Dynamical evolution of correlated spontaneous emission of a single photon from a N atoms cloud

We study the correlated spontaneous emission from a dense spherical cloud of $N$ atoms uniformly excited by absorption of a single photon. We find that the decay of such a state depends on the relation between an effective Rabi frequency $\Omega \propto \sqrt{N}$ and the time of photon flight through the cloud $R/c$. If $\Omega R/c<1$ the state exponentially decays with rate $\Omega ^{2}R/c$ and the state life time is greater then $R/c$. In the opposite limit $\Omega R/c\gg 1$, the coupled atom-radiation system oscillates between the collective Dicke state (with no-photons) and the atomic ground state (with one photon) with frequency $\Omega $ while decaying at a rate $c/R$.