Collective Dynamics in Optomechanical Arrays

Optomechanical systems couple light stored inside an optical cavity to the motion of a mechanical mode. Recent experiments have demonstrated setups, such as photonic crystal structures, that in principle allow one to confine several optical and vibrational modes on a single chip. Here we start to investigate the collective nonlinear dynamics in arrays of coupled optomechanical cells. We show that such “optomechanical arrays” can display synchronization, and that they can be described by an effective Kuramoto-type model.

Figure 1

Optomechanical crystals may be used to build arrays with several localized optical and mechanical modes. (a) Potential setup fabricated as a periodically patterned, freestanding dielectric beam on a chip with laser drive via tapered fibre as in [8, 9]. (b) Schematic array of mechanically coupled optomechanical cells. (c) For a single cell, at sufficient laser drive power, there is a Hopf bifurcation towards self-induced mechanical oscillations with an undetermined phase $\phi$.

Figure 2

Phase locking of an optomechanical cell to an external force. (a) Phase lag $\delta \phi$ between oscillations and force, outside (left) and inside (right) the phase-locked regime. (b) Time-average $⟨\sin \delta \phi ⟩$ as function of $\delta \Omega$ comparing optomechanics (blue, solid) against the Hopf model (magenta, dash-dot) and the Adler phase equation (black, dash). Colored region indicates $\delta \dot{\phi }=0$. (c) “Arnold tongue”; $⟨\sin \delta \phi ⟩$ in the force vs frequency difference plane. Lines show the transition towards phase-locking; styles as in (b). The triangle labels $F_0$ in (b). (Microscopic cell parameters: $\Delta /\Omega =1$, $\kappa /\Omega =1$, $\Gamma /\Omega =0.01$, $\mathcal{P}=\hslash G^2{\alpha }_{\max }^2/m{\Omega }^3=0.36$).

Figure 3

Phase locking of two mechanically coupled optomechanical cells. (a) Phase particle in the effective Kuramoto potential $U(\delta \phi )$; desynchronized (left), phase-locked (right). (b) Time-average $⟨\sin \delta \phi ⟩$ as function of mechanical coupling $k$. When $k$ exceeds a threshold $k_c$ (colored region), the phase difference $\delta \phi (t)$ between the oscillations locks to a constant value despite different bare mechanical frequencies, here $\delta \Omega =0.003{\Omega }_1$. Both in-phase and antiphase synchronization regimes are observed. (c) $⟨\sin (\delta \phi )⟩$ in the plane coupling $k$ vs frequency difference $\delta \Omega$, including a comparison of the critical coupling $k_c$ (white, solid) with the one from a Hopf model with one fit parameter (green, dash) and the effective Kuramoto-type model (blue, dash-dot). (Cell parameters as in Fig. 2).

Figure 4

FEM simulation for two coupled cells localized on the same beam in an optomechanical crystal setup [cf. Figure 1a]. Geometrical parameters for each individual cell as in Ref. 9. (a) Horizontal displacement of vibrational “pinch” modes. (b) Transverse electric fields of optical modes. (c) The intercell couplings decay exponentially as a function of the cells’ distance.

Figure 5

Mechanical frequency spectra $I(\omega )$ of intensity fluctuations, $I(t)=|{\alpha }_1(t){|}^2+|{\alpha }_2(t){|}^2$, for two mechanically coupled cells. (a) Frequency locking upon changing the detuning $\delta \Omega ={\Omega }_2-{\Omega }_1$ between the mechanical frequencies (magenta, dash). Top: spectrum $I(\omega )$ from optomechanics. Bottom: spectral peaks of $\sum _i\cos \mathbf{(}{\phi }_i(t)\mathbf{)}$ from a simulation of the effective Kuramoto-type model, Eq. (5). Depending on initial conditions we find in-phase (blue) and antiphase (red) synchronization. ($k/m{\Omega }_1^2=0.015$). (b) Spectrum $I(\omega )$ and effective Kuramoto coupling $K$ [Eq. (6)] vs laser input power. ($k/m{\Omega }_1^2=0.01$, $\delta \Omega =0.005{\Omega }_1$). (c) Example of trajectories $G{x}_i(t)/\kappa$ displaying strong amplitude modulation, not described by the Kuramoto model [at power indicated by the triangle in (b)]. (Color scale indicates $|I(\omega )|$ in units of the peak height at $\omega =0$ for a system with $G=0$; $\delta$ peaks are broadened for clarity; parameters as in Figs. 2, 3)

Figure 6

Optomechanical array with global coupling. (a) Schematic: global coupling scheme. (b) Order parameter $\chi$ vs coupling $k_g$ for an array of $N=10$ cells, comparing optomechanics (blue) to results from the effective Kuramoto-type model Eq. (5) (red, dashed) using one fit parameter, $\gamma =0.036\Omega$. (${\Omega }_i$ are chosen equidistant from $0.98\Omega$ to $1.02\Omega$.)