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Mathematical Models and Methods in Applied Sciences

DYNAMICS OF THE GROUND STATE AND CENTRAL VORTEX STATES IN BOSE–EINSTEIN CONDENSATION

Weizhu Bao, Yanzhi Zhang

Published Dec 1, 2005• 47 citations
Publisher

Abstract

In this paper, we study dynamics of the ground state and central vortex states in Bose–Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas–Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures.

DOI
10.1142/s021820250500100x

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In this paper, we study dynamics of the ground state and central vortex states in Bose–Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas–Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures.
We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically.We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics.We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum.We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization.Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC.Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime.Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.
In this paper, we study the dynamics of rotating Bose--Einstein condensates (BEC) based on the Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. By formulating the equation in either the two-dimensional polar coordinate system or the three-dimensional cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is time reversible, unconditionally stable, efficient, and accurate for the problem. Moreover, it conserves the position density. We also apply the numerical method to study issues such as the stability of central vortex states and the quantized vortex lattice dynamics in rotating BEC.
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.
We numerically study properties of the dynamics of vortices in nonrotating Bose-Einstein condensates in the Thomas-Fermi regime. On the one hand, we compute the vortex energy as a function of its position and we predict, using the expression of the Magnus force, the vortex precession velocity. On the other hand, we calculate the temporal evolution of the vortex-state and test the accuracy of the previous prediction. We also investigate the validity of analytical formulae of this velocity involving the healing length. In addition, we analyze the velocity field and the angular momentum and we compare them to available analytical expressions.
A time-splitting generalized-Laguerre–Fourier–Hermite pseudospectral method is proposed for computing the dynamics of rotating Bose–Einstein condensates (BECs) in two and three dimensions. The new numerical method is based on the following: (i) the use of a time-splitting technique for decoupling the nonlinearity; (ii) the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient; and (iii) the construction of eigenfunctions for the linear operator by properly scaling the generalized-Laguerre, Fourier, and Hermite functions. The new method enjoys the following properties: (i) it is explicit, time reversible, and time transverse invariant; (ii) it conserves the position density and is spectrally accurate in space and second-order or fourth-order accurate in time; and (iii) it solves the problem in the original whole space instead of in a truncated artificial computational domain. The method is also extended to solve the coupled Gross–Pitaevskii equations for the dynamics of rotating two-component and spin-1 BECs. Extensive numerical results for the dynamics of BECs are reported to demonstrate the accuracy and efficiency of the new method for rotating BECs.
The three-dimensional study of the ring vortex solitons is conducted for both attractive and repulsive BECs subject to harmonic potential confinement. A family of stationary ring vortex solitons, which is defined by the radial excitation number and the winding number of the intrinsic vorticity, are obtained numerically for a given atomic interaction strength. We find that stabilities of the ground and radially excited states of the ring vortex soliton are dependent on the winding number differently. The ground state of the ring vortex soliton with the large winding number is unstable dynamically against random perturbation. The radially excited state of the ring vortex soliton with large winding number corresponds to the increased collapse threshold, and therefore can be made stable for sufficiently small atomic interaction strengths. The ground and radially excited states also demonstrate different dynamical evolutions under large atomic interaction strengths. The former exhibits simultaneously symmetrical splitting in the transverse plane, while the latter displays periodically expand-merge cycles in the longitudinal direction.