Stimulated and Spontaneous Emission of Radiation in a Single Mode for   N-TLMs
  The general expressions for the time development of the ensemble averages of E^- and E^-E^+ are found for N two level molecules (TLMs) interacting with a single mode electromagnetic field. The TLMs are assumed to be at equivalent mode positions in the field. In the main body of the paper, the general results are then simplified by assuming that all TLMs are in the up state at time 0. Examples of spontaneous emission behavior including both Super Radiant behavior and emission suppression are shown in Appendix E while stimulated emission with all TLMs initially in the lower state is examined in Appendix F. Both resonant and non resonant cases are considered in the main body. Analytic results are presented for low numbers of TLMs. In the appendices, it will be shown that for the mean photon number large in comparison to the number of TLMs, that the Average Field Approximation (AFA) is a useful representation with much simpler analytic form. The validity of this approximation will also be examined. In the main paper, a comparison between the exact results and results using the AFA will be presented for a small number of TLMs and a large average photon number. It will be found that the results are in general remarkably similar but for non resonance differ somewhat at a finer level of detail. Of interest is that the ensemble averages for N TLMs is similar to the ensemble average for a single TLM multiplied by the number of TLMs which is somewhat disappointing but is believed to be the consequence of assuming that all TLMs are at equivalent mode positions and large mean photon number. Some results for smaller photon number will also be presented 
Michael T. Tavis and Frederick W. Cummings
quant-ph