Relationships between the absorption and emission of light in multilevel systems

We develop a simple and general framework for understanding the relationships between absorption and emission in multilevel systems (molecules, semiconductors, condensed matter, etc.) in and out of equilibrium. Conditions under which the absorption and emission spectra display mirror image symmetry are determined. The relationship between the absorption and emission cross sections at frequency and temperature T, previously derived by Neporent and McCumber for thermally equilibrated systems, is generalized and a new derivation is presented which extends its range of application to systems out of equilibrium. This derivation requires only microscopic reversibility and that the optically coupled manifold of levels belonging to both initial and final electronic states are each in internal thermal equilibrium at the same temperature T. The initial and final (e.g., ground and excited electronic) states need not be in equilibrium. If more than one excited-electronic-state manifold participates in the dynamics and these excited-electronic-state manifolds are not themselves in equilibrium, a somewhat more complicated relationship between absorption and emission can be obtained, provided that the internal degrees of freedom within the electronic-state manifolds are each in equilibrium at the same temperature. Effects of inhomogeneous broadening are examined. Good agreement is found between the Neporent-McCumber relationship and experimental absorption and emission spectra for the laser dyes rhodamine-6G and 2,5-biphenyloxazol PPO, and the solid-state laser crystal alexandrite. A new relationship between absorption and emission in direct-band-gap semiconductor materials is derived. This relationship allows us to calculate the emission from a direct-band-gap semiconductor provided the dependence of the band-gap energy on excitation level is not large. These relationships hold even when pumping conditions differ for emission and absorption experiments; the dependence of emission on excitation level can thereby be calculated. This permits the gain spectra of semiconductor diode lasers to be determined from their absorption spectra. In the limit that both the absorption and emission are measured for a thermally equilibrated nondegenerate direct-band-gap semiconductor system with identical pumping conditions for the absorption and emission experiments, our new relationship reduces to previous results.