Computing Ground States of Spin-1 Bose–Einstein Condensates by the Normalized Gradient Flow

Abstract

Abstract. In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose-Einstein condensates (BEC) by using the normalized gradient flow or imaginary time method. The key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization. This allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of single component BEC to compute the ground state of spin-1 BEC. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method (BFSP), is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BEC with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.

Analysis and efficient computation for the dynamics of two-component Bose-Einstein condensates

“…Analytical and numerical results for the dynamics of two-component BEC were reviewed. Finally, the analytical results and numerical methods for two-component BEC can be extended to spin-1 BEC [10,8,51].…”

Nonadiabatic quantum phase transition in a trapped spinor condensate

“…The structure of ground states can be found analytically in the Thomas-Fermi approximation (TF) [24,25], or numerically according to [26,27]. We recall it in a regime of parameters such that the spin healing length ξ s = / √ 2mc 2 ρ is much smaller than the size of an atomic cloud determined by the TF radius.…”

“…Dziarmaga Here and below we summarize simple expressions for ground state density profiles of particular components which result from the TF analysis. All of them were checked against numerical results by applying a method proposed by Bao et al [26,27].…”

“…The initial state for the evolution is prepared using the numerical method proposed by Bao et al [26,27] with additional stochastic noise added to mimic quantum fluctuations. In this way an ensemble of initial stochastic fields ψ T = (ψ + ,ψ 0 ,ψ − ) is prepared.…”

Excited spin states and phase separation in spinor Bose-Einstein condensates

“…The Fermi radius of the transverse trapping potential is smaller than the spin healing length, and the nonlinear energy scale is much smaller than the transverse trap energy scale, which allows us to reduce the problem to one spatial dimension [24,31]. The solutions were found numerically using the normalized gradi- ent flow method [32], which is able to find a state which minimizes the total energy for given N and M, and fulfills the phase matching condition (12). The stability of the resulting states was verified using numerical time evolution according to Eqs.…”