  Photonic nonreciprocal components, such as isolators and circulators, provide
highly desirable functionalities for optical circuitry. This motivates the
active investigation of mechanisms that break reciprocity, and pose
alternatives to magneto-optic effects in on-chip systems. In this work, we use
optomechanical interactions to strongly break reciprocity in a compact system.
We derive minimal requirements to create nonreciprocity in a wide class of
systems that couple two optical modes to a mechanical mode, highlighting the
importance of optically biasing the modes at a controlled phase difference. We
realize these principles in a silica microtoroid optomechanical resonator and
use quantitative heterodyne spectroscopy to demonstrate up to 10 dB optical
isolation at telecom wavelengths. We show that nonreciprocal transmission is
preserved for nondegenerate modes, and demonstrate nonreciprocal parametric
amplification. These results open a route to exploiting various nonreciprocal
effects in optomechanical systems in different electromagnetic and mechanical
frequency regimes, including optomechanical metamaterials with topologically
non-trivial properties.
