Einstein condensates, Multiscale Model. Simul., 2 (2004), pp. 210236.
[3]Bao W., Analysis and efficient computation for the dynamics of two-component Bose-Einstien condensates, Contemporary Math., 473 (2008), pp. 126.
[4]Bao W. and Du Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 16741697.
[5]Bao W., Jaksch D. and Markowich P A., Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318342.
[6]Bao W. and Lim F. Y., Computing ground states ofspin-1 Bose-Einstein condensates by the normalized gradient flow, SIAM J. Sci. Comput., 30 (2008), pp. 19251948.
[7]Bao W. and Wang H., A mass and magnetization conservative and energy diminishing numerical method for computing ground state ofspin-1 Bose-Einstein condensates, SIAM J. Numer. Anal., 45 (2007), pp. 21772200.
[8]Bradley C. C., Sackett C. A., Tollett J. J. and Hulet R. G., Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction, Phys. Rev. Lett., 75 (1995), pp. 16871690.
[9]Cafferelli L. and Lin F. H., An optimal partition problem for eigenvalues, J. Sci. Comput., 31 (2007), pp. 518.
[10]Caliari M. and Squassina M., Location andphase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), pp. 117137.
[11]Caliari M., Ostermann A., Rainer S. and Thalhammer M., A minimisation approach for computing the ground state of Gross-Pitaevskii systems, J. Comput. Phys., 228 (2009), pp. 349360.
[12]Chang S. M., Lin W. W. and Shieh S. F., Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. Comput. Phys., 202 (2005), pp. 367390.
[13]Chang S. M., Lin C. S., Lin T. C. and Lin W. W., Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Physica D, 196 (2004), pp. 341361.
[14]Chiofalo M. L., Succi S. and Tosi M. P., Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62 (2000), pp. 74387444.
[15]Davis K. B., Mewes M. O., Andrews M. R., Van Druten N. J., Durfee D. S., Kurn D. M. and Ketterle W., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), pp. 39693973.
[16]Du Q. and Lin F. H., Numerical approximations of a norm preserving gradient flow and applications to an optimal partition problem, Nonlinearity, 22 (2009), pp. 6783.
[17]Hall D. S., Matthews M. R., Ensher J. R., Wieman C. E. and Cornell E. A., Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp 15391542.
[18]Hall D. S., Matthews M. R., Wieman C. E. and Cornell E. A., Measurements of relative phase in two-component Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp 15431546.
[19]Ho T. L. and Shenoy V. B., Binary mixtures of Bose condensates of alkali atoms, Phys. Rev. Lett., 77 (1996), pp. 32763279.
[20]Jaksch D., Gardiner S. A., Schulze K., Cirac J. I. and Zoller P., Uniting Bose-Einstein condensates in optical resonators, Phys. Rev. Lett., 86 (2001), pp. 47334736.
[21]Kasamatsu K. and Tsubota M., Nonlinear dynamics for vortex formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 67 (2003), article 033610.
[22]Kasamatsu K., Tsubota M. and Ueda M., Vortex phase diagram in rotating two-component Bose-Einstein condensates, Phys. Rev. Lett., 91 (2003), article 150406.
[23]Lieb E. H., Seiringer R. and Yngvason J., Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energyfunctional, Phy. Rev. A, 61 (2000), article 043602.
[24]Lieb E. H. and Solovej J. P, Ground state energy of the two-component charged Bose gas, Comm. Math. Phys., 252 (2004), pp. 485534.
[25]Lin T. -C. and Wei J., Ground state of N coupled nonlinear Schrödinger Equations in ℝn, n ≤ 3, Comm. Math. Phys., 255 (2005), pp. 629653.
[26]Lin T. -C. and Wei J., Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Diff. Equ., 229 (2006), pp. 538569.
[27]Liu Z., Two-component Bose-Einstein condensates, J. Math. Anal. Appl., 348 (2008), pp. 274285.
[28]Myatt C. J., Burt E. A., Ghrist R. W., Cornell E. A. and Wieman C. E., Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78 (1997), pp. 586589.
[29]Pitaevskii L. P. and Stringari S., Bose-Einstein condensation, Clarendon Press, 2003.
[30]Schneider J. and Schenzle A., Output from an atom laser: theory vs. experiment, Appl. Phys. B, 69 (1999), pp. 353356.
[31]Schneider J. and Schenzle A., Investigations of a two-mode atom-laser model, Phys. Rev. A, 61 (2000), article 053611.
[32]Schneider B. I. and Feder D. L., Numerical approach to the ground and excited states of a Bose-Einstein condensed gas confined in a completely anisotropic trap, Phys. Rev. A, 59 (1999), pp. 22322242.
[33]Simon L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), pp. 525571.
[34]Stamper-Kurn D. M., Andrews M. R., Chikkatur A. P., Inouye S., Miesner H.-J., Stenger J. and Ketterle W., Optical confinement of a Bose-Einstein condensate, Phys. Rev. Lett., 80 (1998), pp. 20272030.
[35]Wang H., Numerical simulations on stationary states for rotating two-component Bose-Einstein condensates, J. Sci. Comput., 38 (2009), pp. 149163.
[36]Weinstein M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phy. 87 (1983), pp. 567576.
[37]Williams J., Walser R., Cooper J., Cornell E., and Holland M., Nonlinear Josephson-type oscillations of a driven two-component Bose-Einstein condensate, Phys. Rev. A, 59 (1999), article R31-R34.
[38]Zhang Y., Bao W. and Li H., Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Physica D, 234 (2007), pp. 4969.
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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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