dominant source of environmental noise which can heat and decohere the system.

In general, these proposals involve two fields, one for trapping and one for cooling. This may involve an optical cavity mode plus a separate trap, or two optical cavity modes: the so-called “self-trapping” scenario.

Mechanical oscillators in the self-trapping regime differ from other optomechanically-cooled devices in a second fundamental respect (in addition to the absence of mechanical support): the mechanical frequency, $\omega_M$, associated with centre-of-mass oscillations is not an intrinsic feature of the resonator but is determined by the optical field. In particular, it is a function of one or both of the detuning frequencies, $\Delta_1$ and $\Delta_2$, of the optical modes. In previous work [@Opto1], we analysed cooling in the self-trapped regime and found that the optimal condition for cooling occurs where both fields competitively cool and trap the nanosphere. This happens when $\omega_M$ is resonantly red detuned from both the detuning frequencies i.e. $\omega_M(\Delta_1,\Delta_2) \sim -\Delta_{1,2}$ so the relevant resonant frequencies are mutually interdependent. Most significantly, the effective light-matter coupling strength $g$ also depends on the detunings.

The effective coupling strength, ${\tilde g} = g\sqrt{n}$ (the optomechanical coupling rescaled by the square root of photon number) 