Synchronization in coupled mechanical resonator arrays

Abstract

Opto-mechanical systems are based on the nonlinear coupling between the electromagnetic field in a resonator and an array of bulk mechanical resonators such that the frequency of the electromagnetic field resonator depends on the displacement coordinates of each of the mechanical resonators. In this thesis, three cases are considered: the first consists of a single array of mechanical resonators interacting with a common field for which the frequency of the electromagnetic resonance is tuned to depend quadratically (to lowest order) on the displacement of the resonators. Two such systems are coupled together through weak interactions of their common fields to form the second case considered in this thesis. For the third case, two systems, each whose frequency of the electromagnetic resonance is tuned to depend linearly (to lowest order) on the displacement of the two corresponding arrays of resonators, are weakly coupled together. By using the method of amplitude equations around a critical point, the synchronization properties of these three systems are found. For the first system, it is found 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 multiplicity. For the second system, it is shown that the average phases of the two arrays of near-identical bulk mechanical resonators with low driving fail to synchronize with each other unless the natural frequencies of the resonators within each array are identical and their parameters are selected so that these compensate the de-synchronizing effect of the coupling between the fields. For the third system, despite the de-synchronizing effect of the coupling between the fields, it is shown that the average phases of the two arrays of identical bulk mechanical resonators with low driving synchronize with each other unless the natural frequencies within each array are sufficiently non-identical.