A semiclassical optics derivation of Einstein's rate equations

TL;DR

This paper derives Einstein's rate equations using semiclassical optics, establishing their validity limits based on light spectral properties, and confirms results through numerical simulations of stochastic differential equations.

Contribution

It provides a semiclassical derivation of Einstein's rate equations, clarifies their validity conditions, and introduces a phase-diffusion model for verification via numerical simulations.

Findings

ERE validity depends on light spectrum height and width.

Effective Bloch equations incorporate spectral width effects.

Numerical simulations confirm theoretical hypotheses.

Abstract

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 for their validity that depends on the light spectral properties (namely on the height and width of its spectrum). 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 owed to the short correlation time of the light field. The latter is then relaxed, leading to 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, what allows its adiabatic elimination for large enough spectral width. Finally the use of a phase-diffusion model of light allows us to check all the results and hypotheses through numerical simulations of the corresponding stochastic differential equations.