Classical and quantum treatment of amplifier and superradiant free-electron laser dynamics

The quantum initiation for free-electron lasers is defined, and the collective instability threshold starting from the electron position and the momentum quantum fluctuations is derived. The Glauber distribution P(alpha) for the field is obtained, i.e., a displaced Gaussian, which represents the superposition of a coherent field (stimulated emission) and an incoherent one (spontaneous emission). The concept of superradiance for free-electron lasers is defined, both in a classical framework and at a quantum statistical level, stating the conditions under which the effect should be observed.