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Laser Physics

Vortex dynamics in Bose-Einstein condensates: Numerical calculations

Dora M. Jezek, Horacio M. Cataldo

Published Apr 1, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1134/s1054660x09040094

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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.
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.
The center-of-mass dynamics of a vortex in the surface region of a Bose-Einstein condensate is investigated both analytically using a variational calculation and numerically by solving the time-dependent Gross-Pitaevskii equation. We find, in agreement with previous works, that away from the Thomas-Fermi surface, the vortex moves parallel to the surface of the condensate with a constant velocity. We obtain an expression for this velocity in terms of the distance of the vortex core from the Thomas-Fermi surface that fits accurately with the numerical results. We find also that, coupled to its motion parallel to the surface, the vortex oscillates along the direction normal to the surface around a minimum point of an effective potential.
We present a numerical calculation of the velocity field of an off-axis vortex within a harmonic trapping potential. We consider a condensate with a large number of particles which, in spite of being in the Thomas-Fermi regime, possesses a non-totally-negligible vortex core size. Via a simple, yet realistic, modeling of the vortex-state density, we derive, by treating the equation of continuity outside and inside the vortex core, respective equations for the velocity field and the vortex precession velocity. We find that the vortex precession velocity is given by two contributions: the background velocity field evaluated around the border of the core and another term which depends on the core shape. Our findings concerning the velocity field outside the vortex core are in good agreement with previous theoretical predictions in a narrow region around it, while far away from the vortex we observe a field with a large asymmetry with respect to the vortex position. We show that a better approximation may be obtained by adding the velocity field produced by an antivortex located outside the condensate.
One of the key issues related to superfluidity is the existence of quantized vortices. Following recent experiments on Bose–Einstein condensates exhibiting vortices, we investigate the behavior of the wave function which solves the Gross–Pitaevskii equation. For a rotating Bose–Einstein condensate in a harmonic trap, we give a simplified expression of the Gross–Pitaevskii energy in the Thomas–Fermi regime, which only depends on the number and shape of the vortex lines. This allows us to study in detail the structure of the line of a single quantized vortex, which has a U or S shape. S type vortices exist for all values of the angular velocity Ω but are not minimizers of the energy while U vortices are minimizers and exist only for Ω sufficiently large. Finally, we study the drag created by the movement of a detuned laser beam in a condensate and the nucleation of vortices in the low density region close to the boundary layer of the cloud.
The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are investigated by using the two-dimensional Gross-Pitaevskii equation (GPE) with/without an angular momentum rotation term. If all vortices have the same winding number, they would rotate around the trap center but never collide. In contrast, if the winding numbers are different, their interaction highly depends on the initial distance between vortex centers. The analytical results are presented to describe the dynamics of the vortex centers when β=0. While if β≠0, there is no analytical result but some conclusive numerical findings are provided for the further understanding of vortex interaction in BECs. Finally, the dynamic laws describing the relation of vortex interaction in nonrotating and rotating BECs are presented.
For a Bose-Einstein condensate placed in a rotating trap and strongly confined along the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas-Fermi regime for an effective two-dimensional (2D) situation in the $x\ensuremath{-}y$ plane. We investigate an asymptotic expansion of the energy, the critical angular velocities of nucleation of vortices with respect to a small parameter $\ensuremath{\varepsilon},$ and the location of vortices. The limit $\ensuremath{\varepsilon}$ going to zero corresponds to the Thomas-Fermi regime. The nondimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-\ensuremath{\kappa} high-field limit and our estimates rely on techniques developed for this latter problem. We also take advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.