A Model of Interacting Radiation and Matter

Abstract

We investigate the long‐time behavior of a model consisting of N two‐level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual $\int \mathrm{j}·\mathbf{A}\mathrm{dv}$ interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long‐time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is $\mathrm{\gamma ̃N\gamma ̃}$ where $\mathrm{\gamma ̃}$ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self‐consistent field approximation. In the next order, particle‐field correlations appear. We explicitly solve the equations of motion for the particle‐field correlations in terms of the average quantities that appear in the self‐consistent field approximation. We show the self‐consistent field approximation consists of five first‐order differential equations. Next we show the equations of motion for the density matrix of the system correct to order ($\mathrm{\gamma ̃N\gamma ̃}$ )² are equivalent to eight first‐order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest‐order microscopic equations are equivalent to semiphenomenological theories and our higher‐order equations contain only the measurable second‐order moments of the electromagnetic field in addition to the variables that appear in semi‐phenomenological theories.