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arXiv:0711.0568 · SIAM Journal on Scientific Computing

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

Weizhu Bao, Fong Yin Lim

Published Jan 1, 2008• 90 citations
arXivHTMLPDF

Abstract

In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose–Einstein condensates (BECs) 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 a single-component BEC to compute the ground state of spin-1 BECs. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method, is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BECs with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.

DOI
10.1137/070698488

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In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose–Einstein condensates (BECs) 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 a single-component BEC to compute the ground state of spin-1 BECs. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method, is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BECs with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.
Abstract. We propose and analyze an efficient and accurate numerical method for computing ground states of spin-2 Bose–Einstein condensates (BECs) by using the normalized gradient flow (NGF). In order to successfully extend the NGF to spin-2 BECs which have five components in the vector wave function but with only two physical constraints on total mass conservation and magnetization conservation, two important techniques are introduced for designing the proposed numerical method. The first one is to systematically investigate the ground state structure and property of spin-2 BECs within a spatially uniform system, which can be used for how to properly choose initial data in the NGF for computing ground states of spin-2 BECs. The second one is to introduce three additional projection conditions based on the relations between the chemical potentials, together with the two existing physical constraints, such that the five projection parameters used in the projection step of the NGF can be uniquely determined. Then a backward-forward Euler finite difference method is adapted to discretize the NGF. We prove rigorously that there exists a unique solution of the nonlinear system for determining the five projection parameters in the full discretization of the NGF under a mild condition on the time step size. Extensive numerical results on ground states of spin-2 BECs with different types of phases and under different potentials are reported to show the efficiency and accuracy of the proposed numerical method and to demonstrate several interesting physical phenomena on ground states of spin-2 BECs.
The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this paper, we revisit the time discretizations for the GFDN and its generalization to the multicomponent BECs. Several widely used time discretizations are demonstrated to be not accurate for computing the ground state solution in the general case, especially for the multicomponent BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow. Through analysis and numerical computation, we clarify that, in order to accurately compute the ground state solution, the GFDN method must be discretized in very special ways in time such as the linearized backward Euler scheme, while the GFLM can be discretized by various schemes and works for multicomponent BEC with multiple constraints. We propose a simple backward-forward Euler scheme to discretize the GFLM, where only a linear system with constant coefficients needs to be solved at each time step. Taking the spin-1 BEC as an example, we demonstrate how the traditional GFDN approach fails in computing the ground state solution of general multicomponent BECs accurately. The proposed GFLM with backward-forward Euler discretization is shown to be accurate and efficient. Extensive numerical results are provided to verify our claims.
We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.
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.
In this paper, a mass (or normalization) and magnetization conservative and energy-diminishing numerical method is presented for computing the ground state of spin-1 (or F=1F=1F=1 spinor) Bose–Einstein condensates (BECs). We begin with the coupled Gross–Pitaevskii equations, and the ground state is defined as the minimizer of the energy functional under two constraints on the mass and magnetization. By constructing a continuous normalized gradient flow (CNGF) which is mass and magnetization conservative and energy-diminishing, the ground state can be computed as the steady state solution of the CNGF. The CNGF is then discretized by the Crank–Nicolson finite difference method with a proper way to deal with the nonlinear terms, and we prove that the discretization is mass and magnetization conservative and energy-diminishing in the discretized level. Numerical results of the ground state and their energy of spin-1 BECs are reported to demonstrate the efficiency of the numerical method.
The computation of the ground states of spin-FFF Bose--Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. Our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2, and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate interesting physical phenomena.