A Model of Interacting Radiation and Matter. II

We extend the investigation of the long time behavior of a model consisting of N two-level atoms interacting with radiation to include dissipation, pumping, and center-of-mass motion. We show that when the effective expansion parameter γ̃Nγ̃ is small, the self-consistent field approximation remains the solution to lowest order in γ̃Nγ̃. The inclusion of the center-of-mass motion introduces another dimensionless parameter β into the theory. We show that when this parameter is small, the electromagnetic field amplitude varies slowly on the time scale of the center-of-mass motion. We solve the equations of motion in this slowly varying limit by an extension of the Bogoliuboff-Kryloff theory of quasilinearity to the problem of time-dependent integral kernals. We find the unique stable stationary state and show that in the slowly varying limit the stationary state is approached independent of initial conditions. We calculate the frequency shift to second order in β. The first-order frequency shift is the same as that calculated by Lamb. We compare our steady-state solution with recent experiments with lasers. We include the effect of collisions in the steady state.