Light-Mediated Cascaded Locking of Multiple Nano-Optomechanical Oscillators

Collective phenomena emerging from nonlinear interactions between multiple oscillators, such as synchronization and frequency locking, find applications in a wide variety of fields. Optomechanical resonators, which are intrinsically nonlinear, combine the scientific assets of mechanical devices with the possibility of long distance controlled interactions enabled by traveling light. Here we demonstrate light-mediated frequency locking of three distant nano-optomechanical oscillators positioned in a cascaded configuration. The oscillators, integrated on a chip along a common coupling waveguide, are optically driven with a single laser and oscillate at gigahertz frequency. Despite an initial mechanical frequency disorder of hundreds of kilohertz, the guided light locks them all with a clear transition in the optical output. The experimental results are described by Langevin equations, paving the way to scalable cascaded optomechanical configurations.

Figure 1

(a) Side view and (b) top view, scanning electron microscope (SEM) images of a device consisting of 3 GaAs optomechanical disk resonators with identical nominal dimensions (radius of 1.5 micron and thickness of 320 nm) coupled to a common GaAs suspended tapered waveguide. The square-shaped pads are supports for the suspended optical waveguide and have no optomechanical impact. (c) A sketch of the unidirectional cascaded configuration containing multiple optomechanical disks along a common waveguide.

Figure 2

(a) From right (blue curve) to left (red curve), power spectral densities of a single optomechanical oscillator ($2\mu \mathrm{m}$ radius and 320 nm thickness) for increasing laser wavelength $\lambda$ (Supplemental Material [31], Fig. S4 and section on thermo-optical effects) and $P_{\text{laser}}=1\mathrm{mW}$. We first observe the background thermal motion, then the motion is amplified and finally reaches the self-oscillation regime. (b) From right (blue curve) to left (red curve), power spectral density of two optomechanical oscillators ($1.5\mu \mathrm{m}$ radius and 320 nm thickness) for increasing laser wavelength (see Supplemental Material [31], Fig. S4). We successively observe the thermal vibration of both resonators, then their amplification, the self-oscillation, the nonlinear mixing of their resonances, and finally their phase locking. (c) Same as in (b), but with 3 optomechanical oscillators.

Figure 3

(a) Experimental results: gray-scale map of the power spectral density for the system of 3 optomechanical oscillators as a function of the normalized detuning $\mathrm{\Delta }/\mathrm{\Omega }$. Note that the thermomechanical shift has been subtracted for clarity, ${\mathrm{\Omega }}_{\mathrm{th}}$ being the thermally shifted mechanical frequency. (b) Zoom in (a). (c) and (d) Numerical results obtained with the model described in the text. (c) Gray-scale map of the power spectral density for 3 oscillators as a function of the detuning $\mathrm{\Delta }/\mathrm{\Omega }$ for a mechanical disorder $ϵ=5\times {10}^{-4}$. (d) Standard deviation ${\sigma }_{{\chi }_3}$ of the synchronization order parameter as a function of the detuning $\mathrm{\Delta }/\mathrm{\Omega }$ [same vertical axis as in (c)], transiting to zero as phase locking occurs.