3D Wigner model for a quantum free electron laser with a laser wiggler

A three dimensional, time dependent, quantum model for an FEL with a laser wiggler, based on a discrete Wigner function formalism taking into account the longitudinal momentum quantization, is presented. Starting from the exact quantum treatment, a motion equation for the Wigner function coupled to the self-consistent radiation field is derived in the realistic limit in which the normalized electron beam emittance is much larger than the Compton wavelength quantum limit. The model describes the three-dimensional spatial and temporal evolution of the electron and radiation beams, including diffraction, propagation, laser wiggler, emittance and quantum recoil effects. It can be solved numerically and reduces to the three-dimensional Maxwell-VIasov model in the classical limit. We discuss the experimental requirements for a quantum X-ray FEL with a laser wiggler, presenting preliminary numerical results and parameters for a possible future experiment.