content_image/328030.jpg content_image/328031.jpg content_image/328032.jpg content_image/328033.jpg content_image/328034.jpg content_image/328035.jpg
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If we restrict our analysis to the bad cavity regime, we can safely assume that the dynamics is restricted to the subspace spanned by the three states {| g, 0 〉 , | g, 1 〉 , | e, 0 〉} , irrespective of the pumping rate P$_{x}$ . This allows us to obtain steady state solutions for the atomic ( n$_{x}$ = 〈 ˆ σ$_{+}$ ˆ σ$_{−}$ 〉$_{ss}$ ) and the cavity mode ( n$_{a}$ = 〈 ˆ a $^{†}$ˆ a 〉$_{ss}$ ) populations, respectively. We obtain 

where we have introduced the effective rates 

<img src='content_image/328031.jpg'>

<img src='content_image/328032.jpg'>

<img src='content_image/328033.jpg'>

<img src='content_image/328034.jpg'>

In the case where P$_{x}$ ≪ γ + γ ∗ + κ , one obtains for ˜ R the expression of the effective coupling R defined in Eq. (7), making transparent the expression of the atomic population n$_{x}$ . It corresponds indeed to the incoherent pumping of a two-level system connected to two different loss channels, the first being due to the continuum of leaky photonic modes with a rate γ , and the second to the coupling to the cavity mode as it appears schematically in Fig. 1d. The effective loss rate of this second channel is, again, κ ˜ R/ ( κ + ˜ R ), as already showed in the spontaneous emission picture. The latter considerations reinforce the generality of the physical interpretation for the parameter R . In the following, we identify the two coupling strengths and use a unified notation for it, R . 

Finally, the rate of photons emitted in the bad cavity regime can be explicitly given as 

<img src='content_image/328035.jpg'>

In the limiting case of very low pumping rate, i.e. for P$_{x}$ ≪ γ + κR/ ( κ + R ), the rate of photons exiting the cavity can be expressed as N = βP$_{x}$ , where β is the efficiency of the single photon source given in eq.(8), showing that the low incoherent pumping scheme can safely be modeled as a series of spontaneous emission events with a rate P$_{x}$ . This conclusion completely restores the continuity between the spontaneous emission picture and 

