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.

A radiative lifetime depends on the available electromagnetic modes as well as on the atomic transition. In the ideal parallel-perfect conductor geometry of polarization-selective spontaneous emission between conducting plates, an in-plane transition dipole has no coupled propagating electric dipole radiation mode below the relevant waveguide cutoff frequency.

If that idealized electric dipole transition is the only decay channel, its rate is zero and the corresponding lifetime diverges. This is a statement about a restricted channel and geometry, not a general prediction of an immortal excited atom. Finite plates, absorbing surfaces, dipole misalignment and other allowed transitions can restore decay. A normal transition dipole can couple to the surviving transverse electromagnetic mode.

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 $d$.