  The mechanical properties of light have found widespread use in the
manipulation of gas-phase atoms and ions, helping create new states of matter
and realize complex quantum interactions. The field of cavity-optomechanics
strives to scale this interaction to much larger, even human-sized mechanical
objects. Going beyond the canonical Fabry-Perot cavity with a movable mirror,
here we explore a new paradigm in which multiple cavity-optomechanical elements
are wired together to form optomechanical circuits. Using a pair of
optomechanical cavities coupled together via a phonon waveguide we demonstrate
a tunable delay and filter for microwave-over-optical signal processing. In
addition, we realize a tight-binding form of mechanical coupling between
distant optomechanical cavities, leading to direct phonon exchange without
dissipation in the waveguide. These measurements indicate the feasibility of
phonon-routing based information processing in optomechanical crystal
circuitry, and further, to the possibility of realizing topological phases of
photons and phonons in optomechanical cavity lattices.

Optics;Mesoscale and Nanoscale Physics;Quantum Physics