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 ℏq, 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, ρ, that can be interpreted as the average number of photons scattered per atom in the classical limit. When ρ ≫ 1, the semiclassical CARL regime is recovered, with many momentum levels populated at saturation. On the contrary, when ρ ≤ 1, the average momentum oscillates between zero and ℏq, and a periodic train of 2π 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 π 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.