cooling resonances (at three different values of $\Delta_2$), where strong damping is observed (and similarly for each fixed $\Delta_2$). This is in contrast to single optical mode schemes where there is a single cooling resonance for which $\Delta_{1}=-\omega_M$ or $\Delta_{2}=-\omega_M$. For the $R=0.5$ map the three cooling resonances merge, giving a single extended cooling region of about 1MHz width.

For $R=1$ the map has a high degree of symmetry, since the role of the two optical modes is interchangeable. The figures show that the largest cooling rates are found in the double resonance region, making it the most favourable region to work in.

The equilibrium phonon number in Eq. \[neq\] concerns only the idealised situation where there is a very good vacuum, negligible photon recoil heating and thus no mechanical damping or heating effects. For small $r \lesssim 200$nm spheres, we assume recoil heating is negligible [@Zoller] and the dominant source of mechanical damping is background gas collisions, which provide an effective mechanical damping $\Gamma_M=\frac{8}{3} \pi \frac{m_g}{m_s}  r^2 n_g \bar{v}_g$ [@Isart; @Zoller] where $m_g / m_s$ is ratio of the gas particle’s mass to that of the sphere, $n_g$ is the gas number density and $\bar{v}_g$ is the mean gas 