Quantum theory of SASE-FEL with discrete spectrum

We present a proof of principle of the novel regime of quantum SASE with high temporal coherence and discrete spectrum. Using a self-consistent system of Schroedinger-Maxwell equations with propagation effects, we show that the dynamics of the system are determined by a properly defined "quantum Fel-parameter", p, which represents the ratio between the classical momentum spread in the high gain regime and the one photon recoil momentum hk. In the limit p>> 1 the quantum model reproduces the classical SASE regime with random spiking behavior and broad spectrum. In this limit we show that the equation for the quantum Wigner function reduces to the classical Vlasov equation. In the opposite limit, p < 1, we demonstrate "quantum purification" of SASE: the classical chaotic spiking behaviour disappears and the spectrum becomes a series of discrete very narrow lines which correspond to transitions between discrete momentum eigenstates, resulting in high temporal coherence.