A description of the parametric amplifier and frequency converter is presented without introducing the classical (i.e., parametric) approximation for the pumping field. Constants of the motion are found which reduce the solution of the Schrödinger equation to the diagonalization of a matrix. This diagonalization is accomplished numerically, and the eigenvalues and eigen-functions of a system with fixed energy are calculated. The time-dependent behavior of the mean number of photons in the amplified or frequency up-converted field is presented. The time evolution of the probability distributions is illustrated. The technique is extended to the problem of coherent spontaneous emission from a system of $N$ two-level atoms interacting with the radiation field where both the atomic system and the radiation field are quantized.

• Received 23 June 1969

DOI:https://doi.org/10.1103/PhysRevA.1.446