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

Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient $β$ that represents the fraction of photons emitted into the lasing mode. While $β$ is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about $β$ 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 $β$ 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 $β$ and that the laser phase transition-like analogy becomes transparent. In fact, $β$ 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 $β$ and the continuous phase symmetry breaking.