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Nanocavity lasers are commonly characterized by the spontaneous couplingcoefficient $\beta$ that represents the fraction of photons emitted into thelasing mode. While $\beta$ is conventionally discussed in semiconductor laserswhere the photon lifetime is much shorter than the carrier lifetime (class-Blasers), little is known about $\beta$ in atomic lasers where the photonlifetime is much longer than the other lifetimes and only the photon degree offreedom exists (class-A lasers). We investigate the impact of the spontaneouscoupling coefficient $\beta$ on lasing properties in the class-A limit byextending the well-known Scully-Lamb master equation. We demonstrate that, inthe 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 standpointof a quantum dissipative system. Calculating a Liouvillian gap, we clarify therelation between $\beta$ and the continuous phase symmetry breaking.

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