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SIAM Journal on Applied Mathematics

Analysis and Computation for Ground State Solutions of Bose--Fermi Mixtures at Zero Temperature

Yongyong Cai, Hanquan Wang

Published Jan 1, 2013• 8 citations
Publisher

Abstract

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 $L^\infty$ 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.

DOI
10.1137/120873820

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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.
We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions and also contains solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.
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在耦合Gross-Pitaevskii方程组数值解的基础上，我们探讨在绝对零度下Bose-Fermi混合物的动力学规律。为了计算简便，我们首先把最初的耦合Gross-Pitaevskii方程组转化为无量纲形式方程组，然后把无量纲三维形式方程组简化成二维形式再进一步简化为一维形式，并证明了与耦合Gross-Pitaevskii方程组相关的守恒律——模量守恒以及能量守恒。其次，为了探讨Bose-Fermi混合物的动力学规律，我们提出了一种高效的数值方法——时间分裂差分法来求解耦合Gross-Pitaevskii方程组。并证明了该数值方法具有无条件稳定性以及保持耦合Gross-Pitaevskii方程组守恒律等优点。最后利用该方法来数值模拟Bose-Fermi混合物的动力学规律。 In this paper, based on the numerical solutions of coupled Gross-Pitaevskii equations, we investigate the dynamics of Bose-Fermi mixtures at zero temperature. Firstly, to simplify the numerical computation, we reformulate the equations into three-dimensional dimensionless ones, which are also further reduced to two-dimensional ones and one-dimensional ones. Secondly, to compute the numerical solutions, we present a high-efficiency method for the equations, the time-splitting difference method. We have proved that the method is unconditionally stable and keeps well the related conservation laws. Lastly we apply the method into studying the dynamics of Bose-Fermi mixtures.
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 mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.