Two-level laser: Analytical results and the laser transition

The problem of the two-level laser is studied analytically. The steady-state solution is expressed as a continued fraction and allows for accurate approximation by rational functions. Moreover, we show that the abrupt change observed in the pump dependence of the steady-state population is directly connected to the transition to the lasing regime. The condition for a sharp transition to Poissonian statistics is expressed as a scaling limit of vanishing cavity loss and light-matter coupling, kappa -> 0, g -> 0, such that g(2)/kappa stays finite and g(2)/kappa > 2 gamma, where gamma is the rate of nonradiative losses. The same scaling procedure is also shown to describe a similar change to the Poisson distribution in the Scully-Lamb laser model, suggesting that the low-kappa, low-g asymptotics is of more general significance for the laser transition.