  This paper reports the results of a theoretical and experimental study of how
the initial circulation of ring-shaped Bose-Einstein condensates (BECs) can be
probed by time-of-flight (TOF) images. We have studied theoretically the
dynamics of a BEC after release from a toroidal trap potential by solving the
3D Gross-Pitaevskii (GP) equation. The trap and condensate characteristics
matched those of a recent experiment. The circulation, experimentally imparted
to the condensate by stirring, was simulated theoretically by imprinting a
linear azimuthal phase on the initial condensate wave function. The theoretical
TOF images were in good agreement with the experimental data. We find that upon
release the dynamics of the ring--shaped condensate proceeds in two distinct
phases. First, the condensate expands rapidly inward, filling in the initial
hole until it reaches a minimum radius that depends on the initial circulation.
In the second phase, the density at the inner radius increases to a maximum
after which the hole radius begins slowly to expand. During this second phase a
series of concentric rings appears due to the interference of ingoing and
outgoing matter waves from the inner radius. The results of the GP equation
predict that the hole area is a quadratic function of the initial circulation
when the condensate is released directly from the trap in which it was stirred
and is a linear function of the circulation if the trap is relaxed before
release. These scalings matched the data. Thus, hole size after TOF can be used
as a reliable probe of initial condensate circulation. This connection between
circulation and hole size after TOF will facilitate future studies of
atomtronic systems that are implemented in ultracold quantum gases.
