single quantum dot laser case, and its differences and similarities with respect to the two level atom case, are discussed in Subsec. D. When the pumping rate is too large, the atomic emission becomes incoherent, leading to a decrease of the cavity population, and to the corresponding increase of the atomic popula- tion, until total inversion in reached. The emitted field becomes thermal and the parameter g $^{2}$(0) converges to its limiting value, 2. This phenomenon is known as the quenching of the laser, and was first predicted in [30]; it was attributed to the saturation of the two-level emitter. The notion of good cavity regime sheds new light on this feature. As a matter of fact, the atom-cavity coupling R decreases with respect to pump power P$_{x}$ . Once the lasing regime is reached, one can increase P$_{x}$ , and consequently the number of photons in the lasing mode, as long as one remains in the good cavity regime; when this condition is no longer satisfied, the laser gradually switches off. Thus, in this view quenching is due to the transition to the bad cavity regime, induced by power broadening. We have plotted the evolution of the parameter R with respect to pump power P$_{x}$ (inset): as it can be seen in the figure, quenching happens for a typical value of the pump P$_{x}$ ∼ 20 g , for which the atom-cavity coupling constant becomes lower than κ , confirming our initial guess. Before examining the influence of pure dephasing on the lasing signatures, we mention another usual criterium for lasing in conventional devices, namely the narrowing of the spectrum emitted by the cavity mode. In the single atom device, it has been predicted by [13, 37, 42], and maybe experimentally observed in [41], that at low pump power the spectrum consists in a series of peaks (the so-called “Jaynes-Cummings forks” [13]). The authors of [41] have talked about a “coexistence of the strong coupling regime and the lasing regime”. We underline here that the emission of Jaynes-Cummings forks is a natural feature of the single emitter laser [42]. Reaching stimulated emission in the steady state regime simply means that one observes photons coming from radiative transitions between states of the excited manifolds in the Jaynes Cummings ladder, which naturally results in series of peaks centered around the cavity frequency. Increasing the pump power leads to the broadening of the peaks and to their convergence towards a single one at the cavity frequency, as mentioned in [13, 37, 42]. The transition between multi-peaks and single peak emission takes place after the lasing threshold, as evidenced in [41]. 15