g}_{1}$ (and for $n_1\sim
n_2$, also ${\tilde g}_{2}$) reaches a maximum value for $r\simeq 300$nm before falling to zero. Other maxima for larger $r$ do not provide a larger value of ${\tilde g_{1,2}}$. Furthermore, they have the disadvantage that they may enhance photon recoil heating effects. For comparison, the value of $A$ is also shown, overlaid on the experimental and simulated frequencies.

Dynamics
========

Optomechanical damping
----------------------

A previous study [@Opto1], using rescaled coordinates, investigated the full parameter space of two optical mode cooling. Here we investigate more carefully the effect of non-zero mechanical damping. Using linear response theory, we can extract cooling rates from the equations Eq. \[a2s\]:

$$\Gamma_{opt}=
\left[S_1(\omega_M)+S_2(\omega_M)-S_1(-\omega_M)-S_2(-\omega_M)\right],
\label{gamma}$$

where

$$S_j(\omega)=\frac{ |\alpha_j|^2 g^2_j \kappa}{[\Delta_j^x-\omega]^2 +
\frac{\kappa^2}{4}},
\label{gamma12}$$

for $j=1,2$. Net cooling occurs for $\Gamma_{opt} <0$. Although the above is quite similar in form to standard optomechanical expressions, as explained previously, rather different behaviour is observed since here $\omega_M$ and $g_j$ are both dependent on the $\Delta_j^x$.

From quantum perturbation theory we can show that $R_{n\to m}$, the rate of transition from state $n$ to $n+1$ is: $R_{n \to n+1}=(n+1) \left( S_1(\omega_M)+S_2(\omega_M) \right)$ while $R_{n \to n-1}=n  \left( S_1(-\omega_M)+S_2(-\omega_M) \right)$

For $n>>1$, then $R_{n \to n+1}- R_{n \to 