rate κ . In the following, we will keep the spontaneous emission rate in the formulas but shall neglect this parameter in comparison with the cavity decay rate as far as possible. The pure dephasing channel is described by the following Lindblad operator: From the master equation, the evolution of the populations and coherences is described by the following equations of motion: The coherent or quantum regime is characterized by the reversible exchange of a quantum of energy between the atom and the cavity mode. At resonance, this so-called vacuum Rabi oscillation shows up if the vacuum Rabi splitting 2 g essentially overcomes the effective dephasing rate γ + γ ∗ + κ . The opposite case corresponds to a regime where the energy is irreversibly spread between the atom, the cavity and the environment. In this incoherent, classical regime, the crossed terms 〈 ˆ a $^{†}$ˆ σ 〉 and 〈 ˆ σ$_{+}$ ˆ a 〉 , which are responsible for the Rabi oscillation, can be adiabatically eliminated. Out-of-resonance, the adiabatic elimination is valid for δ > g . At this step it is worth pointing out the difference between the coherent regime and the so-called strong coupling regime. Strong coupling is reached when the spontaneous emission spectrum of the system consists in two peaks of distinct frequencies [20], which corresponds, taking into account pure dephasing, to 2 g > | γ + γ ∗ − κ | . If this condition is fulfilled, the width of each peak equals ( κ + γ + γ $^{∗}$) / 2. Coherent coupling is reached when the doublet is resolved, which is the spectral counterpart of the vacuum Rabi oscillation. In the standard CQED picture where the atomic width γ ∗ + γ is negligible, the strong coupling regime perfectly matches the coherent regime. On the contrary, when γ ∗ is of the same order of magnitude as κ , strong coupling is a necessary but not sufficient condition to observe a coherent exchange of energy between the atom and the cavity mode. Coherent regime thus appears as more demanding than strong coupling. d 〈 ˆ a $^{†}$ˆ a 〉 dt = − κ 〈 ˆ a $^{†}$ˆ a 〉 + g 〈 ˆ σ$_{+}$ ˆ a 〉 + g 〈 ˆ a $^{†}$ˆ σ$_{−}$ 〉 d 〈 ˆ σ$_{+}$ ˆ σ$_{−}$ 〉 dt = − γ 〈 ˆ σ$_{+}$ ˆ σ$_{−}$ 〉 − g 〈 ˆ σ$_{+}$ ˆ a 〉 − g 〈 ˆ a $^{†}$ˆ σ$_{−}$ 〉 d 〈 ˆ σ$_{+}$ ˆ a 〉 dt = iδ 〈 ˆ σ$_{+}$ ˆ a 〉 − γ + γ ∗ + κ 2 〈 ˆ σ$_{+}$ ˆ a 〉 + g ( 〈 ˆ σ$_{+}$ ˆ σ$_{−}$ 〉 − 〈 ˆ a $^{†}$ˆ a 〉 ) . (5) 5 L$_{deph}$ = γ ∗ 4 (ˆ σ$_{z}$ρ ˆ σ$_{z}$ − ρ ) . (4)