Note that this effective quality factor gives a symmetrical role to the emitter and to the cavity mode as long as one does not exit the bad cavity regime. In experiments exploiting CQED effects, increasing the Purcell factor is an important goal, giving rise to a quest to increase the quality factor of the cavity mode. According to the generalized expression we have derived, it makes sense to search for the highest possible Q$_{cav}$ (even when it overcomes Q$_{em}$ ) compatible with the bad cavity regime. In particular, by making Q$_{cav}$ bigger than Q$_{em}$ , one can double the effective Purcell factor of the system in the case where the atom is resonant with the cavity mode. On the contrary, increasing pure dephasing γ ∗ leads to a decrease of the emitter’s quality factor Q$_{em}$ , thus to a decrease of the effective quality factor Q$_{eff}$ . Consequently, the spontaneous emission rate is reduced as it appears in Fig. 2a, where we have represented the relaxation of an initially excited atom in the resonant case for different values of the pure dephasing rate. The influence of pure dephasing is dramatically different if the atom and the cavity are detuned, as already pointed out in Ref. [28]. In this case indeed, it makes sense that a decrease of Q$_{em}$ , thus of Q$_{eff}$ , induced by pure dephasing leads to the enhancement of the spontaneous emission rate as it clearly appears in Fig. 2b. To quantify the maximal enhancement we can get, it is worth noticing that the parameter R is maximized when κ + γ + γ ∗ = 2 δ , allowing to reach an optimal value R$_{max}$ = 2 g $^{2}$/δ and a maximal effective Purcell factor F ∗ max = 2 g $^{2}$/δγ . Higher values of pure dephasing rate lead to a decrease of the effective atom-cavity coupling, and consequently of the spontaneous emission rate. By playing on pure dephasing, one can significantly improve the effective Purcell factor by a factor F ∗ $_{max}$/F ∼ δ/κ . Note that we have used state of the art parameters of quantum dots coupled to optical semi-conducting cavities, where coupling strengths like g = 50 µ eV and cavity linewidths κ = 250 µ eV (corresponding to quality factors of Q$_{cav}$ ∼ 5000) are commonly reached. Tuning of the pure dephasing rate can be achieved using temperature or pump power as it clearly appears in the studies described in [25]. The observed behavior was attributed to the fluctuation in the occupancy of electron traps in the QD neighbourhood [9]. In particular, these experiments show that the linewidth of the QD exciton can be tailored almost at will over a large range of experimental values (1-500 µ eV). Nevertheless, non-resonant pumping was used in these studies, thus allowing the trapping of more than a single exciton in the quantum dot and effectively hindering the validity of the two-level atom model used in the 8