Q-function approach to a two-photon laser.

Abstract

A quantum theory of a two-photon laser is developed by making use of the antinormal-ordering Q function. Starting from the microscopic atom-field interaction Hamiltonian for cascade three-level atoms and including saturable absorbers, we first present a master equation for the field-density operator, and then transform the master equation into a Fokker-Planck equation for the Q function. The Q-function approach enables us to study the two-photon laser analytically, obtaining simple expressions for nonlinear gain, mean photon number, frequency pulling, natural linewidth, and photon-number variance in a unified method. We find that the field in a two-photon laser will build up from a vacuum without triggering if its linear gain is larger than the cavity loss. Also hysteresis can occur in the two-photon laser even without triggering. With an overall two-photon resonance, the normalized photon-number variance approaches a common value 11/8 well above ``threshold,'' independent of the one-photon detuning. We compare these results with previous results obtained from an effective interaction Hamiltonian. We find that the domain of validity of the effective Hamiltonian for the photon-number variance is much smaller than that for the mean photon number. Well above ``threshold'' the effective Hamiltonian overestimates both the natural linewidth and the photon-number variance.