In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con­ ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os­ cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra­ diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.