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The paper is organized as follows. We first introduce an effective atom-cavity coupling parameter, and use it to define the good and bad cavity regimes, when pure dephasing is properly taken into account. Focusing our attention onto the bad cavity regime, we derive a generalized Purcell factor, and analyze how to optimize the rate of emitted photons by a broad emitter-based single photon source. We then address the good cavity regime by revisiting the properties of the single emitter laser, paying attention to the influence of pure dephasing. We finally analyze the relevance of the developed model to the case of quantum dots coupled to semiconductor photonic cavities. 

## II. EFFECTIVE ATOM-CAVITY COUPLING 

The system under study is represented in Fig. 1a. A two-level atom of frequency ω$_{x}$ is initially pumped in its excited state. It is coupled to a single mode cavity of frequency ω$_{a}$ , with a strength g . The detuning between the atom and the cavity is denoted δ = ω$_{x}$ − ω$_{a}$ , the losses from the isolated atom and from the cavity mode are respectively γ and κ . The atom undergoes pure dephasing with a rate γ $^{∗}$. One first recalls some results related to the relaxation of the atom-cavity system, the atom having initially been excited [11]. The full quantum evolution of this system is described by the master equation 

where the total Hamiltonian of the system is ($_{ℏ}$ = 1) 

<img src='content_image/361303.jpg'>

<img src='content_image/361302.jpg'>

Here, ˆ a (ˆ a $^{†}$) is the annihilation (creation) operator for a photon in the cavity mode, while ˆ σ$_{−}$ (ˆ σ$_{+}$ ) is the lowering (rising) operator for the atom. The damping part for both the atom and the cavity mode can be described by operators in the Lindblad form within the master equation, written as 

<img src='content_image/361301.jpg'>

We notice that in most of the experimental situations currently accessible, the spontaneous emission rate γ is much lower than the other typical rates, particularly the cavity damping 

