Generalized Einstein Relations Derived for Absorption and Emission Spectra in Dispersive Media

The paper, posted to arXiv's Chemical Physics section, derives quantum mechanical expressions for Einstein-coefficient spectra under the electric-dipole approximation, incorporating an intramolecular Boltzmann distribution and quantized field operators appropriate for isotropic, dispersive media as formulated by Nienhuis and Alkemade. A central contribution is the rigorous definition of dipole-strength spectra in terms of conditional transition probabilities per unit time, which are then shown to be directly related to Einstein-coefficient spectra. For transitions between two broadened bands, the dipole-strength spectra depend on a single total dipole strength, with Einstein's classical degeneracy ratio and transition frequency replaced by a change in standard chemical potential and a single underlying lineshape that manifests differently across the four spectra. At equilibrium, the generalized relations specify the Stokes shift between forward and reverse transitions. Importantly, the relationships between dipole-strength spectra, spontaneous emission spectral densities, and stimulated transition cross sections depend on the refractive index, dielectric constant, and local field, but not on the derivative of the refractive index—a notable simplification. The broadband results are shown to reduce correctly to known expressions for narrow spectra inside materials and for line spectra in vacuum, providing consistency checks. The manuscript is a revised version correcting references to local field treatments and elaborating on the connection to the Strickler-Berg relation.