Quantum effects in spontaneous emission by a relativistic, undulating   electron beam
  Current models of the effect of spontaneous emission on the electron beam dynamics neglect the discreteness of electron recoil associated with photon emission. We present a novel, one-dimensional model of the effect of spontaneous emission on the electron beam dynamics in an undulator both in the classical regime where discrete electron recoil is negligible, and the quantum regime where it is significant. It is shown that in the classical regime, continuous decrease of the average electron energy and diffusive growth of the electron energy spread occurs, in agreement with previous classical models. In the quantum regime, it is shown that the evolution of the electron momentum distribution occurs as discrete momentum groups according to a Poisson distribution. The narrow momentum features of the quantum regime may be useful for generation of coherent radiation, which relies on electron beams having sufficiently narrow momentum/energy distributions. 
G. R. M. Robb and R. Bonifacio
physics.acc-ph quant-ph