increasing photon number by stronger driving (and by implication increasing rescaled coupling strengths) will not prove an effective alternative, since in the present system we show ${\tilde g}
\propto  {n}^{1/4}$ rather than $\sqrt{n}$, so the rescaled coupling increases very slowly with laser input power.

In section 3, we in investigate the cooling and dynamics. In sec. 3.1, we review the corresponding cooling rate expressions obtained from quantum perturbation theory (or linear response theory). In sec. 3.2 we report a study of the corresponding semiclassical Langevin equations and compare them with fully quantum noise spectra; we compare also quantum, semiclassical and perturbation theory results for levitated nanospheres. In section 4 we investigate novel regimes of triple mode hybridization, coincident with static bistabilities, which the present study shows are experimentally accessible given the large optomechanical coupling strengths associated with $r=100-300$nm nanospheres. In section 5 we describe the experimental study which provides data from which the size-dependence of the coupling may be inferred. In Section 6, we conclude.

Theory: Quantum Hamiltonian for a nanosphere in a cavity
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We approximate the equivalent cavity model by a one-dimensional system, with centre-of-mass motion confined to the axial dimension. In this simplified study, we consider only 