Generalization of Kirchhoff's Law: The inherent relations between quantum efficiency and emissivity

Planck's law of thermal radiation depends only on the temperature T and emissivity $\varepsilon$. It is one of the most fundamental discoveries about light-matter interaction that led to the development of quantum physics. Another basic property of a body is its ability to absorb incoming light, characterized by absorptivity $\alpha$. Kirchhoff's law of thermal radiation equals these two properties at thermodynamic equilibrium, i.e., $\varepsilon$=$\alpha$. The generalized Planck's equation extends Kirchhof's law out of equilibrium by scaling the absorptivity with the pump-dependent chemical potential $\mu$, obscuring emissivity as a material property. Quantum efficiency (QE) is a material property, defined out of equilibrium, describing the statistics of absorption followed by emission of a photon. Both emissivity and QE depend on the interplay between radiative and non-radiative rates. Here we theoretically and experimentally demonstrate a prime equation for emissivity as a material property in and out of equilibrium in the form of $\varepsilon$=$\alpha$(1-QE), which at equilibrium is reduced to Kirchhoff's law. Our work lays out the fundamental evolution of non-thermal emission with temperature, which is critical for the development of lighting and energy devices.