0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 g=0.5 g= 0 .1 g=1. 5 g=1 .0  t n  a FIG. 2: The mean photon number in the opto-mechanical cavity versus time: ¯ n = 0 , γ = 0 . 9 , κ = 1 . 0 , µ = 0 . 001 , ω$_{m}$ = 4 . 4 , Δ = 1 . 02 ω$_{m}$, ω$_{m}$ = 4 . 4 and four values of g . is shown in figure 5 where we compare the dynamics for a Fock state in the source cavity n = 5, and two coherent states, | α 〉 with α = √$_{5 and}$ √$_{5}$$_{i}$$_{.}$ We finally include coupling of the mechanical resonator to a nonzero temperature heat bath, here ¯ n = 1000 (see Figure 6). This can be accomplished by starting from different initial conditions that take into account that, prior to the single-photon injection, the optical cavity and the mechanical resonator are in thermal equilibrium. While the signal is of the same magnitude as in the zero-temperature case, there is an added noise that corresponds to the steady-state thermal occupation of the mechanical oscillator. The calculation we have presented is based on a linearization of the intensity dependent force acting on the mechanical element, around a strong coherent steady state field inside the cavity. This means that the average photon numbers we have calculated are in addition to a coherent steady state field. In order to detect the added photon number due to the linearized interaction, over and above the steady state coherent field inside the cavity, we need to subtract off the known steady state field amplitude α$_{0}$ by displacing the output field amplitude from the cavity before sending it to a photodetector. Such displacements III. DISCUSSION AND CONCLUSION 9