The $E1$ & $M1$ Spontaneous Decay Rates for an Emitter Inside a Cavity Within a Medium

TL;DR

This paper derives formulas for the spontaneous decay rates of electric and magnetic dipole emitters inside a cavity within a medium, accounting for local field effects and medium properties.

Contribution

It provides a theoretical expression for decay rate ratios considering local field corrections in a medium with a cavity.

Findings

Decay rate ratio depends on medium's index of refraction and local field correction.

Derived explicit formulas for E1 and M1 transition decay rates in medium.

Highlights the role of medium permittivity and permeability in spontaneous emission rates.

Abstract

We discuss the $E1$ and $M1$ spontaneous decay rates of the an emitter residing inside of a real cavity carved out of a vast, uniform, homogenous, isotropic, linear, lossless, dispersionless, and continuous medium. The ratio of the medium rate to vacuum rate is given by $\Gamma_m/\Gamma_0 = [G(u)]^2 n^3 / u$, where $G(u) = 3u/(2u+1)$ is the local field correction factor, $n = \sqrt{\epsilon\mu/(\epsilon_0\mu_0)}$ is the index of refraction of the medium, $\epsilon(\epsilon_0)$ is the electric permitivity of the medium (vacuum), $\mu(\mu_0)$ is the magnetic permeability of the medium (vacuum), and $u = \epsilon/\epsilon_0$ for $E1$ transitions or $u = \mu_0/\mu$ for $M1$ transitions.