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Ground states and dynamics of rotating Bose-Einstein condensates

  • Weizhu Bao
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Since its realization in dilute bosonic atomic gases [7], [23], Bose-Einstein condensation of alkali atoms and hydrogen has been produced and studied extensively in the laboratory [1], and has permitted an intriguing glimpse into the macroscopic quantum world. In view of potential applications [38], [61], [63], the study of quantized vortices, which are well-known signatures of superfluidity, is one of the key issues. In fact, bulk superfluids are distinguished from normal fluids by their ability to support dissipationless flow. Such persistent currents are intimately related to the existence of quantized vortices, which are localized phase singularities with integer topological charge [39]. The superfluid vortex is an example of a topological defect that is well known in superconductors [52] and in liquid helium [33]. The occurrence of quantized vortices in superfluids has been the focus of fundamental theoretical and experimental work [33]. Different research groups have obtained quantized vortices in Bose-Einstein condensates (BECs) experimentally, e.g., the JILA group [35], [57], the ENS group [56] and the MIT group [1], [32]. Currently, there are at least two typical ways to generate quantized vortices from a BEC ground state: (i) impose a laser beam rotating with an angular velocity on the magnetic trap holding the atoms to create a harmonic anisotropic potential [51], [4], [75]; or (ii) add to the stationary magnetic trap a narrow, moving Gaussian potential, representing a far-blue detuned laser [49], [50], [24], [25], [10], [12].

Keywords

Einstein Condensate Vortex Lattice Ground State Solution Quantized Vortex Rotational Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), p. 476.CrossRefGoogle Scholar
  2. [2]
    S. K. Adhikari and P. Muruganandam, Effect of an implusive force on vortices in a rotating Bose-Einstein condensate, Phys. Lett. A, 301(2002), pp. 333–339.CrossRefGoogle Scholar
  3. [3]
    A. Aftalion and I. Danaila, Three-dimensional vortex configurations in a rotating Bose-Einstein condensate, Phys. Rev. A, 68 (2003), article 023603.Google Scholar
  4. [4]
    A. Aftalion and Q. Du, Vortices in a rotating Bose-Einstein condensate: critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A, 64 (2001), article 063603.Google Scholar
  5. [5]
    A. Aftalion, Q. Du and Y. Pomeau, Dissipative flow and vortex shedding in the Painlevé boundary layer of a Bose Einstein condensate, Phy. Rev. Lett., 91(2003), article 090407.Google Scholar
  6. [6]
    A. Aftalion and T. Riviere, Vortex energy and vortex bending for a rotating Bose-Einstein condensate, Phys. Rev. A, 64 (2001), article 043611.Google Scholar
  7. [7]
    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), p. 198.CrossRefGoogle Scholar
  8. [8]
    W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates, Multiscale Modeling and Simulation, 2 (2004), pp. 210–236.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), pp. 758–786.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    W. Bao and D. Jaksch, An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), pp. 1406–1426.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii Equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318–342.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    W. Bao, D. Jaksch and P. A. Markowich, Three dimensional simulation of jet formation in collapsing condensates, J. Phys. B: At. Mol. Opt. Phys., 37 (2004), pp. 329–343.CrossRefGoogle Scholar
  14. [14]
    W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487–524.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W. Bao, S. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes, SIAM J. Sci. Comput., 25 (2003), pp. 27–64.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    W. Bao, F.Y. Lim and Y. Zhang, Energy and chemical potential asymptotics for the ground state of Bose-Einstein condensates in the semiclassical regime, Trans. Theory Stat. Phys., to appear.Google Scholar
  17. [17]
    W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), pp. 2010–2028.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    W. Bao and W. Tang, Ground state solution of trapped interacting Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003), pp. 230–254.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    W. Bao and H. Wang, An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217 (2006), pp. 612–626.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    W. Bao, H. Wang and P. A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Comm. Math. Sci., 3 (2005), pp. 57–88.MATHMathSciNetGoogle Scholar
  21. [21]
    W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Mod. Meth. Appl. Sci., 15 (2005), pp. 1863–1896.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    I. Bialynicki-Birula and Z. Bialynicki-Birula, Center-of-mass motion in the many-body theory of Bose-Einstein condensates, Phys. Rev. A, 65 (2002), article 063606.Google Scholar
  23. [23]
    C. C. Bradley, C. A. Sackett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75 (1995), p. 1687.CrossRefGoogle Scholar
  24. [24]
    B. M. Caradoc-Davis, R. J. Ballagh and P. B. Blakie, Three-dimensional vortex dynamics in Bose-Einstein condensates, Phys. Rev. A, 62 (2000), article 011602.Google Scholar
  25. [25]
    B. M. Caradoc-Davis, R. J. Ballagh and P. B. Blakie, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), p. 895.CrossRefGoogle Scholar
  26. [26]
    Y. Castin and R. Dum, Bose-Einstein condensates with vortices in rotating traps, Eur. Phys. J. D, 7 (1999), pp. 399–412.CrossRefGoogle Scholar
  27. [27]
    M. M. Cerimele, M. L. Chiofalo, F. Pistella, S. Succi and M. P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose-Einstein condensates, Phys. Rev. E, 62 (2000), pp. 1382–1389.CrossRefGoogle Scholar
  28. [28]
    M. M. Cerimele, F. Pistella and S. Succi, Particle-inspired scheme for the Gross-Pitaevskii equation: An application to Bose-Einstein condensation, Comput. Phys. Comm., 129 (2000), pp. 82–90.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    S.-M. Chang, W.-W. Lin and S.-F. Shieh, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. Comput. Phys., 64 (2001), article 053611.Google Scholar
  30. [30]
    M. L. Chiofalo, S. Succi and M. P. Tosi, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62 (2000), p. 7438.CrossRefGoogle Scholar
  31. [31]
    F. Dalfovo and S. Giorgini, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71(1999), p. 463.CrossRefGoogle Scholar
  32. [32]
    K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), p. 3969.CrossRefGoogle Scholar
  33. [33]
    R. J. Donnelly, Quantizied Vortices in Helium II, Cambridge University Press, London, 1991.Google Scholar
  34. [34]
    P. Engels, I. Coddington, P. Haijan and E. Cornell, Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates, Phy. Rev. Lett., 89 (2002), article 100403.Google Scholar
  35. [35]
    J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman and E. A. Cornell, Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation, Phys. Rev. Lett., 77 (1996), p. 4984.CrossRefGoogle Scholar
  36. [36]
    D. L. Feder, C. W. Clark and B. I. Schneider, Nucleation of vortex arrays in rotating anisotropic Bose-Einstein condensates, Phys. Rev. A, 61(1999), article 011601.Google Scholar
  37. [37]
    D. L. Feder, C. W. Clark and B. I. Schneider, Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps, Phys. Rev. Lett., 82 (1999), p. 4956.CrossRefGoogle Scholar
  38. [38]
    D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark, Anomalous modes drive vortex dynamics in confined Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), pp. 564–567.CrossRefGoogle Scholar
  39. [39]
    A. L. Fetter and A. A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate, J. Phys. Condens. Matter, 13 (2001), R135–194.CrossRefGoogle Scholar
  40. [40]
    J. J. Garcia-Ripoll and V. M. Perez-Garcia, Vortex bending and tightly packed vortex latices in Bose-Einstein condensates, Phys. Rev. A, 64 (2001), article 053611.Google Scholar
  41. [41]
    J. J. Garcia-Ripoll and V. M. Perez-Garcia, Stability of vortices in inhomogeneous Bose-Einstein condensates subject to rotation: A three dimensional analysis, Phys. Rev. A, 60 (1999), pp. 4864–4874.CrossRefGoogle Scholar
  42. [42]
    J. J. Garcia-Ripoll and V. M. Perez-Garcia, Optimizing Schrödinger functionals using Sobolev gradients: application to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23 (2001), pp. 1316–1334.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    J. J. Garcia-Ripoll and V. M. Perez-Garcia, Vortex nucleation and hysteresis phenomena in rotating Bose-Einstein condensate, Phys. Rev. A, 63 (2001), article 041603.Google Scholar
  44. [44]
    J. J. Garcia-Ripoll, V. M. Perez-Garcia and V. Vekslerchik, Construction of exact solutions by spatial translations in inhomogeneous nonlinear Schrödinger equations, Phys. Rev. E, 64 (2001), article 056602.Google Scholar
  45. [45]
    R. Glowinski and P. LeTallec, Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.Google Scholar
  46. [46]
    E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20 (1961), p. 454.MATHCrossRefGoogle Scholar
  47. [47]
    D. S. Hall, M. R. Mattthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp. 1539–1542.CrossRefGoogle Scholar
  48. [48]
    B. Jackson, J. F. McCann and C. S. Adams, Vortex formation in dilute inhomogeneous Bose-Einstein condensates, Phys. Rev. Lett., 80 (1998), p. 3903.CrossRefGoogle Scholar
  49. [49]
    B. Jackson, J. F. McCann and C. S. Adams, Vortex line and ring dynamics in trapped Bose-Einstein condensates, Phys. Rev. A, 61 (1999), article 013604.Google Scholar
  50. [50]
    D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett., 81 (1998), pp. 3108–3111.CrossRefGoogle Scholar
  51. [51]
    K. Kasamatsu, M. Tsubota and M. Ueda, Nonlinear dynamics of vortex lattice formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 67 (2003), article 033610.Google Scholar
  52. [52]
    L. Laudau and E. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, New York, 1977.Google Scholar
  53. [53]
    E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61(2000), p. 3602.CrossRefGoogle Scholar
  54. [54]
    E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, 3rd edition, Pergamon, Oxford, 1980.Google Scholar
  55. [55]
    E. Lundh, C. J. Pethick and H. Smith, Vortices in Bose-Einstein-condensated atomic clouds, Phys. Rev. A, 58 (1998), pp. 4816–4823.CrossRefGoogle Scholar
  56. [56]
    K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000), p. 806.CrossRefGoogle Scholar
  57. [57]
    M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999), p. 2498.CrossRefGoogle Scholar
  58. [58]
    A. Minguzzi, S. Succi, F. Toschi, M. P. Tosi and P. Vignolo, Numerical methods for atomic quantum gases with applications to Bose-Einstein condensates and to ultracold fermions, Phys. Rep., 395 (2004), pp. 223–355.CrossRefGoogle Scholar
  59. [59]
    M. Modugno, L. Pricoupenko and Y. Castin, Bose-Einstein condensates with a bent vortex in rotating traps, Eur. Phys. J. D, 22 (2003), pp. 235–257.Google Scholar
  60. [60]
    A. A. Penckwitt and R.J. Ballagh, The nucleation, growth and stabilization of vortex lattices, Phys. Rev. Lett., 89 (2002), article 268402.Google Scholar
  61. [61]
    C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, London, 2002.Google Scholar
  62. [62]
    L. P. Pitaevskii, Vortex lines in a imperfect Bose gas, Zh. Eksp. Teor. Fiz., 40 (1961), p. 646. (Sov. Phys. JETP, 13 (1961), p. 451).Google Scholar
  63. [63]
    L. P. Pitaevskii and S. Stringari, Bose-Einstein condensation, Clarendon Press, Oxford, New York, 2003.MATHGoogle Scholar
  64. [64]
    C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 87 (2001), article 210402.Google Scholar
  65. [65]
    P. Rosenbuch, V. Bretin and J. Dalibard, Dynamics of a single vortex line in a Bose-Einstein condensate, Phys. Rev. Lett., 89 (2002), article 200403.Google Scholar
  66. [66]
    R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys., 229 (2002), pp. 491–509.MATHCrossRefMathSciNetGoogle Scholar
  67. [67]
    R. Seiringer, Ground state asymptotics of a dilute rotating gas, J. Phys. A: Math. Gen., 36 (2003), pp. 9755–9778.MATHCrossRefMathSciNetGoogle Scholar
  68. [68]
    L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), pp. 525–571.CrossRefGoogle Scholar
  69. [69]
    T. P. Simula, P. Engels, I. Coddington, V. Schweikhard, E. A. Cornell and R. J. Ballagh, Observations on sound propagation in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett., 94 (2005), article 080404.Google Scholar
  70. [70]
    T. P. Simula, A. A. Penckwitt and R. J. Ballagh, Giant vortex lattice deformations in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett., 92 (2004), article 060401.Google Scholar
  71. [71]
    S. Sinha and Y. Castin, Dynamic instability of a rotating Bose-Einstein condensate, Phys. Rev. Lett., 87 (2001), article 190402.Google Scholar
  72. [72]
    G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), pp. 505–517.CrossRefMathSciNetGoogle Scholar
  73. [73]
    A. A. Svidzinsky and A. L. Fetter, Dynamics of a vortex in a trapped Bose-Einstein condensate, Phys. Rev. A, 62 (2000), article 063617.Google Scholar
  74. [74]
    M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 65 (2000), article 023603.Google Scholar
  75. [75]
    J. E. Williams and M. J. Hooand, Preparing topological states of a Bose-Einstein condensate, Nature, 401(1999), p. 568.CrossRefGoogle Scholar
  76. [76]
    Y. Zhang and W. Bao, Dynamics of the center of mass in rotating Bose-Einstein condensates, Appl. Numer. Math., to appear.Google Scholar

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© Birkhäuser Boston 2007

Authors and Affiliations

  • Weizhu Bao
    • 1
  1. 1.Department of Mathematics and Center for Computational Science and EngineeringNational University of SingaporeSingapore

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