Spontaneous emission from a quantum dot in a structured photonic reservoir: phonon-mediated breakdown of Fermi's golden rule

We describe how a structured photonic medium controls the spontaneous emission rate from an excited quantum dot in the presence of electron-phonon coupling. We analyze this problem using a polaron transformed master equation and we consider specific examples of a photonic crystal cavity and a coupled cavity waveguide. We find that when the relaxation times of the photon and phonon baths are comparable, phonons influence spontaneous emission in a non-trivial way. We demonstrate why and how the broadband frequency dependence of the local photon density of states determines the photon emission rate, manifesting in a complete breakdown of Fermi's golden rule. For a single cavity resonance, we generalize Purcell's formula to include the effects of electron-phonon coupling. For a waveguide, we show a suppression and a 200-fold enhancement of the photon emission rate.