Collective Effects in Nanolasers Explained by Generalized Rate Equations

We study the stationary photon output and statistics of micro- and nano-lasers. Our closed-form expressions clarify the contribution of collective effects due to the interaction between quantum emitters. As a first main result we generalize laser rate equations and explain photon trapping: a decrease of the photon number output below the lasing threshold. Our second main result is an expression for the stationary cavity mode autocorrelation function $g_2$, which implies that collective effects may strongly infuence the photon statistics. We identify conditions for coherent, thermal and superthermal radiation, the latter being a unique fingerprint for collective emission in lasers. These generic analytical results agree with recent experiments, complement recent numerical results, and provide insight into and design rules for nanolasers.