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Advances in Applied Mathematics

A Splitting Finite Difference Method for Simulating the Dynamics of Mixture of Bose-Fermi Gas at Zero Temperature

志红 张

Published Jan 1, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.12677/aam.2018.74050

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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.
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 numerically study the nonlocal Gross–Pitaevskii equation (NGPE) which describes the dynamics of Bose–Einstein condensates (BEC) with dipole–dipole interaction at extremely low temperature. In preparation for the numerics, first we reformulate the dimensionless NGPE into a Schrödinger–Poisson system. Then, we discretize the three-dimensional Schrödinger–Poisson system in space by a sixth-order compact finite difference method and in time by a splitting technique. By means of three-dimensional discrete fast Sine transform, we develop a fast solver for the resulting discretized system. Finally, we present numerical examples in three dimensions to demonstrate the power of the numerical methods and to discuss some physics of dipolar BEC. The merits of the proposed method for the NGPE are that it is fast and unconditionally stable. Moreover, the method is of spectral-like accuracy in space, and conserves the particle number and the energy of the system in the discretized level.
In this work we define, analyze, and compare different numerical schemes that can be used to study the ground state properties of Bose-Fermi systems, such as mixtures of different atomic species under external forces or self-bound quantum droplets. The bosonic atoms are assumed to be condensed and are described by the generalized Gross-Pitaevskii equation. The fermionic atoms, on the other hand, are treated individually, and each atom is associated with a wave function whose evolution follows the Hartree-Fock equation. We solve such a formulated set of equations using a variety of methods, including those based on adiabatic switching of interactions and the imaginary time propagation technique combined with the Gram-Schmidt orthonormalization or the diagonalization of the Hamiltonian matrix. We show how different algorithms compete at the numerical level by studying the mixture in the range of parameters covering the formation of self-bound quantum Bose-Fermi droplets. • Split operator method for the system of condensed bosons and degenerate fermions. • Ground state of the Bose-Fermi mixture by the Gram-Schmidt orthonormalization. • Ground state of the Bose-Fermi mixture by diagonalizing the Hamiltonian matrix. • Formation of Bose-Fermi droplets using the split operator method.
We propose a time-splitting spectral method for the generalized Gross–Pitaevskii equations, which describe the dynamics of spinor F=1 Bose–Einstein condensates at a very low temperature. The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space. Moreover, it conserves the position densities at the discretized level. We apply the method for studying both the dynamic generation of vortices and vortex lattice dynamics for the spinor F=1 Bose–Einstein condensates held in an Ioffe-Pritchard magnetic field.
In this work we present a very simple and efficient numerical scheme which can be applied to study the dynamics of bosonic systems like, for instance, spinor Bose–Einstein condensates (BEC) with non-local interactions but equally well works for Fermi gases. The method we use is a modification of well known Split Operator Method (SOM). We carefully examine this algorithm in the case of F=1 spinor BEC without and with dipolar interactions and for strongly interacting two-component Fermi gas. Our extension of the SOM method has many advantages: it is fast, stable, and keeps constant all the physical constraints (constants of motion) at high level.
The year 2025 marks the 100 and 30 years anniversaries of the discovery of Bose--Einstein condensation and its successful experimental realisation. Inspired by these important research achievements, a conceptually simple approach is proposed to facilitate reliable and efficient numerical simulations. The structure of the underlying systems of coupled Gross--Pitaevskii equations suggests the use of optimised high-order operator splitting methods for dynamical evolution and ground state computation. A second-order barrier, however, prevents the applicability of standard operator splitting methods for both, time evolution as well as imaginary time propagation. An innovative alternative approach accomplishes the design of novel modified operator splitting methods that remain stable under moderate smallness assumptions on the time increments. The core idea is to incorporate commutators of the defining differential and nonlinear multiplication operators, since this permits to fulfill the basic stability requirement of positive method coefficients. Further improvements with respect to convergence at the targeted precision arise from automatic adjustments of the time stepsizes by an inexpensive local error control. The presented numerical experiments confirm the favourable performance of a specific fourth-order modified operator splitting method. Amongst others, it is demonstrated that the excellent mass and energy conservation in long-term evolutions, intrinsic attributes of geometric numerical integrators for Hamiltonian systems, is maintained for a sensible variation of the time stepsizes. Moreover, the benefits of adaptive higher-order approximations in ground state computations are illustrated.