Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional
Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional
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DOI：
10.1016/s0021-9991(03)00097-4
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发表时间：
2003-03
影响因子：
4.1
通讯作者：
W. Bao;Weijun Tang
中科院分区：
文献类型：
--
作者：
W. Bao;Weijun Tang
本文提出了一种新的数值方法，通过有限元近似直接极小化能量泛函，计算零温或极低温下囚禁相互作用玻色-爱因斯坦凝聚的基态解。作为准备步骤，我们开始与3D Gross-Pitaevskii方程（GPE），规模得到一个三参数模型，并显示如何减少到2D和1D GPE。基态解是通过在约束条件下最小化能量泛函而得到的，并通过有限元方法进行离散。详细介绍了一维、二维径向对称和三维球柱对称的有限元近似，并给出了GPE在两个极端区域：极弱相互作用和强排斥相互作用下的近似基态解，作为我们实际数值计算极小化问题的初始猜想.报道了凝聚体中数百万个原子的一维、二维径向对称以及三维球对称和柱对称的数值结果，以验证新的数值方法。此外，基态解和它们的M-Fermi近似之间的比较也被报道。本文还介绍了计算GPE激发态的数值方法的推广。
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.