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arXiv:1505.04036 · International Journal of Computer Mathematics

Unified way for computing dynamics of Bose–Einstein condensates and degenerate Fermi gases

Krzysztof Gawryluk, Tomasz Karpiuk, Mariusz Gajda, Kazimierz Rza̧żewski, Mirosław Brewczyk

Published Aug 22, 2017• 23 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1080/00207160.2017.1370545

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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.
In this work we present a very effi cient way of solving a spinor version of a time-dependent Gross-Pitaevskii equation (GP). The method we use is a modification of the well known Spli t Operator Method (SOM) - applied to the case of F = 1 spinor Bose-Einstein condensates. We focus here on F = 1 case for an educational purpose only - the extension to higher values of F is straightforward. Our extension of the SOM to the spinor version has many advantages: it is fast, stable, and keeps constant all the physical constrain ts (constants of motion) at very high level.
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.
Trapped degenerate dipolar Bose and Fermi gases of the cylindrical symmetry with the polarization vector along the symmetry axis are only stable for the strength of dipolar interaction below a critical value. In the case of bosons, the stability of such a dipolar Bose–Einstein condensate (BEC) is investigated for different strengths of contact and dipolar interactions using a variational approximation and a numerical solution of a mean-field model. In the disc shape, with the polarization vector perpendicular to the plane of the disc, the atoms experience an overall dipolar repulsion and this fact should contribute to the stability. However, a complete numerical solution of the dynamics leads to the collapse of a strongly disc-shaped dipolar BEC due to the long-range anisotropic dipolar interaction. In the case of fermions, the stability of a trapped single-component degenerate dipolar Fermi gas is studied including the Hartree–Fock exchange and Brueckner–Goldstone correlation energies in the local-density approximation valid for a large number of atoms. Estimates for the maximum allowed number of polar Bose and Fermi molecules in the BEC and degenerate Fermi gas are given.
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.
Abstract. Trapped degenerate dipolar Bose and Fermi gases of cylindrical symmetry with the polarization vector along the symmetry axis are only stable for the strength of dipolar interaction below a critical value. In the case of bosons, the stability of such a dipolar Bose-Einstein condensate (BEC) is investigated for different strengths of contact and dipolar interactions using variational approximation and numerical solution of a mean-field model. In the disk shape, with the polarization vector perpendicular to the plane of the disk, the atoms experience an overall dipolar repulsion and this fact should contribute to the stability. However, a complete numerical solution of the dynamics leads to the collapse of a strongly disk-shaped dipolar BEC due to the long-range anisotropic dipolar interaction. In the case of fermions, the stability of a trapped single-component degenerate dipolar Fermi gas is studied including the Hartree-Fock exchange and Brueckner-Goldstone correlation energies in the local-density approximation valid for a large number of atoms. Estimates for the maximum allowed number of polar Bose and Fermi molecules in BEC and degenerate Fermi gas are given. PACS numbers: 03.75.Hh,03.75.Ss,03.75.Kk,05.30.Fk ar
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.