  We introduce a hybrid optomechanical system containing an annularly trapped
Bose-Einstein condensate (BEC) inside an optical cavity driven by
Lauguerre-Gaussian (LG) modes. Spiral phase elements serve as the end mirrors
of the cavity such that the rear mirror oscillates torsionally about the cavity
axis through a clamped support. As described earlier in a related system [P.
Kumar et. al., Phys. Rev. Lett. 127, 113601 (2021)], the condensate atoms
interact with the optical cavity modes carrying orbital angular momentum which
create two atomic side modes. We observe three peaks in the output noise
spectrum corresponding to the atomic side modes and rotating mirror
frequencies, respectively. We find that the trapped BEC's rotation reduces
quantum fluctuations at the mirror's resonance frequency. We also find that the
atomic side modes-cavity coupling and the optorotational coupling can produce
bipartite and tripartite entanglements between various constituents of our
hybrid system. We reduce the frequency difference between the side modes and
the mirror by tuning the drive field's topological charge and the condensate
atoms' rotation. When the atomic side modes become degenerate with the mirror,
the stationary entanglement between the cavity and the mirror mode diminishes
due to the suppression of cooling. Our proposal, which combines atomic
superfluid circulation with mechanical rotation, provides a versatile platform
for reducing quantum fluctuations and producing macroscopic entanglement with
experimentally realizable parameters.
