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

TLDR

The study introduces a continuous normalized gradient flow (CNGF) and demonstrates its energy‑diminishing property to justify the imaginary‑time method for computing the ground state of Bose–Einstein condensates. Two discretizations of the CNGF are proposed—a backward‑Euler centered finite‑difference (BEFD) scheme and an explicit time‑splitting sine‑spectral (TSSP) method—whose energy‑diminishing behavior is analytically established. Both BEFD and TSSP preserve the energy‑diminishing property (and BEFD also shows monotonicity in nonlinear cases), outperforming CNFD and FEFD, and numerical experiments in one to three dimensions with magnetic trap and stirrer potentials confirm their effectiveness, including direct computation of the first excited state with odd initial data.

Abstract

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.