Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional

By Weizhu Bao, Weijun Tang. 2003

167 citations recorded by the source corpus.

In this paper, we propose a new numerical method to compute the ground-state solution of trapped interacting Bose-Einstein condensation at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground-state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground-state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground-state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/854af5f6cd49ab6a7131b1ee045dd492db3ccb51

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap (2009, background): “א φ(0) φ(0) [4] xrms xrms [4] μ μ [4] -2.”
Mathematical Models and Numerical Methods for Bose-Einstein Condensation (2012, methodology): “Bao and Tang proposed a finite element method to find the ground state by directly minimizing the energy functional in [43].”
Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model (2010, methodology): “We should mention that there exist other numerical experiments in the literature, for instance, that of GPE for Bose-Einstein condensates [2, 7, 8 ] and Schr˜odinger-Newton equations for quantum state reduction [30, 39], which coincide with our theory, too.”