Quantum effects in the collective light scattering by coherent atomic recoil in a Bose-Einstein condensate

We extend the semiclassical model of the collective atomic recoil laser (CARL) to include the quantum mechanical description of the center-of-mass motion of the atoms in a Bose-Einstein condensate (BEC). We show that when the average atomic momentum is less than the recoil momentum $\hbar\vec q$, the CARL equations reduce to the Maxwell-Bloch equations for two momentum levels. In the conservative regime (no radiation losses), the quantum model depends on a single collective parameter, $\rho$, that can be interpreted as the average number of photons scattered per atom in the classical limit. When $\rho\gg 1$, the semiclassical CARL regime is recovered, with many momentum levels populated at saturation. On the contrary, when $\rho\le 1$, the average momentum oscillates between zero and $\hbar\vec q$, and a periodic train of $2\pi$ hyperbolic secant pulses is emitted. In the dissipative regime (large radiation losses) and in a suitable quantum limit, a sequential superfluorescence scattering occurs, in which after each process atoms emit a $\pi$ hyperbolic secant pulse and populate a lower momentum state. These results describe the regular arrangement of the momentum pattern observed in recent experiments of superradiant Rayleigh scattering from a BEC.