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East Asian Journal on Applied Mathematics

Ground States of Two-Component Bose-Einstein Condensates with an Internal Atomic Josephson Junction

Weizhu Bao, Yongyong Cai

Published Oct 26, 2010• 82 citations
Publisher

Abstract

In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.

DOI
10.4208/eajam.190310.170510a

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In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
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.
静态相耦合的Gross-Pitaevskii方程组的解描述了双原子的玻色爱因斯坦凝聚体在极低温度下的基态现象。我们提出一种十分有效的数值方法——梯度法来求解此基态解。我们提出的梯度法在数值上既保持总模量守恒又能使总能量递减；我们严格地证明我们提出的梯度法是一种获得能量函数在给定限制性条件下的最小值(也即基态解)的十分有效的方法。我们通过大量的例子来显示该方法的优点并且应用该方法去研究处于旋转状态下的双原子–玻色爱因斯坦凝聚体在极低温度下所呈现的复杂涡旋现象。 The ground states of rotating two-component Bose-Einstein condensates (BEC) at extremely low temperature are solutions of time-independent coupled Gross-Pitaevskii equations. To compute the ground states, we propose an efficient numerical method—gradient flows with discrete nor-malization. In linear cases and under properly chosen initial data, we can prove that the gradient flows converge into the ground state which has the lowest energy. We show that the method is quite efficient and apply the method to study complicated vortex structure in the ground state solutions of rotating two-component BEC at extremely low temperature.
We study numerically the time-independent vector Gross--Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose--Einstein condensate (BEC) at zero or a very low temperature. In preparation for the numerics, we scale the three-dimensional (3d) VGPEs, approximately reduce it to lower dimensions, present a continuous normalized gradient flow (CNGF) to compute ground states of multicomponent BEC, prove energy diminishing of the CNGF, which provides a mathematical justification, and discretize it by the backward Euler finite difference (BEFD), which is monotone in linear and nonlinear cases and preserves energy diminishing property in the linear case. Then we use a time-splitting sine-spectral (TSSP) method to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multicomponent BEC. The merits of the TSSP method for VGPEs are that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, has "good" resolution in the semiclassical regime, is of spectral-order accuracy in space and second-order accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in three dimensions for ground states and dynamics of multicomponent BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multicomponent BEC.
Previous numerical studies on the ground state structure of Bose--Fermi mixtures mostly relied on Thomas--Fermi (TF) approximation for the Fermi gas. In this paper, we establish the existence and uniqueness of ground state solutions of Bose--Fermi mixtures at zero temperature for both a coupled Gross--Pitaevskii (GP) equations model and a model with TF approximation for fermions. To prove the uniqueness, the key is to estimate the L∞L^\inftyL∞ bounds of the ground state solution. By implementing an efficient method---gradient flow with discrete normalization with backward Euler finite difference discretization---to compute the coupled GP equations, we report extensive numerical results in one and two dimensions. The numerical experiments show that we can also extract many interesting phenomena without reference to TF approximation for the fermions. Finally, we numerically compare the ground state solutions for the coupled GP equations model and the model with TF approximation for fermions as well as for the model with TF approximations for both bosons and fermions.
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.