  We theoretically examine the optomechanical interaction between a rotating
nanoparticle and an orbital angular momentum-carrying optical cavity mode.
Specifically, we consider a dielectric nanosphere rotating uniformly in a
ring-shaped optical potential inside a Fabry-Perot resonator. The motion of the
particle is probed by a weak angular lattice, created by introducing two
additional degenerate Laguerre-Gaussian cavity modes carrying equal and
opposite orbital angular momenta. We demonstrate that the rotation frequency of
the nanoparticle is imprinted on the probe optical mode, via the Doppler shift,
and thus may be sensed experimentally using homodyne detection. We show
analytically that the effect of the optical probe on the particle rotation
vanishes in the regime of linear response, resulting in an accurate frequency
measurement. We also numerically characterize the degradation of the
measurement accuracy when the system is driven in the nonlinear regime. Our
results are relevant to rotational Doppler velocimetry and to studies of
rotational Brownian motion in a periodic lattice.
