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Experimental exploration of synchronization in scalable oscillator microsystems has unfolded a deeper understanding of networks, collective phenomena, and signal processing. Cavity optomechanical devices have played an important role in this scenario, with the perspective of bridging optical and radio frequencies through nonlinear classical and quantum synchronization concepts. In its simplest form, synchronization occurs when an oscillator is entrained by a signal with frequency nearby the oscillator’s tone, and becomes increasingly challenging as their frequency detuning increases. Here, we experimentally demonstrate entrainment of a silicon-nitride optomechanical oscillator driven up to the fourth harmonic of its 32 MHz fundamental frequency. Exploring this effect, we also experimentally demonstrate a purely optomechanical RF frequency divider, where we performed frequency division up to a 4:1 ratio, i.e., from 128 MHz to 32 MHz. Further developments could harness these effects towards frequency synthesizers, phase-sensitive amplification and nonlinear sensing.

We propose and numerically validate a modified perturbation theory that captures non-Hermitian features present in dissipative optomechanical systems. Our theory predicts different behaviors than commonly used perturbation theories derived assuming purely Hermitian dynamics.

We demonstrate dissipative optomechanical transduction and backaction in coupled nanobeams. Compared to previous demonstrations, our system corresponds to a hundredfold increase in mechanical frequency and displays a record-high dissipative optomechanical coupling.