Cooperative effects in the single-mode laser

We apply Lugiato's theory of open systems to the single-mode laser model of Haken and co-workers, in order to investigate the effect on laser dynamics of processes involving one photon and two atoms. A Fokker-Planck time-evolution differential equation is obtained for the field quasiprobability distribution $P(\beta , {\beta }^{*}, t)$, which includes terms proportional to the squared number of atoms $N^2$. A detailed analysis reveals a twofold effect of these terms. The main effect that becomes relevant well above threshold is a broadening of the stationary field-intensity distribution. This broadening is due to a "cooperative noise," whose source is a squared polarization created by the two-atom processes. These processes also give rise to a very weak loss mechanism, which affects the maximum of the stationary intensity distribution in the threshold region, slightly increasing the threshold for the laser emission.