A semiclassical optics derivation of Einstein’s rate equations

We provide a semiclassical optics derivation of Einstein’s rate equations (ERE) for a two-level system illuminated by a broadband light field, setting a limit on their validity that depends on the spectral properties of the light field. Starting from the optical Bloch equations for individual atoms, the ensemble averaged atomic inversion is shown to follow ERE under two concurrent hypotheses: (i) the decorrelation of the inversion at a given time from the field at later times and (ii) a Markov approximation owing to the short correlation time of the light field. When the latter hypothesis is relaxed, we find effective Bloch equations for the ensemble average in which the atomic polarization decay rate is increased by an amount equal to the width of the light spectrum, which allows its adiabatic elimination for a large enough spectral width. The use of a phase-diffusion model of light allows us to check our results and hypotheses using numerical simulations of the corresponding stochastic differential equations. We take into account both light bandwidth and atomic linewidth, which allows us to discuss the differences existing between rate equations in the limits of large light bandwidth or large atomic linewidth. Only in the former case does one obtain ERE with the correct expression for Einstein’s B coefficient.