Generalization of Kirchhoff's law of thermal radiation: the inherent relations between quantum efficiency and emissivity.

Planck's law of thermal radiation depends on the temperature, , and the emissivity, , of a body, where emissivity is the coupling of heat to radiation that depends on both phonon-electron nonradiative interactions and electron-photon radiative interactions. Another body property is absorptivity, , which only depends on the electron-photon radiative interactions. At thermodynamic equilibrium, nonradiative interactions are balanced, resulting in Kirchhoff's law of thermal radiation that equals these two properties, i.e., =. For non-equilibrium, quantum efficiency () describes the probability of emitting an absorbed photon, which, like emissivity, depends on both radiative and nonradiative interactions. Past generalized Planck's equation extends Kirchhoff's law out of equilibrium by scaling the emission with the pump-dependent chemical potential, , obscuring the relations between the body properties. Here we theoretically and experimentally demonstrate a prime equation relating these properties in the form of =(1-), which agrees with a recent universal modal radiation law for all thermal emitters. At equilibrium, these relations are reduced to Kirchhoff's law. Our study focused on photoluminescence, and future extensions to electroluminescence and other quantum emission processes will further reveal the fundamental principles of radiation.