Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods. (English) Zbl 1380.81496
Summary: We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned steepest descent (PSD) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.
Keywords:
Bose-Einstein condensation; rotating Gross-Pitaevskii equation; stationary states; Fourier spectral method; steepest descent; conjugate gradient; optimization algorithms on Riemannian manifolds; preconditionerReferences:
| [1] | Abo-Shaeer, J. R.; Raman, C.; Vogels, J. M.; Ketterle, W., Observation of vortex lattices in Bose-Einstein condensates, Science, 292, 5516, 476-479 (2001) |
| [2] | Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization Algorithms on Matrix Manifolds (2009), Princeton University Press · Zbl 1147.65043 |
| [3] | Adhikari, S. K., Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms, Phys. Lett. A, 265, 1-2, 91-96 (2000) |
| [4] | Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 5221, 198-201 (1995) |
| [5] | Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 12, 2621-2633 (2013) · Zbl 1344.35130 |
| [6] | Antoine, X.; Duboscq, R., GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations I: computation of stationary solutions, Comput. Phys. Commun., 185, 11, 2969-2991 (2014) · Zbl 1348.35003 |
| [7] | Antoine, X.; Duboscq, R., Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates, J. Comput. Phys., 258, 509-523 (2014) · Zbl 1349.82027 |
| [8] | Antoine, X.; Duboscq, R., GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations II: dynamics and stochastic simulations, Comput. Phys. Commun., 193, 95-117 (2015) · Zbl 1344.82004 |
| [9] | Antoine, X.; Duboscq, R., Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity, (Besse, C.; Garreau, J. C., Nonlinear Optical and Atomic Systems: at the Interface of Physics and Mathematics. Nonlinear Optical and Atomic Systems: at the Interface of Physics and Mathematics, Lect. Notes Math., vol. 2146 (2015)), 49-145 · Zbl 1344.35114 |
| [10] | Bai, Z.; Demmel, J.; Dongarra, J.; Ruhe, A.; van der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (2000), SIAM · Zbl 0965.65058 |
| [11] | Bao, W., Ground states and dynamics of multi-component Bose-Einstein condensates, SIAM J. Multiscale Model. Simul., 2, 2, 210-236 (2004) · Zbl 1062.82034 |
| [12] | Bao, W.; Cai, Y., Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math., 1, 49-81 (2011) · Zbl 1290.35236 |
| [13] | Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1, 1-135 (2013) · Zbl 1266.82009 |
| [14] | Bao, W.; Cai, Y.; Wang, H., Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229, 20, 7874-7892 (2010) · Zbl 1198.82036 |
| [15] | Bao, W.; Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 5, 1674-1697 (2004) · Zbl 1061.82025 |
| [16] | Bao, W.; Jiang, S.; Tang, Q.; Zhang, Y., Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296, 72-89 (2015) · Zbl 1354.65200 |
| [17] | Bao, W.; Tang, W., Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187, 1, 230-254 (2003) · Zbl 1028.82500 |
| [18] | Barrett, R.; Berry, M.; Chan, T.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; Van der Vorst, H., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (1994), SIAM |
| [19] | Baye, D.; Sparenberg, J. M., Resolution of the Gross-Pitaevskii equation with the imaginary-time method on a Lagrange mesh, Phys. Rev. E, 82, 5 (2010) |
| [20] | Bradley, C. C.; Sackett, C. A.; Tollett, J. J.; Hulet, R. G., Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75, 9, 1687-1690 (1995) |
| [21] | Bretin, V.; Stock, S.; Seurin, Y.; Dalibard, J., Fast rotation of a Bose-Einstein condensate, Phys. Rev. Lett., 92, 5 (2004) |
| [22] | Byrnes, T.; Wen, K.; Yamamoto, Y., Macroscopic quantum computation using Bose-Einstein condensates, Phys. Rev. A, 85, 4 (2012) |
| [23] | Caliari, M.; Ostermann, A.; Rainer, S.; Thalhammer, M., A minimisation approach for computing the ground state of Gross-Pitaevskii systems, J. Comput. Phys., 228, 2, 349-360 (2009) · Zbl 1159.82311 |
| [24] | Cances, E.; Defranceschi, M.; Kutzelnigg, W.; Le Bris, C.; Maday, Y., Computational quantum chemistry: a primer, Handb. Numer. Anal., 10, 3-270 (2003) · Zbl 1070.81534 |
| [25] | Cerimele, M. M.; Chiofalo, M. L.; Pistella, F.; Succi, S.; Tosi, M. P., Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates, Phys. Rev. E, 62, 1, 1382-1389 (2000) |
| [26] | Chiofalo, M. L.; Succi, S.; Tosi, M. P., Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62, 5, 7438-7444 (2000) |
| [27] | Courant, R.; Friedrichs, K.; Lewy, H., On the partial difference equations of mathematical physics, IBM J. Res. Dev., 11, 2, 215-234 (1967) · Zbl 0145.40402 |
| [28] | Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 3, 463-512 (1999) |
| [29] | Danaila, I.; Hecht, F., A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates, J. Comput. Phys., 229, 19, 6946-6960 (2010) · Zbl 1198.82035 |
| [30] | Danaila, I.; Kazemi, P., A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32, 5, 2447-2467 (2010) · Zbl 1216.35006 |
| [31] | David, K. B.; Mewes, M. O.; Andrews, M. R.; Vandruten, N. J.; Durfee, D. S.; Kurn, D. M.; Ketterle, W., Bose-Einstein condensation in gas of sodium atoms, Phys. Rev. Lett., 75, 22, 3969-3973 (1995) |
| [32] | Dion, C. M.; Cances, E., Ground state of the time-independent Gross-Pitaevskii equation, Comput. Phys. Commun., 177, 10, 787-798 (2007) · Zbl 1196.81017 |
| [33] | Edelman, A.; Arias, T. A.; Smith, S. T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 2, 303-353 (1998) · Zbl 0928.65050 |
| [34] | Fetter, A. L.; Jackson, B.; Stringari, S., Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap, Phys. Rev. A, 71, Article 013605 pp. (2005) |
| [35] | Jeng, B.-W.; Wang, Y.-S.; Chien, C.-S., A two-parameter continuation algorithm for vortex pinning in rotating Bose-Einstein condensates, Comput. Phys. Commun., 184, 3, 493-508 (2013) · Zbl 1302.65232 |
| [36] | Knyazev, A. V., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23, 2, 517-541 (2001) · Zbl 0992.65028 |
| [37] | Madison, K. W.; Chevy, F.; Bretin, V.; Dalibard, J., Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation, Phys. Rev. Lett., 86, 20, 4443-4446 (2001) |
| [38] | Madison, K. W.; Chevy, F.; Wohlleben, W.; Dalibard, J., Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84, 5, 806-809 (2000) |
| [39] | Matthews, M. R.; Anderson, B. P.; Haljan, P. C.; Hall, D. S.; Wieman, C. E.; Cornell, E. A., Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83, 13, 2498-2501 (1999) |
| [40] | Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D., Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients, Rev. Mod. Phys., 64, 4, 1045-1097 (1992) |
| [41] | Raman, C.; Abo-Shaeer, J. R.; Vogels, J. M.; Xu, K.; Ketterle, W., Vortex nucleation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 87, 21 (2001) |
| [42] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042 |
| [43] | Saad, Y., Numerical Methods for Large Eigenvalue Problems (2011), SIAM · Zbl 1242.65068 |
| [44] | Saad, Y.; Chelikowsky, J. R.; Shontz, S. M., Numerical methods for electronic structure calculations of materials, SIAM Rev., 52, 1, 3-54 (2010) · Zbl 1185.82004 |
| [45] | Teter, M. P.; Payne, M. C.; Allan, D. C., Solution of Schrödinger’s equation for large systems, Phys. Rev. B, 40, 12255-12263 (1989) |
| [46] | Wang, Y.-S.; Jeng, B.-W.; Chien, C.-S., A two-parameter continuation method for rotating two-component Bose-Einstein condensates in optical lattices, Commun. Comput. Phys., 13, 442-460 (2013) · Zbl 1373.82083 |
| [47] | Wu, X.; Wen, Z.; Bao, W., A regularized Newton method for computing ground states of Bose-Einstein condensates (2015) |
| [48] | Yuce, C.; Oztas, Z., Off-axis vortex in a rotating dipolar Bose-Einstein condensate, J. Phys. B, At. Mol. Opt. Phys., 43, 13 (2010) |
| [49] | Zeng, R.; Zhang, Y., Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates, Comput. Phys. Commun., 180, 6, 854-860 (2009) · Zbl 1198.82007 |
| [50] | Zhou, Y.; Chelikowsky, J. R.; Gao, X.; Zhou, A., On the preconditioning function used in planewave DFT calculations and its generalization, Commun. Comput. Phys., 18, 1, 167-179 (2015) · Zbl 1388.65195 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
