  Synthetic magnetism has been used to control charge neutral excitations for
applications ranging from classical beam steering to quantum simulation. In
optomechanics, radiation-pressure-induced parametric coupling between optical
(photon) and mechanical (phonon) excitations may be used to break time-reversal
symmetry, providing the prerequisite for synthetic magnetism. Here we design
and fabricate a silicon optomechanical circuit with both optical and mechanical
connectivity between two optomechanical cavities. Driving the two cavities with
phase-correlated laser light results in a synthetic magnetic flux, which in
combination with dissipative coupling to the mechanical bath, leads to
nonreciprocal transport of photons with 35dB of isolation. Additionally,
optical pumping with blue-detuned light manifests as a particle non-conserving
interaction between photons and phonons, resulting in directional optical
amplification of 12dB in the isolator through direction. These results indicate
the feasibility of utilizing optomechanical circuits to create a more general
class of nonreciprocal optical devices, and further, to enable novel
topological phases for both light and sound on a microchip.
