Quantum theory of an atom near partially reflecting walls

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Abstract
We consider first a dielectric medium of identical two-state atoms coupled by the radiation field to an initially excited atom outside the dielectric. From the Schrödinger equation follows a delay-differential equation describing how the atom interacts with the dielectric by virtual photon exchanges. In the macroscopic limit of a continuous distribution of atoms in the dielectric, we derive a simpler delay-differential equation in which a Fresnel reflection coefficient appears. We apply our results to a model of an atom in a multimode Fabry-Perot resonator, and obtain a general delay-differential equation for the probability amplitude of the initially excited state. This equation predicts well-known Rabi oscillations when the round-trip photon propagation time is negligible compared with the inverse of the Rabi frequency and the mirrors are highly reflective. For low mirror reflectivities we recover Purcell’s prediction that the emission rate is enhanced by the cavity Q factor. When the photon bounce time is large compared with the inverse Rabi frequency, Rabi oscillations do not occur. We discuss the Ewald-Oseen extinction theorem from the standpoint of quantum mechanics.