Fermi's golden rule for spontaneous emission in absorptive and amplifying media

TL;DR

This paper reveals that the traditional Fermi's golden rule for spontaneous emission fails in absorptive and amplifying media, providing a corrected quantum model that accurately predicts emission dynamics in complex nanophotonic environments.

Contribution

It introduces a rigorous quantization approach leading to a corrected Fermi's golden rule and master equation for systems in gain-loss media, addressing limitations of the local density of states approximation.

Findings

Standard formulas fail in gain-loss media

Corrected decay rates include quantum pumping terms

Numerical simulations show discrepancies with traditional models

Abstract

We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formula derived from Fermi's golden rule, in absorptive and amplifying media, where one assumes the SE rate scales with the local photon density of states, an approach often used in more complex, semiclassical nanophotonics simulations. Using a rigorous quantization of the macroscopic Maxwell equations in the presence of arbitrary linear media, we derive a corrected Fermi's golden rule and master equation for a quantum two-level system (TLS) that yields a quantum pumping term and a modified decay rate that is net positive. We show rigorous numerical results of the temporal dynamics of the TLS for an example of two coupled microdisk resonators, forming a gain-loss medium, and demonstrate the clear failure of the commonly adopted formulas based solely on the local density of states.