Derivation of the stationary fluorescence spectrum formula for molecular systems from the perspective of quantum electrodynamics

Numerical simulations of stationary fluorescence spectra of molecular systems usually rely on the relation between the photon emission rate and the system’s dipole–dipole correlation function. However, research papers usually take this relation for granted, and standard textbook expositions of the theory of fluorescence spectra also tend to leave out this important relation. In order to help researchers with less theoretical training gain a deeper understanding of the emission process, we perform a step-by-step derivation of the expression for the fluorescence spectrum, focusing on rigorous mathematical treatment and the underlying physical content. Right from the start, we employ quantum description of the electromagnetic field, which provides a clear picture of emission that goes beyond the phenomenological treatment in terms of the Einstein A coefficient. Having obtained the final expression, we discuss the relation of the latter to the present level of theory by studying a simple two-level system. From the technical perspective, the present work also aims at familiarizing the reader with the density matrix formalism and with the application of the double-sided Feynman diagrams.