Methods for obtaining the solution for the steady state correlation functions of this system, under conditions of stability, i.e. where all the eigenvalues of ${\bf A}$ have negative real parts, are well-known [@Milburn]. The required noise spectra, or autocorrelation functions, in frequency space are:

$$\begin{aligned}
{\bf S}(\omega)=\frac{1}{2\pi} _{-\infty}\int^{\infty} e^{-i\omega \tau} \langle
{\bf X}(t+\tau) {\bf X^T}(t)\rangle d\tau,\end{aligned}$$

where,

$$\begin{aligned}
{\bf S}(\omega)=\frac{1}{2\pi} \left({\bf A}+ i \omega{\bf I}\right)^{-1}
               {\bf B}{\bf B^T}\left({\bf A^T}- i \omega{\bf I}\right)^{-1},
\label{SSC}\end{aligned}$$

where the diagonal matrix ${\bf B}{\bf B^T}$ has elements $\left((n_B+\frac{1}{2}){\Gamma_M},
(n_B+\frac{1}{2}){\Gamma_M},\frac{\kappa}{2},\frac{\kappa}{2},\frac{\kappa}{2},
\frac{\kappa}{2}\right)$. From the above, the noise spectra of all modes may be calculated. Eq.[SSC]{} yields semiclassical sideband spectra, symmetrical in $\omega$.

In Fig. \[Noise\], we compare semiclassical displacement spectra calculated from Eq. \[SSC\] with corresponding quantum results obtained from Eq. \[qnoise\]. At high pressures (and hence high phonon occupancy) there is excellent agreement between semiclassical and quantum results. At low pressures (near ground state cooling) however, the quantum spectrum shows a characteristic asymmetry, such as was observed recently in experiments on photonic cavities [@Asymm].

![Maps of the displacement noise spectra $S_{xx}(\omega)$, showing mode splitting for similar parameters to Fig. \[Noise\], near the quantum limit, except that here $\Delta_1 = -1.5$MHz is held fixed while $\Delta_2$ (vertical 