Low- and high-$\beta$ lasers in Class-A limit: photon statistics, linewidth, and the laser-phase transition analogy

Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient $\beta$ that represents the fraction of photons emitted into the lasing mode. While $\beta$ is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about $\beta$ in atomic lasers where the photon lifetime is much longer than the other lifetimes and only the photon degree of freedom exists (class-A lasers). We investigate the impact of the spontaneous coupling coefficient $\beta$ on lasing properties in the class-A limit by extending the well-known Scully-Lamb master equation. We demonstrate that, in the class-A limit, all the photon statistics are uniquely characterized by $\beta$ and that the laser phase transition-like analogy becomes transparent. In fact, $\beta$ perfectly represents the "system size" in phase transition. Finally, we investigate the laser-phase transition analogy from the standpoint of a quantum dissipative system. Calculating a Liouvillian gap, we clarify the relation between $\beta$ and the continuous phase symmetry breaking.