axes) is swept. (a) Shows the semiclassical spectrum which is symmetric in frequency $\omega$. (b) Shows the quantum spectrum which is asymmetric. In both cases, triple hybridization appears clearly and is seen near $\Delta_2 \approx -1.0$MHz. (c) shows the corresponding behaviour for $A=\kappa/2$, showing that the triple peak structure has disappeared. Note that $\log{ S_{xx}(\omega)}$ is plotted.[]{data-label="SPLIT"}](SPLIT.eps){height="3in"}

Comparison between perturbation theory, semiclassical and quantum results 
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The equilibrium variance (and hence the final phonon number) of the mechanical oscillator is,

$$\begin{aligned}
\langle x^2\rangle=\frac{1}{2\pi} _{-\infty}\int^{\infty} \langle
|x(\omega)|^2\rangle d\omega,\end{aligned}$$

thus, the final equilibrium temperature of the mechanical oscillator after optomechanical cooling is $k_BT_{eq}=1/2m \omega_M^2 \langle X^2\rangle$. Noting the rescaling $ \langle X^2\rangle= 2X_{ZPF} \langle x^2\rangle$ and setting $k_BT_{eq}=(\langle n \rangle +1/2) \hbar \omega_M$, we can write $\langle x^2\rangle= \langle n \rangle+1/2$.

Using Eqs. (\[QM\]), (\[SSC\]) and (\[PT\]), we can investigate final equilibrium phonon numbers (and the minimum achievable for levitated self-trapped spheres) comparing quantum, semiclassical and perturbation theory respectively. In Fig. \[Phonon\], we compare the corresponding equilibrium phonon numbers, $\langle n \rangle_{QM}$, $\langle n \rangle_{SC}$ and $\langle n \rangle_{PT}$ respectively for the unusual triple cooling resonance region shown in Fig. \[Cool\]. Cooling to near the ground state $ \langle n \rangle \sim 