Synchronization in a mechanical resonator array coupled quadratically to a common electromagnetic field mode

Optomechanical systems are based on the nonlinear coupling between the electromagnetic (EM) field in a resonator and one or more bulk mechanical resonators such that the frequency of the EM field resonator depends on the displacement coordinates of each of the mechanical resonators. In this paper we consider the case of multiple mechanical resonators interacting with a common field for which the frequency of the EM resonance is tuned to depend quadratically (to lowest order) on the displacement of the resonators. By using the method of amplitude equations around a critical point, it is shown that groups of near-identical bulk mechanical resonators with low driving fail to synchronize unless their natural frequencies are identical, in which case the resulting system can exhibit multistability.