A quantum model for collective recoil lasing

Free Electron Laser (FEL) and Collective Atomic Recoil Laser (CARL) are described by the same model of classical equations for properly defined scaled variables. These equations are extended to the quantum domain describing the particle's motion by a Schr\"{o}dinger equation coupled to a self-consistent radiation field. The model depends on a single collective parameter $\bar \rho$ which represents the maximum number of photons emitted per particle. We demonstrate that the classical model is recovered in the limit $\bar \rho\gg 1$, in which the Wigner function associated to the Schr\"{o}dinger equation obeys to the classical Vlasov equation. On the contrary, for $\bar \rho\le 1$, a new quantum regime is obtained in which both FELs and CARLs behave as a two-state system coupled to the self-consistent radiation field and described by Maxwell-Bloch equations.