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
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.