The equations of motion for a fully quantized system of atoms interacting with traveling waves is solved to terms of fourth order in the field strengths. From these, criteria are developed for the stability of traveling waves in solid and gaseous media. Expressions are given for the power outputs for both traveling and standing waves. The results for the latter agree with those obtained by Lamb in his semiclassical treatment of standing waves.

• Received 15 October 1964

DOI:https://doi.org/10.1103/PhysRev.137.A1651