L at pump = P$_{x}$ 2 (2ˆ σ$_{+}$ρ ˆ σ$_{−}$ − ˆ σ$_{−}$ ˆ σ$_{+}$ρ − ρ ˆ σ$_{−}$ ˆ σ$_{+}$ ) . (11) n$_{x}$ = P$_{x}$ P$_{x}$ + ( γ + κ ˜ R κ + ˜ $_{R}$) (12) n$_{a}$ = ˜ R κ + ˜ R n$_{x}$ , (13) Γ 2 = P$_{x}$ + γ + γ ∗ + κ 2 (14) ˜ R = 4 g 2 Γ 1 1 + (2 δ/ Γ) 2 . (15) N = κn$_{a}$ = κR κ + R P$_{x}$ $^{(}$P$_{x}$ + γ + κR κ + R ) . (16) 10 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 Finally, the rate of photons emitted in the bad cavity regime can be explicitly given as 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 . where we have introduced the effective rates 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