Transition from incoherence to coherence in the spontaneous emission of extended systems

Abstract

Using the results of a previous paper, we study the evolution of a completely excited system of $N$ two-level atoms, distributed in a circular cylinder with dimensions much larger than the wavelength, interacting with all modes of radiation field. We emphasize the distinction between $r$-conserving ($r\mathrm{C}$) and $r$-nonconserving ($r\mathrm{NC}$) processes ($r$ is the so-called cooperation number). Considering the number of photons emitted as the discrete independent variable, this evolution is described by a partial difference equation. Numerical solutions of this equation show the transition from $r\mathrm{NC}$ dominance at the beginning to $r\mathrm{C}$ dominance later. This is also a transition from incoherent to coherent emission of radiation.