Computing the Ground State Solution of Bose-Einstein Condensates by a Normalized Gradient Flow

By Weizhu Bao, Qiang Du. SIAM Journal on Scientific Computing, 2003

526 citations recorded by the source corpus.

In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/a31e7977159ef974aa2363b316ff2b28a7af72cb

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model (2010, methodology): “We shall mention that when the ground state solution is unique (up to sign), some convergence analysis and upper error bounds of finite dimensional approximations for the nonlinear eigenvalue problems resulting from BEC and the standard TFW model are presented in [54] and [55] respectively.”
Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates (2004, methodology): “Recently, a generalization of the normalized gradient flow (NGF) method [4] and the time-splitting spectral method[5] have been developed in [3] for computing the ground state solutions of (1.”
Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations (2007, methodology): “Part of this work was done when J.Y. was visiting Zhou Pei-Yuan Center for Applied Mathematics at Tsinghua University.”
Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates (2005, methodology): “Furthermore for the above CNGF, as observed in [6] for a non-rotating BEC, the solution of (5.”
A Multiscale Neural Network Based on Hierarchical Matrices (2018, methodology): “We employ the 357 method in [3] for the numerical solution, which solves a time-dependent NLSE by a normalized 358 gradient flow.”
Local spectral time splitting method for first- and second-order partial differential equations (2005, unknown): “: +1 517 353 4689. ail address: wei@math.msu.edu (G.W. Wei).”
Two-dimensional vortex quantum droplets (2018, methodology): “(7) with ﬁxed N and S were obtained by means of the imaginary-time-integration method (which may work for vortex conﬁgurations as well as for ground states [58,59]), applied to Eq.”
(INVITED) Vortex solitons: Old results and new perspectives (2019, background): “(29), by means of imaginary-time simulations [236]-[239], starting from the Gaussian input,”
Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap (2009, background): “Most of the previous studies [3,4,5,8,9,12,18,20,21,24,25,33,34,35,39,45,46] on the numerical solution of the GP equation are confined to a consideration of stationary states only.”
Energy-diminishing integration of gradient systems (2014, background): “…(Chafee & Infante (1974); Elliott & Zheng (1986)), the porous medium equation (Otto (2001)), models in image processing (Strzodka et al. (2004); Michailovich et al. (2007)), in differential geometry (Droske & Rumpf (2004); Bobenko & Schröder (2005)), and in quantum systems (Bao & Du (2004)).”
Accelerated Imaginary‐time Evolution Methods for the Computation of Solitary Waves (2007, methodology): “Implicit methods allow larger time steps without causing divergence to the iterations.”
Mathematical Models and Numerical Methods for Bose-Einstein Condensation (2012, methodology): “(9.32) For the above CNGF, we have [19] Theorem 9.4.”