{\hat a}_j -iE_j +i A {\hat a}_j\cos^2 (k
{\hat x}-\phi_j) -\frac{\kappa}{2} {\hat a}_j,
\label{Heisenberg}\end{aligned}$$

where $j=1,2$ for the two optical-mode realisation. Additional damping terms have also been added: $\frac{\kappa}{2} {\hat a}_i$ accounts for photon losses due to mirror imperfections and the $\Gamma_M {\dot  {\hat x}}$ term for mechanical damping. The above should also include quantum noise terms arising from (say) shot noise or gas collisions: for brevity, the quantum noise terms are left out until sec.3.

We consider here the linearised dynamics; we replace operators by their expectation values and perform the shifts about equilibrium values such as ${\hat a}_j(t) \to \alpha_j
+ {\hat a}_j(t)$, and ${\hat x} \to x_0 + {\hat x}(t)$. The values for the equilibrium photon fields (e.g. for the two-mode case) are $\alpha_{1} = -iE_1 \left[\frac{\kappa}{2} - i\Delta_{1}^x \right]^{-1} \ $ and $\alpha_{2}  =  -iRE_1\left[\frac{\kappa}{2}- i\Delta_{2}^x \right]^{-1}$. The equilibrium position is then given by the relation $-\frac{\sin
2(kx_0-\phi_1)}{\sin 2(kx_0-\phi_2)} = {|\alpha_2|^2}/{|\alpha_1|^2}$, by numerical solution of the equation,

$$\begin{aligned}
-\frac{\sin 2(kx-\phi_1)}{\sin 2(kx-\phi_2)}= R^2\frac{|\frac{\kappa}{2} -
i\Delta^x_{1}|^2}{|\frac{\kappa}{2} - i\Delta^x_{2}|^2},
\label{equil}\end{aligned}$$

where $\Delta_j^x=\Delta_j +A \cos^2 (kx_0-\phi_j)$. As usual we consider the dynamics of the fluctuations via the linearised equations. To first order, the linearised equations of motion, in the shifted 