Coherent Dynamics of Vortex Formation in Trapped Bose-Einstein Condensates

Abstract

Obtaining this paper

This paper has been published in Physical Review Letters, volume 83, issue 5, pages 895-898, 2 August 1999. If you have a subscription to the online form of this journal, you can retrieve a copy of the paper from the Physical Review Letters Online server. A preprint is also available (as an 88 kB gzipped PostScript file) from the LANL preprint server. The purpose of this page to elaborate on the results presented in our paper, and so this page should be read in conjunction with the paper.

MPEG movies

These MPEG movies provide a more detailed account of the simulations presented in the various figures of the paper. There are some slight differences in terminology between the paper and the MPEG movies. The quantity omega (upper or lower case) is the angular frequency of the stirrer.

Figure 1

Fast stirring Probability density MPEG (1571 kB). Note that even after the stirrer is removed, vortex pair creation and annihilation events continue to occur.
 
An alternative version of the probability density MPEG (1631kB) marks the vortices with coloured symbols, rather than coloured dots, according to their sign.

Figures 2 and 3

In these MPEGs, the angular momentum expectation value <L> is also given.

omega_f = 0.6

Probability density (799 kB) and phase (1221 kB) MPEGs. In this case, the final angular speed of stirring is a little above the critical angular speed, and numerous vortices are drawn into the condensate. This simulation corresponds to the upper dashed line in Figure 3.

omega_f = 0.5

If you see nothing else here, get this! ==>  Probability density (674 kB)  <==  and phase (901 kB) MPEGs. This simulation shows the almost completely modulated cycling of a single vortex when the condensate is stirred just a smidgen below the critical angular frequency. This is analogous to on resonance Rabi cycling. The frames of Figure 2 are taken from this simulation, which corresponds to the solid line in Figures 3 and 4.

omega_f = 0.4

Probability density (706 kB) and phase (988 kB) MPEGs. The lower dashed line in Figure 3.

Notes

Speed of sound

Our early results for stirring at r_s=3, (C=88.13) showed that the critical linear speed for vortex formation was about v=1.5, which is quite close to the speed of sound ahead of the stirrer (c=1.62). However, upon investigating different values of C and radii of stirring, we were able to discount importance of the speed of sound in the situations we have considered. For example;

• For lower values of C the speed of sound goes down, but the critical linear speed goes up (at a given radius).

• For smaller radii the critical linear speed goes down, but the speed of sound goes up (for a given value of C).

The two-state model predicts omega_c=0.54 for C=88.13, so a stirrer at radius r_s=3 would have a critical linear speed of v=3*0.54=1.62, which is very close to the speed of sound (c=1.62) ahead of the stirrer. This is entirely coincidental.

Vortex detection

Vortices are detected numerically by a phase circulation algorithm which tells us if a given mesh point is near a vortex. To remove the effects of numerical noise (and spurious low-density features), we do not mark as a vortex any mesh point where the curvature of the density is less than 0.001.

GP equation simulations

We solve the time-dependent Gross-Pitaevskii equation in two dimensions using a modified split-step Fourier method. The simulations we present here are calculated using a 512x512 point grid which ranges from -20 to +20 in each dimension. Note that only a small fraction of this spatial region is displayed in the MPEG movies [(x,y) each from -10 to 10, or a quarter of the area], and much less in the images in Figures 1 and 2.

Two-state model simulations

For a given value of n_v, Equations (4) can be solved efficiently by optimisation techniques to give mu_s, mu_v, phi_s, and phi_v. Knowing the form of W', we calculate delta_s, delta_v, and Omega. Because these quantities vary slowly with n_v, we precalculated them as a look-up table. We solve Equations (5) numerically, using the look-up table to determine mu_s, mu_v, delta_s, delta_v and Omega.

Other results

Below we present a small sample of our other results which may be of interest, but have not yet been published except for mention of the stability of the k=+1 vortex eigenstate.

Vortex stability

An efficient optimisation technique was used to find the first (k=+1) and second (k=+2) excited vortex eigenstates. In these simulations, we studied the stability of these vortex eigenstates by inserting and withdrawing a stationary Gaussian stirrer. In each case, a large perturbation was induced, but the difference in the stability of the central vortex is quite striking.

Stability of a k=+1 central vortex eigenstate

Probability density MPEG (498 kB) showing that the central vortex is stable to a gross perturbation.

Stability of a k=+2 central vortex eigenstate

Probability density MPEG (482 kB) illustrates that the slightest perturbation causes the immediate dissociation of the central k=+2 vortex into two unit vortices. These remain near the centre of the condensate, precessing slowly about the centre.

Rotating collisions

We have simulated a number of rotating collisions between condensates displaced from the trap centre and given an azimuthal velocity. These demonstrate the vortices predicted by Feynman (for a flow in Helium II) because of the phase shear between the two condensates.

Rotating collisions

Probability density MPEG (540 kB) for two condensates separated by 10, each given a velocity kick (delta v = 2), up for the right hand, down for the left hand, and then allowed to evolve freely. They collide, causing vortices and curved interference fringes. Initially, the phase difference between the condensates is set to pi.

Condensate slicing

This series of simulations studies the effect of a stirrer moving in a straight line passing through an initially stationary condensate. In each case the condensate is greatly disturbed, causing vortex formation.

Fast (v=3) stirrer, on axis

Probability density MPEG (540 kB)

Medium (v=1.5) stirrer, on axis

Probability density MPEG (527 kB)

Fast (v=3) stirrer, off axis

Probability density MPEG (536 kB)