  We present numerical simulations of the cavity optomechanical detection of
persistent currents and bright solitons in an atomic Bose-Einstein condensate
confined in a ring trap. This work describes a novel technique that measures
condensate rotation in situ, in real-time, and with minimal destruction, in
contrast to currently used methods, all of which destroy the condensate
completely. For weakly repulsive inter-atomic interactions, the analysis of
persistent currents extends our previous few-mode treatment of the condensate
[P. Kumar et al. Phys. Rev. Lett. 127, 113601 (2021)] to a stochastic
Gross-Pitaevskii simulation. For weakly attractive atomic interactions, we
present the first analysis of optomechanical detection of matter-wave soliton
motion. We provide optical cavity transmission spectra containing signatures of
the condensate rotation, sensitivity as a function of the system response
frequency, and atomic density profiles quantifying the effect of the
measurement backaction on the condensate. We treat the atoms at a mean-field
level and the optical field classically, account for damping and noise in both
degrees of freedom, and investigate the linear as well as nonlinear response of
the configuration. Our results are consequential for the characterization of
rotating matter waves in studies of atomtronics, superfluid hydrodynamics, and
matter-wave soliton interferometry.
