the mechanical oscillator.

There is considerable interest in achieving strong-coupling, which leads to regimes of light-matter hybridization. The corresponding mode splitting has been observed experimentally [@strong]. In typical set-ups, these regimes are reached if the rescaled effective optomechanical coupling exceeds the damping rates i.e. ${\tilde g} = g \sqrt{n} \gtrsim \kappa,
\Gamma_M$, where $n \sim |\alpha|^2$ is the cavity photon number. Even if the unnormalized coupling is weak, strong-coupling regimes may be achieved by increasing the driving power and thus increasing intracavity photon numbers.

In the present levitated system, a particularly interesting regime would involve triple-mode hybridization enabling, for example, the coupling of the two modes of light via the mechanical mode. However, here, mode hybridization (for which ${\tilde g}_{1,2}= g_{1,2} \alpha_{1,2} \gtrsim \kappa, \Gamma_M$) depends non-trivially on the detunings.

We argue that large coupling cannot be easily achieved by increasing the driving power, since ${\tilde g}_{1,2} \propto n_{1,2}^{1/4}$ and thus increases slowly with the driving strength. We can show, by a simple argument, that increasing the nanosphere size provides the most effective means to attain strong coupling.

For the self-trapped system, optomechanical coupling strengths are $g_j=
\sqrt{2}kA X_{zpf} \sin 2(kx_0-\phi_j)$ and depend on the detunings via $x_0$. Note also 