  We investigate collective nonlinear dynamics in a blue-detuned optomechanical
cavity that is mechanically coupled to an undriven mechanical resonator. By
controlling the strength of the driving field, we engineer a mechanical gain
that balances the losses of the undriven resonator. This gain-loss balance
corresponds to the threshold where both coupled mechanical resonators enter
simultaneously into self-sustained limit cycle oscillations regime. Rich sets
of collective dynamics such as in-phase and out-of-phase synchronizations
therefore emerge, depending on the mechanical coupling rate, the optically
induced mechanical gain and spring effect, and the frequency mismatch between
the resonators. Moreover, we introduce the quadratic coupling that induces
enhancement of the in-phase synchronization. This work shows how phonon
transport can remotely induce synchronization in coupled mechanical resonator
array and opens up new avenues for metrology, communication, phonon-processing,
and novel memories concepts.
