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Planck's law of thermal radiation depends on the temperature, $T$, and theemissivity, $\epsilon$, of a body, where emissivity is the coupling of heat toradiation that depends on both phonon-electron nonradiative interactions andelectron-photon radiative interactions. Another property of a body isabsorptivity, $\alpha$, which only depends on the electron-photon radiativeinteractions. At thermodynamic equilibrium, nonradiative interactions arebalanced, resulting in Kirchhoff's law of thermal radiation that equals thesetwo properties, i.e., $\epsilon = \alpha$. For non-equilibrium, quantumefficiency ($QE$) describes the statistics of photon emission, which likeemissivity depends on both radiative and nonradiative interactions. Pastgeneralized Planck's equation extends Kirchhoff's law out of equilibrium byscaling the emissivity with the pump-dependent chemical-potential $\mu$,obscuring the relations between the body properties. Here we theoretically andexperimentally demonstrate a prime equation relating these properties in theform of $\epsilon = \alpha(1-QE)$, which is in agreement with a recentuniversal modal radiation law for all thermal emitters. At equilibrium, theserelations are reduced to Kirchhoff's law. Our work lays out the fundamentalevolution of non-thermal emission with temperature, which is critical for thedevelopment of lighting and energy devices.

Generalization of Kirchhoff's Law of Thermal Radiation: The Inherent Relations Between Quantum Efficiency and Emissivity