  Rapidly scanning magnetic and optical dipole traps have been widely utilised
to form time-averaged potentials for ultracold quantum gas experiments. Here we
theoretically and experimentally characterise the dynamic properties of
Bose-Einstein condensates in ring-shaped potentials that are formed by scanning
an optical dipole beam in a circular trajectory. We find that unidirectional
scanning leads to a non-trivial phase profile of the condensate that can be
approximated analytically using the concept of phase imprinting. While the
phase profile is not accessible through in-trap imaging, time-of-flight
expansion manifests clear density signatures of an in-trap phase step in the
condensate, coincident with the instantaneous position of the scanning beam.
The phase step remains significant even when scanning the beam at frequencies
two orders of magnitude larger than the characteristic frequency of the trap.
We map out the phase and density properties of the condensate in the scanning
trap, both experimentally and using numerical simulations, and find excellent
agreement. Furthermore, we demonstrate that bidirectional scanning eliminated
the phase gradient, rendering the system more suitable for coherent matter wave
interferometry.
