Quantum-optical master equations—exemplified by the Jaynes-Cummings model with damping—are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.

• Received 4 January 1994

DOI:https://doi.org/10.1103/PhysRevA.50.2667