modes. The normal mode frequencies are given by ω 2 ± = 1 2 ( Δ 2 + ω 2 m ± √ (Δ 2 − ω 2 $_{m}$) 2 + 16 g $^{2}$Δ ω$_{m}$ ) (4) The normal mode splitting occurs for any g > 0 in the undamped case, but when damping is included a minimum coupling strength is required and given by g > κ (see the Supplementary Information in [12]). In the undamped case, oscillatory solutions will occur in the sought single-photon detection probability when ω$_{±}$ are real, i.e. 4 g 2 ≤ Δ ω$_{m}$ . In the limit that 4 g 2 << ω$_{m}$ Δ we can chose the detuning Δ to make particular resonant terms (Δ = ± ω$_{m}$ ) dominate the interaction. To identify the resonant terms we first move to an interaction picture defined by the free Hamiltonian, H$_{0}$ = ℏ Δ a $^{†}$a + ℏ ω$_{m}$b $^{†}$b . The resulting time-dependent Hamiltonian may be approximated by a time-independent Hamil- tonian using the appropriate resonance condition. For example, if Δ = ω$_{m}$ , the interaction Hamiltonian can be approximated by the red sideband coupling H$_{r}$ = ℏ g ( ab † + a $^{†}$b ) (5) which leads to cooling of the mechanical motion if the optical cavity is rapidly damped[17– 19]. The validity of the rotating wave approximation implicit in this ‘side-band’ Hamiltonian depends on the ratio 2 g/ Δ << 1. We note that in general the coupling constant can be quite large[12], and one may have to take the full interaction, Eq.(3) into account. This can lead to entanglement between the optical and mechanical degrees of freedom[20], and also to heating of the mechanical motion[11] and eventually to an instability of the steady state resulting in self sustained oscillation on a limit cycle[21]. The red side-band approximation, Eq.(5) describes a reversible swap of a single-photon excitation from the cavity field to the mechanical system. In the general case this exchange will not be perfect due to the counter rotating terms ab and a $^{†}$, b$^{†}$. Before the single photon enters the cavity, the opto-mechanical system is in a dynamical steady state with a large circulating power in the cavity due to the coherent field α$_{0}$ . On top of this, there will be addi- tional photons due to the excitation of both the cavity and the mechanical resonator mainly by the counter rotating terms. A single photon then enters the cavity at a random time and the system moves away from the steady state through damped oscillations (provided 4 g 2 ≤ Δ ω$_{m}$ ) corresponding to exchange of energy between the cavity and the mechanical resonator. This additional excitation can be lost either through the mechanical damping 5