Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow

SIAM Journal on Scientific Computing

Volumn 25, Issue 5, 2004, Pages 1674-1697

  Bao, Weizhu ^a   Du, Qiang ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Bose Einstein condensate; Continuous normalized gradient flow; Energy diminishing; Gross Pitaevskii equation; Ground state; Monotone scheme; Nonlinear Schr dinger equation; Time splitting spectral method

Indexed keywords

COMPUTATIONAL METHODS; FINITE DIFFERENCE METHOD; GRADIENT METHODS; GROUND STATE; LASER BEAMS; NONLINEAR EQUATIONS; PROBLEM SOLVING;

BOSE-EINSTEIN CONDENSATES (BEC); CONTINUOUS NONRMALIZED GRADIENT FLOW; ENERGY DIMINISHING; GROSS-PITAEVSKII EQUATIONS; MONOTONE SCHEME; NONLINEAR SCHRÖDINGER EQUATIONS (NLS); TIME SPLITTING SPECTRAL METHOD;

GAS CONDENSATES;

EID: 6344289363     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827503422956     Document Type: Article

References (36)

1. S.K. ADHIKARI, Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms, Phys. Lett. A, 265 (2000), pp. 91-96.
2. 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.
3. J.R. ANGLIN AND W. KETTERLE, Bose-Einstein condensation of atomic gases, Nature, 416 (2002), pp. 211-218.
4. W. BAO, S. JIN, AND P.A. MARKOWICH, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524.
5. W. BAO, S. JIN, AND P.A. MARKOWICH, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), pp. 27-64.
6. 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.
7. 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.
8. W. BAO AND D. JAKSCH, An explicit unconditionally stable numerical methods for solving damped nonlinear Schrödinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), pp. 1406-1426.
9. B.M. CARADOC-DAVIS, R.J. BALLAGH, AND K. BURNETT, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), pp. 895-898.
10. 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.
11. 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.
12. M.M. CERIMELE, F. PISTELLA, AND S. SUCCI, Particle-inspired scheme for the Gross-Pitaevskii equation: An application to Base-Einstein condensation, Comput. Phys. Comm., 129 (2000), pp. 82-90.
13. 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), pp. 7438-7444.
14. E. CORNELL, Very cold indeed: The nanokelvin physics of Bose-Einstein condensation, J. Res. Natl. Inst. Stan., 101 (1996), pp. 419-434.
15. F. DALFOVO, S. GIORGINI, L.P. PITAEVSKII, AND S. STRINGARI, Theory of Bose-Einstein condensation in trapped gases, Rev, Modern Phys., 71 (1999), pp. 463-512.
16. F. DALFOVO AND S. STRINGARI, Bosons in anisotropic traps: Ground state and vortices, Phys. Rev. A, 53 (1996), pp. 2477-2485.
17. R.J. DODD, Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates, J. Res. Natl. Inst. Stan., 101 (1996), pp. 545-552.
18. Q. Du, Numerical computations of quantized vortices in Bose-Einstein condensate, in Recent Progress in Computational and Applied PDEs, T. Chan et al., eds., Kluwer Academic, Dordrecht, The Netherlands, 2002, pp. 155-168.
19. M. EDWARDS AND K. BURNETT, Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms, Phys. Rev. A, 51 (1995), pp. 1382-1386.
20. A. GAMMAL, T. FREDERICO, AND L. TOMIO, Improved numerical approach for the time-independent Gross-Pitaevskii nonlinear Schrödinger equation, Phys. Rev. E, 60 (1999), pp. 2421-2424.
21. G.H. GOLUB AND C.F. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1989.
22. M. GREINER, O. MANDEL, T. ESSLINGER, T.W. HÄNSCH, AND I. BLOCH, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 415 (2002), pp. 39-45.
23. E.P. GROSS, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), pp. 454-477.
24. B. JACKSON, J.F. MCCANN, AND C.S. ADAMS, Vortex formation in dilute inhomogeneous Bose-Einstein condensates, Phys. Rev. Lett., 80 (1998), pp. 3903-3906.
25. 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.
26. L. LANDAU AND E. LIFSCHITZ, Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, New York, 1977.
27. I.N. LEVINE, Quantum Chemistry, 5th ed., Prentice-Hall, New York, 1991.
28. 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), article 043602.
29. F.-H. LIN AND Q. DU, Ginzburg-Landau vortices: Dynamics, pinning, and hysteresis, SIAM J. Math. Anal., 28 (1997), pp. 1265-1293.
30. E. LUNDH, C.J. PETHICK, AND H. SMITH, Vortices in Bose-Einstein-condensed atomic clouds, Phys. Rev. A, 59 (1998), pp. 4816-4823.
31. L.P. PITAEVSKII, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), pp. 451-454.
32. D.S. ROKHSAR, Vortex stability and persistent currents in trapped Bose gas, Phys. Rev. Lett., 79 (1997), pp. 2164-2167.
33. P.A. RUPRECHT, M.J. HOLLAND, K. BURRETT, AND M. EDWARDS, Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A, 51 (1995), pp. 4704-4711.
34. B.I. SCHNEIDER AND D.L. FEDER, 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. 2232-2242.
35. L. SIMON, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math. (2), 118 (1983), pp. 525-571.
36. C. SULEM AND P.L. SULEM, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, Springer-Verlag, New York, 1999.