A splitting compact finite difference method for computing the dynamics of dipolar Bose–Einstein condensate

References

• K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Bose–Einstein condensation of Erbium, Phys. Rev. Lett. 108 (2012), pp. 210401. doi: 10.1103/PhysRevLett.108.210401
   PubMed Web of Science ®Google Scholar
• W. Bao, Mathematical models and numerical methods for Bose–Einstein condensation, Proc. Int. Congr. Math. IV (2014), pp. 971–996.
   Google Scholar
• W. Bao, X. Antoine, and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations, Comput. Phys. Commun., 184 (2013), pp. 2621–2633. doi: 10.1016/j.cpc.2013.07.012
   Web of Science ®Google Scholar
• W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation, Kinet. Relat. Mod. 6 (2013), pp. 1–135. doi: 10.3934/krm.2013.6.1
   Web of Science ®Google Scholar
• W. Bao, Y. Cai, and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose–Einstein condensates, J. Comput. Phys. 229 (2010), pp. 7874–7892. doi: 10.1016/j.jcp.2010.07.001
   Web of Science ®Google Scholar
• W. Bao, S. Jiang, Q. Tang, and Y. Zhang, Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys. 296 (2010), pp. 7874–7892. doi: 10.1016/j.jcp.2010.07.001
   Web of Science ®Google Scholar
• W. Bao, Q. Tang, and Y. Zhang, Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT, Commun. Comput. Phys. 19 (2016), pp. 1141–1166. doi: 10.4208/cicp.scpde14.37s
   Web of Science ®Google Scholar
• M.A. Baranov, Theoretical progress in many-body physics with ultracold dipolar gasses, Phys. Rep. 464 (2008), pp. 71–111. doi: 10.1016/j.physrep.2008.04.007
   Web of Science ®Google Scholar
• P.B. Blakie, C. Ticknor, A.S. Bradley, A.M. Martin, M.J. Davis, and Y. Kawaguchi, Numerical method for evolving the dipolar projected Gross-Pitaevskii equation, Phys. Rev. E 80 (2009), pp. 016703. doi: 10.1103/PhysRevE.80.016703
   Web of Science ®Google Scholar
• Y. Cai, M. Rosenkranz, Z. Lei, and W. Bao, Mean-field regime of trapped dipolar Bose–Einstein condensates in one and two dimensions, Phys. Rev. A 82 (2010), pp. 043623.
   Web of Science ®Google Scholar
• M. Ciment and S. Leventhal, Higher order compact implicit schemes for the wave equation, Math. Comp. 132 (1975), pp. 985–994. doi: 10.1090/S0025-5718-1975-0416049-2
   Google Scholar
• M. Dehghan and A. Taleei, A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comp. Phys. Comm. 181 (2010), pp. 43–51. doi: 10.1016/j.cpc.2009.08.015
   Web of Science ®Google Scholar
• C. Eberlein, S. Giovanazzi, and D. O'Dell, Exact solution of the Thomas–Fermi equation for a trapped Bose–Einstein condensate with dipole–dipole interactions, Phys. Rev. A 71 (2005), pp. 033618. doi: 10.1103/PhysRevA.71.033618
   Web of Science ®Google Scholar
• M.S. Ellio, J.J. Valentini, and D.W. Chandler, Subkelvin cooling NO molecules via ‘billiard-like’ collisions with argon, Science 302 (2003), pp. 1940–1943. doi: 10.1126/science.1090679
   PubMed Web of Science ®Google Scholar
• S. Giovanazzi, P. Pedri, L. Santos, A. Griesmaier, M. Fattori, T. Koch, J. Stuhler, and T. Pfau, Expansion dynamics of a dipolar Bose–Einstein condensate, Phys. Rev. A 74 (2006), pp. 013621. doi: 10.1103/PhysRevA.74.013621
   Web of Science ®Google Scholar
• K. Góral, K. Rzayewski, and T. Pfau, Bose–Einstein condensation with magnetic dipole–dipole forces, Phys. Rev. A 61 (2000), pp. 051601(R). doi: 10.1103/PhysRevA.61.051601
   Web of Science ®Google Scholar
• K. Góral and L. Santos, Ground state and elementary excitations of single and binary Bose–Einstein condensates of trapped dipolar gases, Phys. Rev. A 66 (2002), pp. 023613.
   Web of Science ®Google Scholar
• A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose–Einstein Condensation of Chromium, Phys. Rev. Lett. 94 (2005), pp. 160401. doi: 10.1103/PhysRevLett.94.160401
   PubMed Web of Science ®Google Scholar
• R. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975), pp. 90–109. doi: 10.1016/0021-9991(75)90118-7
   Web of Science ®Google Scholar
• S. Jiang, L. Greengard, and W. Bao, Fast and accurate evaluation of nonlocal coulomb and dipole–dipole interactions via the nonuniform FFT, SIAM J. Sci. Comput. 36 (2014), pp. B777–B794. doi: 10.1137/130945582
   Web of Science ®Google Scholar
• T.F. Jiang and W.C. Su, Ground state of the dipolar Bose–Einstein condensate, Phys. Rev. A 74 (2006), pp. 063602.
   Web of Science ®Google Scholar
• M. Klawunn, R. Nath, P. Pedri, and L. Santos, Transverse instability of straight vortex lines in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 100 (2008), pp. 240403. doi: 10.1103/PhysRevLett.100.240403
   PubMed Web of Science ®Google Scholar
• T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72 (2009), pp. 126401. doi: 10.1088/0034-4885/72/12/126401
   Web of Science ®Google Scholar
• T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, d-Wave Collapse and Explosion of a Dipolar Bose–Einstein Condensate, Phys. Rev. Lett. 101 (2008), pp. 080401. doi: 10.1103/PhysRevLett.101.080401
   PubMed Web of Science ®Google Scholar
• S. Lele, Compact finite difference scheme with spectral-like resolution, J. Comput. Phys. 103 (1992), pp. 16–42. doi: 10.1016/0021-9991(92)90324-R
   Web of Science ®Google Scholar
• C. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations, Math. Comput. 77 (2008), pp. 2141–2153. doi: 10.1090/S0025-5718-08-02101-7
   Web of Science ®Google Scholar
• M. Lu, N.Q. Burdick, and L.B. Lev, Quantum degenerate dipolar Fermi gas, Phys. Rev. Lett. 108 (2012), pp. 215301.
   PubMed Web of Science ®Google Scholar
• D.H. O'Dell and C. Eberlein, Vortex in a trapped Bose–Einstein condensate with dipole-dipole interactions, Phys. Rev. A 75 (2007), pp. 013604.
   Web of Science ®Google Scholar
• N.G. Parker, C. Ticknor, A.M. Martin, and D.H.J. O'Dell, Structure formation during the collapse of a dipolar atomic Bose–Einstein condensate, Phys. Rev. A 79 (2009), pp. 013617. doi: 10.1103/PhysRevA.79.013617
   Web of Science ®Google Scholar
• P. Pedri and L. Santos, Two-dimensional bright solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 95 (2005), pp. 200404. doi: 10.1103/PhysRevLett.95.200404
   PubMed Web of Science ®Google Scholar
• L.P. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Oxford University, New York, 2003.
   Google Scholar
• S. Ronen, D.C.E. Bortolotti, and J.L. Bohn, Bogoliubov modes of a dipolar condensate in a cylindrical trap, Phys. Rev. A 74 (2006), pp. 013623.
   Web of Science ®Google Scholar
• J.M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Optical production of ultracold polar molecules, Phys. Rev. Lett. 94 (2005), pp. 203001. doi: 10.1103/PhysRevLett.94.203001
   PubMed Web of Science ®Google Scholar
• W. Spotz and G. Carey, High-order compact scheme for the steady stream-function vorticity equations, Int. J. Num. Meth. Eng. 38 (1995), pp. 3497–3512. doi: 10.1002/nme.1620382008
   Web of Science ®Google Scholar
• I. Tikhonenkov, B.A. Malomed, and A. Vardi, Anisotropic solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 100 (2008), pp. 090406. doi: 10.1103/PhysRevLett.100.090406
   PubMed Web of Science ®Google Scholar
• D. Wang, J. Qi, M.F. Stone, O. Nikolayeva, H. Wang, B. Hattaway, S.D. Gensemer, P.L. Gould, E.E. Eyler, and W.C. Stwalley, Photoassociative production and trapping of Ultracold KRb molecules, Phys. Rev. Lett. 93 (2004), pp. 243005.
   PubMed Web of Science ®Google Scholar
• H. Wang, Y. Zhang, X. Ma, J. Qiu, and Y. Liang, An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions, Comput. Math. Appl. 71 (2016), pp. 1843–1860. doi: 10.1016/j.camwa.2016.02.022
   Web of Science ®Google Scholar
• R.M. Wilson, S. Ronen, and J.L. Bohn, Stability and excitations of a dipolar Bose–Einstein condensate with a vortex, Phys. Rev. A 79 (2009), pp. 013621. doi: 10.1103/PhysRevA.79.013621
   Web of Science ®Google Scholar
• R.M. Wilson, S. Ronen, J.L. Bohn, and H. Pu, Manifestations of the Roton mode in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 100 (2008), pp. 245302. doi: 10.1103/PhysRevLett.100.245302
   PubMed Web of Science ®Google Scholar
• B. Xiong, J. Gong, H. Pu, W. Bao, and B. Li, Symmetry breaking and self-trapping of a dipolar Bose–Einstein condensate in a double-well potential, Phys. Rev. A 79 (2009), pp. 013626. doi: 10.1103/PhysRevA.79.013626
   Web of Science ®Google Scholar
• S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A 61 (2000), pp. 041604(R). doi: 10.1103/PhysRevA.61.041604
   Web of Science ®Google Scholar
• S. Yi and L. You, Expansion of a dipolar condensate, Phys. Rev. A 67 (2003), pp. 045601. doi: 10.1103/PhysRevA.67.045601
   Web of Science ®Google Scholar
• Y. Zhang, Optimal error estimates of compact finite difference discretizations for the Schrödinger–Poisson system, Commun. Comput. Phys. 13 (2013), pp. 1357–1388. doi: 10.4208/cicp.251011.270412a
   Web of Science ®Google Scholar
• X. Zhong, High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition, J. Comput. Phys. 144 (1998), pp. 662–709. doi: 10.1006/jcph.1998.6010
   Web of Science ®Google Scholar