Stimulated and Spontaneous Emission of Radiation in a Single Mode for N-TLMs

TL;DR

This paper derives and compares exact and approximate models for the emission behavior of N two-level molecules interacting with a single electromagnetic mode, covering spontaneous and stimulated emission, with insights into the validity of the Average Field Approximation.

Contribution

It provides analytic expressions for emission dynamics of multiple molecules, compares exact results with the Average Field Approximation, and explores conditions for their validity.

Findings

Average field approximation is valid for large photon numbers.

Results show ensemble averages scale linearly with the number of molecules.

Differences between exact and approximate models are minor at high photon numbers.

Abstract

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