A Wigner function model for free electron lasers

We derive a 1D quantum model for the free electron laser in terms of a Wigner function for the electron beam. We consider both the case of an unbounded space coordinate, for which the momentum is continuous, and the case of a periodic space coordinate, for which the momentum is discrete. The Wigner model extends the Schrödinger model, previously considered, since it also describes the evolution of a mixed state. Furthermore, the Wigner model shows explicitly the classical limit when the quantum FEL parameter ρ¯ is large. This model is also the starting point for a future extension to a 3D description of the electron dynamics. The results obtained here are also valid for the quantum description of the collective atomic recoil laser.