0$ is possible for a pressure of order $10^{-6}$mBar, even for modest driving powers of 2mW and values of $A \simeq 3 \times 10^5$Hz corresponding to spheres of order $r \simeq 100$nm.

![Mode mixing and bistability for $A=3\kappa=6 \times 10^{5}$Hz. We consider a relatively low input power of 0.37mW. (a) plots the optomechanical cooling rate (blue indicates cooling, brown indicates heating). The red dashed line indicates the locus of bistability as a function of the detunings $\Delta_1$ and $\Delta_2$. The discontinuity in the cooling can be discerned near the strong doubly resonant cooling region. (b) shows displacement noise spectra $S_{xx}(\omega)$ as a function of $\omega$ plotted along $\Delta_1=-1$MHz (along black horizontal line in upper panel). We sweep in increasing $\Delta_2$. At the point where $\Delta_1=-1$MHz intersects the bistability, a discontinuity in the noise spectra is apparent. On one side of the discontinuity there is strong hybridization between the mechanical mode and optical mode 2; this changes abruptly across the discontinuity to hybridization between the mechanical mode and optical mode 1 for larger $\Delta_2$. This may allow for control of entanglement between the modes. The map corresponds to near-vacuum conditions so the system is near the quantum ground state in this 