  Non-reciprocal devices, with one-way transport properties, form a key
component for isolating and controlling light in photonic systems.
Optomechanical systems have emerged as a potential platform for optical
non-reciprocity, due to ability of a pump laser to break time and parity
symmetry in the system. Here we consider how the non-reciprocal behavior of
light can also impact the transport of sound in optomechanical devices. We
focus on the case of a quasi one dimensional optical ring resonator with many
mechanical modes coupled to light via the acousto-optic effect. The addition of
disorder leads to finite diffusion for phonon transport in the material,
largely due to elastic backscattering between clockwise and counter-clockwise
phonons. We show that a laser pump field, along with the assumption of high
quality-factor, sideband-resolved optical resonances, suppresses the effects of
disorder and leads to the emergence of chiral diffusion, with
direction-dependent diffusion emerging in a bandwidth similar to the
phase-matching bandwidth for Brillouin scattering. A simple diagrammatic theory
connects the observation of reduced mechanical linewidths directly to the
associated phonon diffusion properties, and helps explain recent experimental
results.
