Spontaneous vortices in the formation of Bose-Einstein condensates

C. N. Weiler,1 T. W. Neely,1 D. R. Scherer,1 A. S. Bradley,2 M. J. Davis,2 and B. P. Anderson1

1 College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
2 ARC Centre of Excellence for Quantum-Atom Optics, School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia

Accepted to appear as a letter in Nature, 11 July 2008.

Main text

Manuscript, figures, and supplementary information (1044 kB, pdf)

Supplementary videos

Condensate formation movies in a harmonic trap: Quench A.
From 300 simulations of Quench A, 229 contained no vortex cores after 1.5 s. The following five movies show a selection of outcomes, including cases with 0, 1, and 2 long-lived vortices.

Supplementary Video 1 (2.1 Mb)
The first movie is an example of BEC formation in which there are no vortices trapped in the final condensate, and is provided for comparison purposes. At early times there is some indication of vortices in the condensate; however, these do not survive at longer times. In this situation the condensate appears to grow adiabatically in its ground state, as was assumed in the condensate formation calculations of Refs. [34–39,42,43]. The reader should note that cases such as this are typical for the data set corresponding to Quench A.

Supplementary Video 2 (2.1 Mb)
This movie provides an example of BEC formation with a single vortex line that survives at long times and remains close to the centre of the condensate. The vortex line remains approximately vertical, and is usually easily visible in the column density plot. The reader should note that there are many examples similar to this where the vortex line
is not so close to the centre, and instead slowly spirals towards the boundary of the condensate over a time scale of several seconds.

Supplementary Video 3 (2.1 Mb)
In this example there are three clear vortices at early times. Two of these arise near the edge of the condensate and these damp out relatively quickly, leaving a single vortex line of opposite charge near the centre that survives to the end of the simulation. This is an example of a vortex that does not directly align with the z axis, and the column density often shows an elongated density dip. This movie uses the same simulation data as in Figure 5 of the main text.

Supplementary Video 4 (2.1 Mb)
This simulation results in two oppositely charged vortices that remain well separated, precessing about the centre in opposite directions. It should be noted that only 3 out of 300 simulations in this data set clearly exhibited more than one core at long times.

Supplementary Video 5 (2.2 Mb)
The final movie for the data set of Quench A shows a number of vortex cores that undergo some rather complicated dynamics as they move about within the condensate, including vortices that cross each other and reconnect in a different configuration, and an example where a vortex line flips (and hence changes colour). This example shows the most complicated vortex dynamics observed in this data set.

Condensate formation movies in a toroidal trap: Quench C.
From the 300 simulations of Quench C, 147 contained no vortices after 1.5 s. The following movies show five examples where vortices arose.

Supplementary Video 6 (2.2 Mb)
The first movie from this data set shows the formation of a single vortex. This is not readily seen in the isodensity surface, as the 2pi phase winding is located in the centre of the trap where there is no condensate density. However, the phase winding about the entire condensate is clear in the bottom-right plot of the phase in the z = 0 plane. Whereas in the harmonic trap all vortices will eventually make their way to the edge of the condensate and disappear, here the gaussian barrier pins the vortex to the centre — this is an example of a persistent current.

Supplementary Video 7 (2.2 Mb)
In this example it is possible to see a pair of oppositely charged vortices at t = 0.8 s, one of which becomes pinned to the central barrier while the other ends up precessing about the centre in the region of maximum condensate
density.

Supplementary Video 8 (2.2 Mb)
Here there are some rather complicated early dynamics that result in a single vortex precessing about the outside of the condensate, but with no persistent current.

Supplementary Video 9 (2.2 Mb)
There are a number of vortices early on, and near 0.6 s it seems that there is a doubly charged persistent current. However, only one vortex ends up being trapped on the central gaussian barrier and the other vortex of the same charge remains within the bulk of the condensate, precessing rapidly about the centre. The rate of precession should be compared with the example from Supplementary Video 7, where the vortex near the edge is of the opposite charge to that pinned by the barrier.

Supplementary Video 10 (2.2 Mb)
The final example shows a stable doubly-charged persistent current. This is particularly interesting given the energy difference between this and the thermodynamic ground state of the condensate with no current. This is the only example from the data set where a stable doubly-charged persistent current was seen.