Electromagnetic Field Interaction with Fast-Moving Dipoles

Abstract

The two conventional Hamiltonians of quantum electrodynamics, namely, the minimal-coupling (P⃗·A⃗) and the multipolar (µ⃗·E⃗) Hamiltonians are known to yield the same rates of resonance transitions (absorption or emission) for stationary atoms. These two Hamiltonians, related by the Power-Zienau canonical transformation, are shown to produce identical rates for moving dipoles, in the nonrelativistic limit (v/c≪l) only if the magnetic moment associated with the moving dipole is included in the multipolar interaction. The effect of the motion is then, regardless of the chosen Hamiltonian, a reduction of the dipole-field coupling constant by a factor of 1-v⃗ q̂/c, where q̂ is the unit vector in the direction of the emitted or absorbed radiation. In the case of dipoles associated with relativistic spin −1/2 particles (e.g. electrons or positrons channeled in crystals), the coupling constant obtained from the Dirac Hamiltonian differs from that of the nonrelativistic limit by a factor of 2γ/(γ+1), where γ=(1-v²/c²)^−1/2. The wave equation and generalized N-particle Bloch equations are written in terms of electric-field and polarization operators of the active modes, with the motionally-reduced field-dipole coupling constants, thus providing the framework for the analysis of spontaneous emission, superfluorescence and lasing from ensembles of fast-moving dipoles. The first-order perturbative result for the γ-dependence of gain obtainable by single-mode field stimulation of channeling radiation from relativistic electrons is retrieved from this analysis in the small — T2, steady-state, small-signal regime. Substantial modifications of this γ-dependence are predicted in other regimes.