Multi-Stability and Synchronization of Many Nano-Mechanical Resonators Coupled via a Cavity Field

Abstract

Using amplitude equations, we describe the classical dynamics of N nano-mechanical resonators inside a single co-planar microwave cavity. The nano-mechanical resonators are not directly coupled to one another, but an effective all-to-all coupling is mediated through their capacitive coupling to a common field mode of the microwave cavity. The coupling of each nanomechanical resonator shifts the frequency of the cavity field proportional to the displacement of each resonator. We show that groups of identical nano-mechanical oscillators synchronize to form a single mechanical mode which couples to the microwave cavity with a strength dependent on the square sum of the individual mechanical-microwave couplings. Classically this system has a rich local and global bifurcation structure dominated by periodic orbits which, when analyzed using amplitude equations, can be shown to exhibit multi-stability. Further we show that if the nano-mechanical oscillators are nonidentical the mechanism by which synchronization is lost resembles that for large amplitude forcing which is not of the Kuramoto form.