Cooperative spontaneous emission via renormalization approach: Classical versus semi-classical effects

We address the many-atom emission of a dilute cloud of two-level atoms through a renormalized perturbation theory. An analytical solution for the truncated coupled-dipole equations is derived, which contains an effective spectrum associated to the initial conditions. Our solution is able to distinguish precisely classical from semi-classical predictions for large atomic ensembles. This manifests as a reduction of the cooperativity in the radiated power for higher atomic excitation, in disagreement with the fully classical prediction from linear optics. Moreover, the second-order cooperative emission appears accurate over several single atom lifetimes and for interacting regimes stronger than those permitted in conventional perturbation theory. We can compute the semiclassical dynamics of hundred thousand of interacting atoms with ordinary computational resources, which makes our formalism particularly promising to probe the nonlinear dynamics of quantum many-body systems that emerge from cumulant expansions.