Polarization-selective spontaneous emission between conducting plates

Wide perfectly conducting plates with $d\ll c/{\omega }_0$ admit only a normal-electric-field transverse electromagnetic mode at the transition frequency. The electric-dipole interaction therefore couples to the normal component of the atomic transition dipole. A normal dipole has enhanced available mode density; an in-plane dipole has no allowed electric-dipole radiation channel below the higher-mode cutoffs. The conclusion assumes negligible changes to the atom and negligible absorption, leakage and symmetry-breaking mixing.

For a normal transition dipole of magnitude $D$, the Fermi's golden rule, one-polarization mode density $\omega /(2\pi c^{2}d)$ and single-photon squared field $\hslash \omega /(2{ϵ}_{0}V)$ give $A_{\perp }=D^2{\omega }_0^2/(2{ϵ}_0\hslash c^{2}d)$. Dividing by the free-space electric-dipole spontaneous emission rate gives the title. The factor three relative to the scalar density-of-states ratio comes from the isotropic free-space polarization average. This ideal weak coupling formula assumes wide plates and unperturbed atomic structure, with atomic size much smaller than $d$.