Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates

We present a new method to propagate rotating Bose-Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose-Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.Comment: 15 pages, 4 figures, as submitted to journa

A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate

On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation \cite{EMZ2015}. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1} and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE

Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT

In this paper, we propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.Comment: 26 pages, 5 figure

Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates

New efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a dipolar interaction potential. Due to the high singularity in the dipolar interaction potential, it brings significant difficulties in mathematical analysis and numerical simulations of dipolar BECs. In this paper, by decoupling the two-body dipolar interaction potential into short-range (or local) and long-range interactions (or repulsive and attractive interactions), the GPE for dipolar BECs is reformulated as a Gross-Pitaevskii-Poisson type system. Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blowup of the dynamics in different parameter regimes of dipolar BECs. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adaption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. Extensive numerical examples in 3D are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar BECs

Quantized vortices in superfluid helium and atomic Bose-Einstein condensates

This article reviews recent developments in the physics of quantized vortices in superfluid helium and atomic Bose-Einstein condensates. Quantized vortices appear in low-temperature quantum condensed systems as the direct product of Bose-Einstein condensation. Quantized vortices were first discovered in superfluid 4He in the 1950s, and have since been studied with a primary focus on the quantum hydrodynamics of this system. Since the discovery of superfluid 3He in 1972, quantized vortices characteristic of the anisotropic superfluid have been studied theoretically and observed experimentally using rotating cryostats. The realization of atomic Bose-Einstein condensation in 1995 has opened new possibilities, because it became possible to control and directly visualize condensates and quantized vortices. Historically, many ideas developed in superfluid 4He and 3He have been imported to the field of cold atoms and utilized effectively. Here, we review and summarize our current understanding of quantized vortices, bridging superfluid helium and atomic Bose-Einstein condensates. This review article begins with a basic introduction, which is followed by discussion of modern topics such as quantum turbulence and vortices in unusual cold atom condensates.Comment: 99 pages, 20 figures, Review articl

The Finite Element Method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation

We consider the time-dependent Gross-Pitaevskii equation describing the dynamics of rotating Bose-Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the $L^2$- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross-Pitaevskii equation. We demonstrate the performance of the method in numerical experiments