  Recently, a method has been proposed to detect the rotation of a ring
Bose-Einstein condensate, in situ, in real-time and with minimal destruction,
using a cavity driven with optical fields carrying orbital angular momentum.
This method is sensitive to the magnitude of the condensate winding number but
not its sign. In the present work, we consider simulations of the rotation of
the angular lattice formed by the optical fields and show that the resulting
cavity transmission spectra are sensitive to the sign of the condensate winding
number. We demonstrate the minimally destructive technique on persistent
current rotational eigenstates, counter-rotating superpositions, and a soliton
singly or in collision with a second soliton. Conversely, we also investigate
the sensitivity of the ring condensate, given knowledge of its winding number,
to the rotation of the optical lattice. This characterizes the effectiveness of
the optomechanical configuration as a laboratory rotation sensor. Our results
are important to studies of rotating ring condensates used in atomtronics,
superfluid hydrodynamics, simulation of topological defects and cosmological
theories, interferometry using matter-wave solitons, and optomechanical
sensing.
