Nanomechanical structures can be coupled to light in order
to control their mechanical vibrations. Recent experimental developments
in this field of "cavity optomechanics" have led to
laser-cooling of such vibrations to their quantum ground state
and entanglement between mechanics and the radiation field.
A particularly promising platform consists in photonic crystal
structures with optical and vibrational modes localized at
defect sites. If these sites were to be arranged periodically,
they would form an "optomechanical array" that can be described via
a tight-binding model of interacting photons and phonons.
I will describe our theoretical predictions, including
light-induced synchronization of mechanical oscillators,
quantum many-body dynamics of photons and phonons,
Dirac physics on an optomechanical honeycomb lattice, and
the possibility of generating artificial photonic gauge fields.