for each emitter, because of inhomogeneous broadening in the solid-state environment, or the atomic motion in gas lasers. In such cases, pure dephasing could provide an effective tool to overcome this problem. As a matter of fact, increasing the homogenous linewidth of a bunch of detuned emitters would not only help reaching the good cavity regime, which is less critical to fulfill in the N emitter’s case, but would also increase their individual β factors so that low-threshold lasing could be favourably achieved. We stress that pure dephasing here is nothing but a very effective model for the broadening of solid-state emitters because of their interaction with the solid-state matrix. Still, such a mechanism may explain some unconventional lasing characteristics of few quantum dots detuned from high-quality photonic crystal cavity modes [43], and again, it looks promising in the context of high β multi-emitters lasers. 

## D. Single quantum dot lasers 

The potential of a single QD coupled to a semiconducting cavity to achieve a solid-state single emitter laser has been explored theoretically [13, 31, 37, 44], and recent experimental demonstrations tend to show that laser gain at the single quantum dot level is within reach [35, 40, 41]. However, as it was underlined by the authors themselves [35, 41], a laser based on single QD emission does not simply maps the physics of the single atom-laser into a solid- state system. One main source of differences is that solid-state cavities are always coupled to a bunch of background emitters that can efficiently feed the mode even if the emitters and the cavity are detuned, as it was experimentally evidenced in [15–18], and theoretically explored in [10, 11, 19]. Cavity feeding is generically attributed to the Purcell enhancement of relaxation processes that are resonant with the cavity mode, the very existence of these processes being due to phonon-assisted decay [19] or pure dephasing [10, 11] that broaden the emitters’ linewidths. Because of this background, cavity field at low pumping rate is not totally antibunched, as it also appears in [35, 40, 41], giving a quantitative measurement of the contribution of the single QD to the cavity emission [38]. 

Most importantly, even in the ideal situation where only a single QD is coupled to the cavity mode, the usually employed non-resonant pumping scheme to operate lasers allows to pump more than one exciton in the QD, making the physics of the single QD laser essen- tially different from a single atom laser. As multiexcitonic transitions happen at different 

