Laser equations for the photon distribution

In order to characterize the 2P vs 1P lasing regions, the complete master equation of the system can be approximately reduced to one for the total photon probability distribution, $\mathrm{p}[n]$ , as we did in Section 5.4.3, Eqs. (5.42), for the JCM:

$\begin{multline} \partial_t\mathrm{p}[n]=-\Big[\gamma_an+(n+1)(l_\mathrm{1PG}... ...\mathrm{p}[n+1] +n(l_\mathrm{1PG}-n l_\mathrm{S})\mathrm{p}[n-1] \end{multline}$

Following the method of Scully & Zubairy (2002), one can obtain the laser one and two photon gains, $l_\mathrm{1PG}$ and $l_\mathrm{2PG}$ , and the self saturation $l_\mathrm{S}$ in the lasing regime (where statistics are basically Poissonian). These equations can provide the conditions in which the 2P lasing dominates over the one-photon lasing.