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with ˜ κ = i Δ + κ/ 2 and ˜ µ = iω$_{m}$ + µ/ 2, and the noise matrix is defined by 

The noise matrix elements obey a separate set of equations given by 

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(and the complex conjugate equations). We have defined σ = 2 i Δ + ( κ + γ ) / 2 and τ$_{±}$ = i ( ω$_{m}$ ± Δ) + ( µ + γ ) / 2. The equations of motion for the source cavity alone are 

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If there is a number state, | n 〉 $_{c}$, prepared in the source cavity at t = 0, one imme- diately sees that 〈 c $^{2}$〉 ( t ) is zero for all time, while 〈 c $^{†}$c 〉 ( t ) = ne − $^{γt}$. Note that the case for an initial coherent state, | β 〉 , in the source cavity is different, as for that case 〈 c $^{2}$〉 ( t ) = β $^{2}$e − (2 i Δ+ γ ) $^{t}$, 〈 c $^{†}$c 〉 ( t ) = | β | $^{2}$e − $^{γt}$, which makes the dynamics dependent on the phase of β . The number state case, in contrast, has no similar phase reference. The equa- tion for the noise matrix may be solved directly in both cases and substituted into the equation for the correlation matrix C . Note that the intracavity photon number will depend on the time dependent correlations between a and c . This starts at zero, rises to a maximum as the single photon excitation begins to grow in the cavity and then decays to zero. 

We discuss our results in direct comparison with present experimental parameters. An experiment reported by Gr¨ oblacher et al[12], demonstrated the strong coupling regime with a linewidth of the optical cavity and the mechanical resonator of κ = 2 π × 215kHz and µ = 2 π × 140Hz, respectively and with an effective coupling strength of g = 2 π × 325kHz. The mechanical resonator frequency was 2 π × 947kHz. In units such that κ = 1 these are equivalent to ω$_{m}$ = 4 . 4 , µ = 6 . 5 × 10 − $^{4}$, g = 1 . 5. In the following we set the detuning 

