strongest peaks have been truncated in height. In (b) three avoided crossings are apparent. The dominant character of each normal mode is indicated by the colour (black is mechanical, blue is optical mode 1, red is optical mode 2). When $-\Delta_2$ is large, there is no mixing. However as $\Delta_2 \to 0$, there is strong mixing and the dominant character of each normal mode changes from light to matter (or vice versa) as an avoided crossing is encountered. Panel (c) shows the cooling and indicates strong cooling at each of the avoided crossings. []{data-label="eigensplit"}](NMODE1.eps){height="3.5in"}

We compare the quantum displacement with corresponding semiclassical solutions in the steady state. The linearised two mode system Eq. \[LinHam\], in matrix form corresponds to a standard problem [@Milburn]. Inclusion of the noise arising from gas collisions or laser shot noise yields a set of corresponding Langevin equations: $ \frac{d{\bf X(t)}}{dt}= {\bf A}{\bf X}+ {\bf B E}(t)$, where ${\bf A}$ is termed the drift matrix. Its eigenvalues give the stabilities and eigenfrequencies of the system’s normal modes, while the noise is determined by ${\bf B}$, a constant diagonal matrix. The elements of the random noise matrix are assumed to be $\delta$-correlated $\langle E_i(t) E_j(t')\rangle= \delta(t-t') \delta_{ij}$. 