Analytical solution to the master equation for a quantized cavity mode

Abstract

We solve analytically the master equation for a quantized cavity mode, finding the time-dependent reduced density operator. The formalism applies to both amplifiers and attenuator, permitting the results to be converted between them by introducing a temperature that can be negative. We obtain a quasi-probability distribution function similar to that obtained by Glauber [ GlauberR. J., ed., Quantum Optics ( Academic, New York, 1969)] from our reduced density matrix and equate his parameters with the coefficients appearing in the master equation. We illustrate the formalism by calculating variances, expectation values, and photon statistics for the damped simple harmonic oscillator and the linear amplifier.