lasing, whatever the type of device (conventional lasers, high β lasers, nanolasers). Namely, in this highly non-linear regime, the cavity population diverges with respect to the atomic one. As it appears in the figure, this can only happen in the good cavity regime, confirming the prediction above. Two strategies can be adopted to enter the good cavity regime: decreasing the cavity losses, i.e. κ , or increasing the effective atom-cavity coupling, R . The first approach has been explored to study the potential of a quantum dot coupled to a high Q microsphere to show lasing [31]. The other approach is more promising, as it allows to simultaneously increase the fraction β of photons spontaneously emitted in the cavity mode. In the limit where β → 1, the device has a perfect quantum efficiency, even before stimulated emission is reached. In such kind of devices, no kink can be observed in the input-output curve (namely, the rate of emitted photons with respect to pump power), thus justifying the denomination of “thresholdless laser” [32, 33]. In the following we restrict the study to the case of a high β single atom laser, and define signatures of the lasing regime. We have studied the evolution of three main properties of the single emitter device with respect to pump power: the cavity population, n$_{a}$ , the atomic population, n$_{x}$ , and the auto- correlation function of the field at zero time delay, defined as g $^{2}$(0) = 〈 ˆ a $^{†}$ˆ a $^{†}$ˆ a ˆ a 〉 /n 2 $_{a}$. To this end, we have numerically solved the master equation, Eq. (1), for the model hamiltonian in Eq. (2) and the Lindblad operators in Eqs. (3), (4), and (11). The operators are explicitly built in matrix form on a Fock basis of occupation numbers for the atom and the cavity mode, respectively. For any given set of model parameters, the steady state density matrix can be obtained by numerically searching for the eigenvector | ρ 〉〉$_{ss}$ corresponding to the eigenvalue λ$_{ss}$ = 0 of the linear operator equation ˆ L | ρ 〉〉 = λ | ρ 〉〉 . In the latter, | ρ 〉〉 is essentially the density operator mapped into vectorial form, and ˆ L is the linear matrix corresponding to the Liouvillian operator in the right-hand side of the master equation. If it exists, as it is always the case for the parameters considered, the steady state solution is unique [34]. After recasting the vector | ρ 〉〉$_{ss}$ in matrix form, the relevant observable quantities can be calculated as 〈 O 〉$_{ss}$ = T r { ˆ Oρ$_{ss}$ } . In the following simulations, we kept up to 30 photons in the basis, which is largely sufficient for convergence. We show the results in Figs. 4a,b,c 13 B. Single two-level emitter lasing criteria