Skip to main content Accessibility help

Accurate and Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates via the Nonuniform FFT

  • Weizhu Bao (a1), Qinglin Tang (a2) (a3) (a4) and Yong Zhang (a5) (a6)


We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.


Corresponding author

*Corresponding author. Email Bao), (Q. Tang), Zhang)


Hide All
[1]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover, 1965.
[2]Aikawa, K., Frisch, A., Mark, M., Baier, S., A., Rietzler, Grimm, R. and Ferlaino, F., Bose-Einstein condensation of Erbium, Phys. Rev. Lett., 108 (2012), 210401.
[3]andersen, J. O., Theory of the weakly interacting Bose gas, Rev. Mod. Phys., 76 (2004), 599639.
[4]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), 198201.
[5]Antoine, X., Bao, W. and Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 26212633.
[6]Bao, W., Ben Abdallah, N. and Cai, Y., Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 17131741.
[7]Bao, W. and Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1135.
[8]Bao, W., Cai, Y. and Wang, H., Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 78747892.
[9]Bao, W., Chern, I-L. and Lim, F., Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates, J. Comput. Phys., 219 (2006), 836854.
[10]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), 16741697.
[11]Bao, W. and Jaksch, D., An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equation with a focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 14061426.
[12]Bao, W., Jaksch, D. and Markowich, P. A., Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), 318342.
[13]Bao, W., Jaksch, D. and Markowich, P. A., Three dimensional simulation of jet formation in collapsing condensates, J. Phys. B: At. Mol. Opt. Phys., 37 (2004), 329343.
[14]Bao, W., Marahrens, D., Tang, Q. and Zhang, Y., A simple and efficient numerical method for computing dynamics of rotating dipolar Bose-Einstein condensation via a rotating Lagrange coordinate, SIAM J. Sci. Comput., 35 (2013), A2671A2695.
[15]Bao, W., Jian, H., Mauser, N. J. and Zhang, Y., Dimension reduction of the Schrödinger equation with Coulomb and anisotropic confining potentials, SIAM J. Appl. Math., 73 (6) (2013) 21002123.
[16]Bao, W., Jiang, S., Tang, Q. and Zhang, Y., Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296 (2015), 7289.
[17]Baranov, M. A., Theoretical progress in many body physics of dipolar gases, Phys. Rep., 464 (2008), 71111.
[18]Blakie, P. B., Ticknor, C., Bradley, A. S., Martin, A. M., Davis, M. J. and Kawaguchi, Y., Numerical method for evolving the dipolar projected Gross-Pitaevskii equation, Phys. Rev. E, 80 (2009), aritcle 016703.
[19]Bloch, I., Dalibard, J. and Zwerger, W., Many body physics with ultracold gases, Rev. Mod. Phys., 80 (2008), 885965.
[20]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), 16871690.
[21]Cai, Y., Rosenkranz, M., Lei, Z. and Bao, W., Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), article 043623.
[22]Carles, R., Markowich, P. A. and Sparber, C., On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 25692590.
[23]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), 39693973.
[24]FETTER, A. L., Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys., 81 (2009), 647691.
[25]Góral, K., Rzayewski, K. and Pfau, T., Bose-Einstein condensation with magnetic dipole-dipole forces, Phys. Rev. A, 61 (2000), article 051601(R).
[26]Griesmaier, A., Werner, J., Hensler, S., Stuhler, J. and Pfau, T., Bose-Einstein condensation of Chromium, Phys. Rev. Lett., 94 (2005), article 160401.
[27]Haken, H., Brewer, W. D. and Wolf, H. C., Molecular Physics and Elements of Quantum Chemistry, Springer, 1995.
[28]Huang, Z., Markowich, P. A. and Sparber, C., Numerical simulation of trapped dipolar quantum gases: collapse studies and vortex dynamics, Kinet. Relat. Mod., 3 (2010), 181194.
[29]Jiang, S., Greengard, L. and Bao, W., Fast and accurate evaluation of nonlocal Coulomb and dipole-dipole interactions via the nonuniform FFT, SIAM J. Sci. Comput., 36 (2014), B777B794.
[30]Jiang, T. F. and Su, W. C., Ground state of the dipolar Bose-Einstein condensate, Phys. Rev. A, 74 (2006), article 063602.
[31]Kawaguchi, Y. and Ueda, M., Spinor Bose-Einstein condensates, Phys. Rep., 520 (2012), 253381.
[32]Kumar, R. K., Young-S., L.E., Vudragović, D., Balaz, A., Muruganandam, P. and Adhikari, S.K., Fortran and C programs for the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap, Comput. Phys. Commun., 195 (2015), 117128.
[33]Lahaye, T., Koch, T., Fröhlich, B., Fattori, M., Metz, J., Griesmaier, A., Gio-Vanazzi, S. and Pfau, T., Strong dipolar effects in a quantum ferrofluid, Nature, 448 (2007), 672675.
[34]Lahaye, T., Menotti, C., Santos, L., Lewenstein, M. and Pfau, T., The physics of dipolar bosonic quantum gases, Rep. Prog. Phys., 72 (2009), 126401.
[35]Lahaye, T., Metz, J., Fröhlich, B., Koch, T., Meister, M., Griesmaier, A., Pfau, T., Saito, H., Kawaguchi, Y. and Ueda, M., D-wave collapse and explosion of a dipolar Bose-Einstein condensate, Phys. Rev. Lett., 101 (2008), article 080401.
[36]Leggett, A. J., Bose-Einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys., 73 (2001), 307356.
[37]Levitt, M. H., Spin Dynamics: Basics of Nuclear Magnetic Resonance, Wiley, 2008.
[38]Lu, M., Burdick, N. Q., Youn, S. H. and Lev, B. L., Strongly dipolar Bose-Einstein condensate of Dysprosium, Phys. Rev. Lett., 107 (2011), article 190401.
[39]Mauser, N. J. and Zhang, Y., Exact artificial boundary condition for the Poisson equation in the simulation of the 2D Schrödinger-Poisson system, Commun. Comput. Phys., 16 (3) (2014), 764780.
[40]Morsch, O. and Oberthaler, M., Dynamics of Bose-Einstein condensates in optical lattices, Rev. Mod. Phys., 78 (2006), 179215.
[41]Ni, K.-K., Ospelkaus, S., de miranda, M. H. G., Pe'er, A., Neyenhuis, B., Zirbel, J. J., Kotochigova, S., Julienne, P. S., Jin, D. S. and Ye, J., A high phase-space-density gas of polar molecules, Science, 322 (2008), 231235.
[42]O'dell, D. H. J., Giovanazzi, S. and Eberlein, C., Exact hydrodynamics of a trapped dipolar Bose-Einstein condensate, Phys. Rev. Lett., 92 (2004), article 250401.
[43]Parker, N. G., Ticknor, C., Martin, A. M. and O'dell, D. H. J., Structure formation during the collapse of a dipolar atomic Bose-Einstein condensate, Phys. Rev. A, 79 (2009), article 013617.
[44]Pitaevskii, L. P. and Stringari, S., Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.
[45]Pollack, S. E., Dries, D., Junker, M., Chen, Y. P., Corcovilos, T. A. and Hulet, R. G., Extreme tunability of interactions in a 7 Li Bose-Einstein condensate, Phys. Rev. Lett., 102 (2009), 090402.
[46]Santos, L., Shlyapnikov, G., Zoller, P. and Lewenstein, M., Bose-Einstein condensation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), 17911797.
[47]Shuman, E. S., Barry, J. F. and Demille, D., Laser cooling of a diatomic molecule, Nature, 467 (2010), 820823.
[48]Ticknor, C., Parker, N.G., Melatos, A., Cornish, S.L., O'dell, D.H.J. and Martin, A.M., Collapse times of dipolar Bose-Einstein condensates, Phys. Rev. A, 78 (2008), article 061607.
[49]Vengalattore, M., Leslie, S. R., Guzman, J. and Stamper-Kurn, D. M., Spontaneously modulated spin textures in a dipolar spinor Bose-Einstein condensate, Phys. Rev. Lett., 100 (2008), 170403.
[50]Yi, S. and You, L., Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63 (2001), article 053607.
[51]Zhang, Y. and Dong, X., On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), 26602676.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed