Theory of the interaction of a single-mode radiation field with $N$ two-level atoms. II. Time evolution of the statistics of the system

In an earlier paper, we considered a Hamiltonian trilinear in boson operators, which described several physical systems of interest such as interaction of $N$ two-level atoms with a single-mode resonant radiation field, parametric amplification, frequency conversion, Raman and Brillouin scattering, etc. We found that the dynamics of the system can almost exactly be determined if either one of the three modes is strong for all times or any two modes are separately strong in two complementary time domains. In the present paper, we determine the statistical behavior of the system in each case. We give explicit results for a number of initial states (e.g., coherent, chaotic, a mixture of coherent and chaotic, number state, etc.) of the modes 1, 2, and 3.