Fast and accurate evaluation of nonlocal coulomb and dipole-dipole interactions via the nonuniform FFT

SIAM Journal on Scientific Computing

Volumn 36, Issue 5, 2014, Pages B777-B794

  Jiang, Shidong ^a   Greengard, Leslie ^b   Bao, Weizhu ^c  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^b NEW YORK UNIVERSITY   (United States)

^c NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Coulomb interaction; Dipole dipole interaction; Interaction energy; Nonlocal; Nonuniform FFT; Poisson equation

Indexed keywords

CONVOLUTION; DEGREES OF FREEDOM (MECHANICS); DIPOLE MOMENT; DISTRIBUTION FUNCTIONS; ELECTRIC DIPOLE MOMENTS; ELECTROSTATICS; FUNCTION EVALUATION; INTEGRAL EQUATIONS; PHASE SPACE METHODS; POISSON EQUATION; WAVE FUNCTIONS;

COMPLEX-VALUED; DIPOLE DIPOLE INTERACTIONS; FAST AND ACCURATE ALGORITHMS; FOURIER INTEGRALS; FOURIER METHODS; FREE SPACES; INTERACTION ENERGIES; NONLOCAL; NONUNIFORM FFT; PHASE SPACES;

FAST FOURIER TRANSFORMS;

EID: 84911435399     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/130945582     Document Type: Article

References (69)

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
2. B. K. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), pp. 1551-1584.
3. W. Bao, N. Ben Abdallah, and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), pp. 1713-1741.
4. W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), pp. 1-135.
5. 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.
6. 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.
7. 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.
8. W. Bao, H. Jian, N. J. Mauser, and Y. Zhang, Dimension reduction of the Schrödinger equation with Coulomb and anisotropic confining potentials, SIAM J. Appl. Math., 73 (2013), pp. 2100-2123.
9. W. Bao, D. Marahrens, Q. Tang, and Y. Zhang, A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35 (2013), pp. A2671-A2695.
10. W. Bao, N. J. Mauser, and H. P. Stimming, Effective one particle quantum dynamics of electrons: A numerical study of the Schrödinger-Poisson-Xa model, Commun. Math. Sci., 1 (2003), pp. 809-831.
11. C. Bardos, L. Erd?os, F. Golse, N. J. Mauser, and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum N-particle Coulomb problem, C. R.Math. Acad. Sci. Paris, 334 (2002), pp. 515-520.
12. C. Bardos, F. Golse, and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), pp. 275-293.
13. N. Ben Abdallah, F. Castella, F. Delebecque-Fendt, and F. Méhats, The strongly confined Schrödinger-Poisson system for the transport of electrons in a nanowire, SIAM J. Appl. Math., 69 (2009), pp. 1162-1173.
14. G. Beylkin, On the fast Fourier transform of functions with singularities, Appl. Comput. Harmon. Anal., 2 (1995), pp. 363-383.
15. 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), 016703.
16. J. Bremer, Z. Gimbutas, and V. Rokhlin, A nonlinear optimization procedure for generalized Gaussian quadratures, SIAM J. Sci. Comput., 32 (2010), pp. 1761-1788.
17. Y. Cai, M. Rosenkranz, Z. Lai, and W. Bao, Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), 043623.
18. R. Carles, P. A. Markowich, and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), pp. 2569-2590.
19. H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys., 155 (1999), pp. 468-498.
20. B. Cichocki and B. U. Felderhof, Electrostatic interactions in thin-film Coulomb systems with periodic boundary conditions, Molecular Phys., 67 (1989), pp. 1373-1384.
21. J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comput., 19 (1965), pp. 297-301.
22. A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368-1393.
23. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975.
24. A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data. II, Appl. Comput. Harmon. Anal., 2 (1995), pp. 85-100.
25. H. Fehske, R. Schneider, and A. Weibe, Computational Many-Particle Physics, Lecture Notes in Phys. 739, Springer, New York, 2008.
26. J. A. Fessler and B. P. Sutton, Nonuniform fast Fourier transforms using minmax interpolatio, IEEE Trans. Signal Process., 51 (2003), pp. 560-574.
27. K. Fourmont, Non-equispaced fast Fourier transforms with applications to tomography, IEEE Trans. Signal Process., 9 (2003), pp. 431-450.
28. Z. Gimbutas, L. Greengard, and M. Minion, Coulomb interactions on planar structures: Inverting the square root of the Laplacian, SIAM J. Sci. Comput., 22 (2001), pp. 2093-2108.
29. Z. Gimbutas and V. Rokhlin, A generalized fast multipole method for nonoscillatory kernels, SIAM J. Sci. Comput., 24 (2002), pp. 796-817.
30. L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), pp. 443-454.
31. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325-348.
32. L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer., 6 (1997), pp. 229-269.
33. A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose-Einstein condensation of chromium, Phys. Rev. Lett., 94 (2005), 160401.
34. H. Haken, W. D. Brewer, and H. C. Wolf, Molecular Physics and Elements of Quantum Chemistry, Springer, New York, 1995.
35. R. A. James, The solution of Poisson equation for isolated source distributions, J. Comput. Phys., 25 (1977), pp. 71-93.
36. F. John, Partial Differential Equations, 4th ed., Springer, New York, 1982.
37. M. Kiffer, H. Park, W. Li, and T. F. Gallagher, Dipole-dipole-coupled double-Rydberg molecules, Phys. Rev. A, 86 (2012), 031401(R).
38. M. Kiffer, W. Li, and D. Jaksch, Magnetic monopoles and synthetic spin-orbit coupling in Rydberg marcrodimers, Phys. Rev. Lett., 110 (2013), 170402.
39. M. Kiffer, W. Li, and D. Jaksch, Abelian and non-Abelian gauge fields in dipole-dipole interacting Rydberg atoms, J. Phys. B, 46 (2013), 134008.
40. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), pp. 1133-1138.
41. P. Mccorquodale, P. Colella, G. T. Balls, and S. B. Baden, A scalable parallel Poisson solver in three dimensions with infinite-domain boundary conditions, in Proceedings of the 7th International Workshop on High Performance Scientific and Engineering Computing, Oslo, Norway, 2005, pp. 814-822.
42. T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Progr. Phys., 72 (2009), 126401.
43. J. Y. Lee and L. Greengard, The type 3 nonuniform FFT and its applications, J. Comput. Phys., 206 (2005), pp. 1-5.
44. J. Y. Lee, L. Greengard, and Z. Gimbutas, NUFFT Version 1.2 Software Release, available from http://www.cims.nyu.edu/cmcl/nufft/nufft.html.
45. M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, Wiley, New York, 2008.
46. D. Lindbo and A.-K. Tornberg, Spectral accuracy in fast Ewald-based methods for particle simulations, J. Comput. Phys., 230 (2011), pp. 8744-8761.
47. D. Lindbo and A.-K. Tornberg, Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems, J. Chem. Phys., 136 (2012), 164111.
48. M. Lu, N. Q. Burdick, S.-H. Youn, and B. L. Lev, A strongly dipolar Bose-Einstein condensate of dysprosium, Phys. Rev. Lett., 107 (2011), 190401.
49. J. Ma, V. Rokhlin, and S. Wandzura, Generalized Gaussian quadrature rules for systems of arbitrary functions, SIAM J. Numer. Anal., 33 (1996), pp. 971-996.
50. T. Miyakawa, T. Sogo, and H. Pu, Phase-space deformation of a trapped dipolar Fermi gas, Phys. Rev. A, 77 (2008), 061603(R).
51. D. H. J. Odell, S. Giovanazzi, and C. Eberlein, Exact hydrodynamics of a trapped dipolar Bose-Einstein condensate, Phys. Rev. Lett., 92 (2004), 250401.
52. D. H. Odell and C. Eberlein, Vortex in a trapped Bose-Einstein condensate with dipoledipole interactions, Phys. Rev. A, 75 (2007), 013604.
53. D. Potts, G. Steidl, and M. Tasche, Fast Fourier transforms for nonequispaced data: A tutorial, in Modern Sampling Theory, Appl. Numer. Harmon. Anal., J. J. Benedetto and P. Ferreira, eds., Birkhäuser, Boston, 2001, pp. 247-270.
54. 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), 013617.
55. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, UK, 1989.
56. M. Rosenkranz, Y. Cai, and W. Bao, Effective dipole-dipole interactions in multilayered dipolar Bose-Einstein condensates, Phys. Rev. A, 88 (2013), 013616.
57. L. Santos, G. Shlyapnikov, P. Zoller, and M. Lewenstein, Bose-Einstein condensation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), pp. 1791-1797.
58. D. E. Sheely and J. Schmalian, Quantum critical scaling in graphene, Phys. Rev. Lett., 99 (2007), 226803.
59. D. E. Sheely and J. Schmalian, Optical transparency of graphene as determined by the fine structure constant, Phys. Rev. B, 80 (2009), 193411.
60. G. Strang, Introduction to Linear Algebra, 4th ed., Wellesley-Cambridge Press, Wellesley, MA, 2009.
61. C. Ticknor, N. G. Parker, A. Melatos, S. L. Cornish, D. H. J. O'Dell, and A. M. Martin, Collapse times of dipolar Bose-Einstein condensates, Phys. Rev. A, 78 (2008), 061607.
62. I. Tikhonenkov, B. A. Malomed, and A. Vardi, Anisotropic solitons in dipolar Bose-Einstein condensates, Phys. Rev. Lett., 100 (2008), 090406.
63. L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
64. 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), 013626.
65. S. Yi and H. Pu, Vortex structures in dipolar condensates, Phys. Rev. A, 73 (2006), 061602.
66. S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000), 041604.
67. S. Yi and L. You, Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63 (2001), 053607.
68. L. Ying, G. Biros, and D. Zorin, A kernel-independent adaptive fast multipole algorithm in two and three dimensions, J. Comput. Phys., 196 (2004), pp. 591-626.
69. Y. Zhang and X. Dong, On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), pp. 2660-2676.