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Physical Review A

Quantum theory of an atom near partially reflecting walls

R. J. Cook, Peter W. Milonni

Published Jun 1, 1987• 109 citations
Publisher

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 Schr\"odinger 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.

DOI
10.1103/physreva.35.5081

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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 Schr"odinger 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.
A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).
We revisit the problem of spontaneous emission by a two-level system in a cavity, concentrating in particular on those aspects associated with the finite propagation time of light between the atom and the cavity boundaries. We find that spontaneous emission always occurs first at the free space rate and there is then an abrupt change in these dynamics when light reflected by the cavity mirrors returns to the atom, leading to interference effects that result either in an enhancement or a reduction of this rate or in a periodic exchange of excitation between the atom and the cavity field. From this point of view, reversible spontaneous emission in terms of single-mode ``vacuum Rabi oscillations'' can be interpreted as resulting from the interference between the multimode partial waves reflected at the cavity mirrors and the instantaneously radiated light. \textcopyright{} 1996 The American Physical Society.
We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field and a single two-level system in front of the dielectric. The dielectric is modelled as a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf–Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenologicalassumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.
A nonperturbative fully quantum-theoretical analysis describing the transient spontaneous emission of an initially excited two-level atom in a one-dimensional cavity with output coupling is presented. A delay-differential equation for the probability amplitude of the upper atomic state is strictly derived from the Schrodinger equation. With this equation, both the time evolution of the atomic state and the emission spectrum are calculated. Furthermore, the mode function makes the calculations of emitted fields both inside and outside the cavity possible. The calculated result of the vacuum Rabi oscillation in an underdamped cavity and the result of enhanced spontaneous emission rate in an overdamped cavity are presented. A retardation-governed revival of the upper atomic state is predicted. The calculated results also show the inhibition of spontaneous emission by detuning or by setting the atom at a node.>
Starting from the quantized version of Maxwell's equations for the electromagnetic field in an arbitrary linear Kramers-Kronig dielectric, the Heisenberg equations of motion for a two-level atom resonantly coupled to the radiation field in the presence of dispersive and absorbing dielectric bodies are derived. The theory is applied to the problem of spontaneous decay of excited atoms in absorbing media. The decay rate is calculated for the (Glauber-Lewenstein) real-cavity model, and a comparison with the recently studied (Clausius-Mosotti) virtual-cavity model [S. Scheel, L. Kn\"oll, D.-G. Welsch, and S. M. Barnett, Phys. Rev. A 60, 1590 (1999)] is given. It is shown that owing to nonradiative decay associated with absorption, the rate of spontaneous decay sensitively depends on the cavity radius, particularly when the atomic transition frequency approaches an absorption band of the medium. Only when the effect of absorption is fully disregarded, is the familiar local-field correction factor recovered.
We study the transient regime of spontaneous emission by an atom excited in a planar high-Q microcavity. The variation of the spatial distribution of the radiation outside the cavity is determined as a function of the increasing number of internal reflections for transition dipole moments parallel and perpendicular to the mirrors and for mirror separations equal to $\ensuremath{\lambda}/2$ and $\ensuremath{\lambda}/4.$ Particular attention is given to the parallel dipole in the $\ensuremath{\lambda}/2$ cavity, where the number of reflections is closely proportional to the elapsed time. It is shown that the transient regime extends over times of the order of the cavity decay time, in contrast to the effectively one-dimensional cavities previously studied, where the steady state is achieved after the cavity round-trip time. The field distribution inside the cavity tends to a steady-state form whose dimension parallel to the mirrors is of the order of the transverse coherence length. Despite this transverse localization of the field excitation, it is shown that the conditions for the achievement of strong atom-field coupling and the observation of Rabi oscillations cannot be met in the planar microcavity.