Quantum theory of stimulated microwave emission from E×B drifting electrons

The quantum theory of stimulated microwave emission from electrons drifting in crossed electric and magnetic fields inside slow wave cavities is considered. The energy and momentum of the emitted radiation quanta come from the electron electrostatic and vector potentials, respectively. The kinematic momentum and energy are invariant during the transition. This “recoilless” emission explains the gain symmetry relative to the resonant frequency. The difference in absorption and emission rates responsible for the gain is caused by the transverse gradients in the waveguide mode structure and the collective beam field. The per-pass gain formula in the low-gain (nonexponential) regime is obtained and found to agree with recent classical results