Quantum Coherence and Giant Enhancement of Positron Channeling Radiation

Abstract

We present a quantum-mechanical treatment of positron channeling radiation in a planar harmonic potential that explicitly accounts for interference between transition amplitudes from different transverse energy levels. Because the planar channel potential for positrons in diamond~(110) is well approximated by a parabola, the transverse spectrum is equidistant, $\varepsilon_n = Ω(n+\tfrac{1}{2})$, and all $n \to n{-}j$ transitions radiate at the same Doppler-shifted frequency. The sudden-approximation entry of the positron into the crystal produces a \emph{Glauber coherent state}~\cite{Glauber1963} with Poisson-distributed level populations $|c_n|^2 = e^{-n_0}n_0^n/n!$ and mean occupation $n_0 \propto θ_{\rm in}^2$. Phase synchronization between the $c_n$ and the dipole matrix elements ensures constructive interference of all contributing amplitudes. Three exact scaling laws follow: (i)~$I_{\rm incoh}\propto n_0\proptoθ_{\rm in}^2$; (ii)~$I_{\rm coh}\propto n_0^2\proptoθ_{\rm in}^4$; (iii)~$\mathcal{G}\equiv I_{\rm coh}/I_{\rm incoh}\approx n_0 \proptoθ_{\rm in}^2$. Numerically, $\mathcal{G} = 12\text{--}31$ for positron energies of $4\text{--}14$~GeV in diamond~(110) at $θ_{\rm in}=31\;μ$rad, in agreement with the experimental first-harmonic peak positions of Avakyan \emph{et al.}~\cite{Avakyan1982} to within 15\%. The transition from $N$- to $N^2$-scaling of radiated intensity, driven by quantum coherence, opens a route toward high-intensity monochromatic gamma-ray sources.