gain and dispersion due to the collective strong coupling of atoms to the cavity and the change in the dynamics of the system is explained. Here, we took the decay of the atomic excited state phenomenologically similar to B. R. Mollow [@mollow_stimulated_1972]. Theoretically, it would be interesting to know how the laser properties change if a full quantum and multimode description including the vacuum modes [@grynberg_central_1993] is used. The vacuum modes and the drive field can be then integrated out similar to the approach of open quantum optics [@scully_quantum_1997] giving an effective non-linear interaction between the photons in the cavity and the atoms. Future work can be taken up to observe the properties of this type of non-linear laser, such as photon statistics, phase diffusion and squeezing, as predicted by single mode theories of Zakrzewski *et.al* [@zakrzewski_theory_1991; @zakrzewski_theory_1991-1; @zakrzewski_theory_1991-2] and Agarwal [@agarwal_theory_1990; @agarwal_dressed-state_1990]

Details of calculations {#calculations}
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The Hamiltonian of a system of driven $N$ two-level atoms interacting with a cavity driven by a probe laser can be written as,

$$\begin{aligned}
\hspace{-0.3 cm}\hat{H}&=\hbar \omega_c \ \hat{a}^{\dagger}\hat{a} +\hbar\left\{\eta^* \hat{a} e^{-i\omega_p t}+\eta \hat{a}^{\dagger} e^{i\omega_p t} \right\} + \sum^{N}_{j}\hbar\left\{\frac{\omega_a}{2}\hat{\sigma}^{z}_j   + g_j \left(\hat{a}^{\dagger} \hat{\sigma}^{-}_j + \hat{a} \hat{\sigma}^{+}_j \right)\right\} +\sum^{N}_{j}\hbar\left\{\Omega^{*} \hat{\sigma}^{-}_j e^{i\omega_d t}+ \Omega 