Classical Purcell factors and spontaneous emission decay rates in a linear gain medium

TL;DR

This paper develops a classical theory to accurately predict spontaneous emission rates in media with linear gain, correcting the traditional Purcell factor and aligning with quantum results, applicable to practical cavity structures.

Contribution

It introduces a classical correction to the Purcell factor for gain media, derived in multiple forms and linked to quantum quasinormal modes, enabling efficient analysis of complex photonic structures.

Findings

Classical correction to Purcell factor matches quantum predictions.

Explicit analytical formulas using quasinormal modes are provided.

Numerical results agree well with full dipole simulations.

Abstract

Recently the photonic golden rule, which predicts that the spontaneous emission rate of an atom depends on the projected local density of states (LDOS), was shown to fail in an optical medium with a linear gain amplifier. We present a classical light-matter theory to fix this widely used spontaneous emission rate, fully recovering the quantum mechanical rate reported in Franke et al., Phys. Rev. Lett. 127, 013602 (2021). The corrected classical Purcell factor, for media containing linear amplifiers, is obtained in two different forms, both of which can easily be calculated in any standard classical Maxwell solver. We also derive explicit analytical results in terms of quasinormal modes, which are useful for studying practical cavity structures in an efficient way, including the presence of local field effects for finite-size dipole emitters embedded inside lossy or gain materials (using a real cavity model). Finally, we derive a full classical correspondence from the viewpoint of quantized quasinormal modes in the bad cavity limit. Example numerical calculations are shown for coupled loss-gain microdisk resonators, showing excellent agreement between few mode expansions and full numerical dipole simulations.