g_1(\Delta_1^x,\Delta_2^x)\alpha_1( {\hat a}_1+  {\hat a}_1^\dagger)(
{\hat b} +{\hat b}^\dagger)   +g_2(\Delta_1^x,\Delta_2^x)
\alpha_2( {\hat a}_2+  {\hat a}_2^\dagger)( {\hat b} +{\hat b}^\dagger).
\label{LinHam}\end{aligned}$$

![(Colour online) Size dependent effects in the magnitude of the optomechanical coupling parameter ${\tilde g}_{1}$. It is assumed that cavity parameters would correspond to ${\tilde
g}_{1}=10^6$Hz at $r=150$nm, for photon numbers $n_1=10^9$. For comparison, the value of $A$ is also shown, as are the experimental and simulated frequencies $\omega_a \equiv \omega_M =2\pi f_M$. The $\omega_M$ are scaled by a factor of 10 for clarity and in fact, values of $\omega_M \sim 1$MHz are quite realistic in optical cavities.[]{data-label="beadg"}](BEADG.eps){height="3.3in"}

Towards strong light-matter coupling with two optical cavity modes
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The Hamiltonian in Eq. \[LinHam\] appears analogous in form to standard, well-studied optomechanical Hamiltonians, albeit with two optical modes rather than one. However, it differs in one important respect: in this case, both the mechanical frequency $\omega_M(\Delta_1^x, \Delta_2^x)$ and the optomechanical coupling strengths $g_{1,2} \equiv g_{1,2}(\Delta_1^x,\Delta_2^x)$ are not fixed and depend on the detunings. The fact that the frequencies of the three modes (two optical, one mechanical) are interdependent makes the dynamics different from other optomechanical set-ups, where the equilibrium mechanical frequency (i.e. excluding shifts arising from the fluctuations) is intrinsic to 