  In this paper, we investigate the evolution of the energy spread and the
divergence of electron beams while they interact with different laser pulses at
intensities where quantum effects and radiation reaction are of relevance. The
interaction is modeled with a QED-PIC code and the results are compared with
those obtained with a standard PIC code with the addition of a classical
radiation reaction module and with theoretical predictions. While classical
radiation reaction is a continuous process, in QED, radiation emission is
stochastic. The two pictures reconcile in the limit when the emitted photons
energy is small compared to the energy of the emitting electrons. The energy
spread of the electron distribution function always tends to decrease with
classical radiation reaction, whereas the stochastic QED emission can also
enlarge it. These two tendencies compete in the QED-dominated regime. Our
analysis, supported by the QED module, reveals an upper limit to the maximal
attainable energy spread due to stochasticity that depends on laser intensity
and the electron beam average energy. Beyond this limit, the energy spread
decreases. These findings are verified for different laser pulse lengths
ranging from short ~ 30 fs pulses presently available to the long ~ 150 fs
pulses expected in the near-future laser facilities, and compared with a
theoretical model. Our results also show that near future experiments will be
able to probe this transition and to demonstrate the competition between
enhanced QED induced energy spread and energy spectrum narrowing from classical
radiation reaction.
