  Optical isolation, non-reciprocal phase transmission and topological phases
for light based on synthetic gauge fields have been raising significant
interest in the recent literature. Cavity-optomechanical systems that involve
two optical modes coupled to a common mechanical mode form an ideal platform to
realize these effects, providing the basis for various recent demonstrations of
optomechanically induced non-reciprocal light transmission. Here, we establish
a unifying theoretical framework to analyze optical non-reciprocity and
breaking of time-reversal symmetry in multimode optomechanical systems. We
highlight two general scenarios to achieve isolation, relying on either optical
or mechanical losses. Depending on the loss mechanism, our theory defines the
ultimate requirements for optimal isolation and the available operational
bandwidth in these systems. We also analyze the effect of sideband resolution
on the performance of optomechanical isolators, highlighting the fact that
non-reciprocity can be preserved even in the unresolved sideband regime. Our
results provide general insights into a broad class of parametrically modulated
non-reciprocal devices, paving the way towards optimal non-reciprocal systems
for low-noise integrated nanophotonics.
