A. Why the good cavity regime is mandatory Here we show that the good cavity regime is a necessary condition to reach stimulated emission. A heuristic demonstration has been developed in [21, 31] and is based on the search of proper conditions to reach a steady state cavity population n$_{a}$ ∼ 1: in this view, the production rate of photons in the cavity, which is at most βP$_{x}$ , should overcome the cavity mode dissipation rate κ . As the cycling rate P$_{x}$ is limited by the typical spontaneous emission rate R , a necessary condition can be formulated as R > κ , which is the condition for good cavity regime. This reasoning is confirmed by analyzing the rate equations for the incoherently pumped atom, which can be derived from Eqs. (5) and (11), respectively. One finds where ˆ σ$_{z}$ = | e 〉 〈 e | − | g 〉 〈 g | stands for the population inversion, while the parameter R is still the effective atom-cavity coupling defined in Eq. (15). As it appears in Eqs. (17- 18), the evolution of the population inversion depends on three terms respectively due to spontaneous emission, pumping, and stimulated emission, whereas the evolution of the cavity mode involves spontaneous emission, stimulated emission and cavity losses. In steady state, the atomic population inversion I = 〈 ˆ σ$_{z}$ 〉$_{ss}$ and the cavity population n$_{a}$ = 〈 ˆ a $^{†}$ˆ a 〉$_{ss}$ are coupled in the following way: We have represented in Fig. 3 the behavior of the steady state cavity population, n$_{a}$ , with respect to the atomic inversion, I , by keeping the parameter R constant, both in the good and bad cavity regimes. When the population is not inverted ( I → 0), which happens for low pumping rates, n$_{a}$ evolves linearly with respect to I : this is the spontaneous emission regime. On the contrary, if stimulated emission can be reached, cavity population evolves linearly with the pump rate, while the atomic population inversion remains clamped to a value where the optical gain compensates for the losses. This behavior is characteristics of 12 n$_{a}$ = 1 2 R ( I + 1) κ − R I . (20) I = P$_{x}$ − ( R + γ ) R (1 + 2 n$_{a}$ ) + γ + P$_{x}$ (19) d 〈 ˆ a $^{†}$ˆ a 〉 d t = R 2 (1 + 〈 ˆ σ$_{z}$ 〉 ) + R 〈 ˆ a $^{†}$ˆ a 〉〈 ˆ σ$_{z}$ 〉 − κ 〈 ˆ a $^{†}$ˆ a 〉 , (18) d 〈 ˆ σ$_{z}$ 〉 d t = − ( R + γ )( 〈 1 + ˆ σ$_{z}$ 〉 ) + P$_{x}$ (1 − 〈 ˆ σ$_{z}$ 〉 ) − 2 R 〈 ˆ a $^{†}$ˆ a 〉〈 ˆ σ$_{z}$ 〉 (17)