From sub-Poissonian to super-Poissonian pumping in the micromaser: Corrections to reservoir theory

We present a quantum theory of the micromaser which allows for arbitrary (sub- as well as super-Poissonian) fluctuations of the pumping beam. It handles the interaction with the active atoms (gain) and cavity decay (loss) on an equal footing. In conventional reservoir theory the rate of change of the cavity field is a sum of the changes due to separate interactions with the individual reservoirs, i.e., the interactions are uncorrelated. In our approach, corrections to reservoir theory arise. They contain the commutator of the gain and loss operators. The magnitude of these additional terms is characterized by the quantity p/${\mathit{N}}_{\mathrm{ex}}$ where p is a parameter describing pump-beam fluctuations. The parameter is, in fact, the negative of the Mandel Q parameter of the pump beam so that p=1 corresponds to regular pumping, p=0 to Poissonian one and p<0 to super-Poissonian pump beam fluctuations. ${\mathit{N}}_{\mathrm{ex}}$ is the number of excited atoms passing through the cavity during the lifetime of the intracavity field. Thus we recover the conventional reservoir limit if p=0 and/or ${\mathit{N}}_{\mathrm{ex}}$ is large. In all other cases the interactions with the gain and loss reservoirs are correlated. We exploit some of the consequences of the additional terms, presenting analytical as well as numerical results for the steady-state photon statistics (mean photon number and photon-number fluctuations, in particular) without resorting to the p expansion.