Cooperative spontaneous emission of N atoms: Many-body eigenstates, the effect of virtual Lamb shift processes, and analogy with radiation of N classical oscillators

TLDR: In this paper, the collective emission of a single photon from a cloud of two-level atoms (one excited, one ground state) is considered and the problem is reduced to finding eigenfunctions and eigenvalues of an integral equation.

Abstract: We consider collective emission of a single photon from a cloud of $N$ two-level atoms (one excited, $N\ensuremath{-}1$ ground state). For a dense cloud the problem is reduced to finding eigenfunctions and eigenvalues of an integral equation. We discuss an exact analytical solution of this many-atom problem for a spherically symmetric atomic cloud. Some eigenstates decay much faster then the single atom decay rate, while the others undergo very slow decay. We show that virtual processes yield a small effect on the evolution of rapidly decaying states. However, they change the long time dynamics from exponential decay into a power-law behavior which can be observed experimentally. For trapped states virtual processes are much more important yielding additional decay channels which results in a slow decay of the otherwise trapped states. We also show that quantum mechanical treatment of spontaneous emission of weakly excited atomic ensemble is analogous to emission of $N$ classical harmonic oscillators.