The two-level atom laser: analytical results and the laser transition

The problem of the two-level atom 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 with 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, $\kappa \to 0$, $g \to 0$, such that $g^2/\kappa$ stays finite and $g^2/\kappa > 2 \gamma$, where $\gamma$ is the rate of atomic losses. The same scaling procedure is also shown to describe a similar change to Poisson distribution in the Scully-Lamb laser model too, suggesting that the low-$\kappa$, low-$g$ asymptotics is of a more general significance for the laser transition.