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Laser Physics

Recoil and Photon Momentum in a Dielectric

Peter W. Milonni, Robert W. Boyd

Published Jan 1, 2005• 25 citations
OpenAlex

Abstract

We consider the recoil momentum of a spontaneously emitting atom in a dispersive dielectric, first from a microscopic approach, in which the dielectric is treated as a collection of atoms and the field is quantized in vacuum , and, then, from the macroscopic approach of quantizing the field in the dielectric. The atom's recoil momentum differs from both the Abraham and Minkowski forms of the field momentum, and this difference is explained in terms of a dispersive contribution to the momentum acquired by the medium.

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We consider the recoil momentum of a spontaneously emitting atom in a dispersive dielectric, first from a microscopic approach, in which the dielectric is treated as a collection of atoms and the field is quantized in vacuum , and, then, from the macroscopic approach of quantizing the field in the dielectric. The atom's recoil momentum differs from both the Abraham and Minkowski forms of the field momentum, and this difference is explained in terms of a dispersive contribution to the momentum acquired by the medium.
An atom recoils when it undergoes spontaneous decay. In this paper we present a microscopic calculation of the recoil of a source atom imbedded in a dielectric medium. We find that the source atom recoils with the canonical photon momentum nℏk0,n\hbar k_{0,}nℏk0,​ where nnn is the index of refraction and ℏk0\hbar k_{0}ℏk0​ is the photon momentum calculated at the source atom atomic frequency ω0ω_{0}ω0​. We also show explicitly how the energy is conserved with the photon inside the medium.
An ad hoc quantization scheme for the electromagnetic field in a weakly dispersive, transparent dielectric leads to the definition of canonical and kinetic forms for the momentum of the electromagnetic field in a dispersive medium. The canonical momentum is uniquely defined as the operator that generates spatial translations in a uniform medium, but the quantization scheme suggests two possible choices for the kinetic momentum operator, corresponding to the Abraham or the Minkowski momentum in classical electrodynamics. Another implication of this procedure is that a wave packet containing a single dressed photon travels at the group velocity through the medium. The physical significance of the canonical momentum has already been established by considerations of energy and momentum conservation in the atomic recoil due to spontaneous emission, the Cerenkov effect, the Doppler effect, and phase matching in nonlinear optical processes. In addition, the data of the Jones and Leslie radiation pressure experiment is consistent with the assignment of one $\ensuremath{\hbar}\mathbf{k}$ unit of canonical momentum to each dressed photon. By contrast, experiments in which the dielectric is rigidly accelerated by unbalanced electromagnetic forces require the use of the Abraham momentum.
We consider the spontaneous emission in an absorbing medium by calculating the quantum flow of energy from an emitting atom. For this purpose, we extend the analysis by Power and Thirunamachandran of quantized Maxwell fields in the vicinity of an atom in the free space to a situation where the emitter is in a dielectric medium. We represent the quantum flow in terms of the non-equal time commutators involving electric and magnetic fields, as well as the local displacement field. The commutators are calculated applying a previously developed microscopic theory of quantization of radiation in dielectrics. Consequently, we derive from first principles the rate of spontaneous emission in an absorbing medium containing local field factors and other refractive contributions. The results support the previous introduction of the (virtual cavity) local field correction by Barnett et al on a phenomenological basis, as well as our previous approach in which the spontaneous emission is considered as a resonance energy transfer to the far-zone atoms. A distinctive feature of the present formalism is that it deals with the spontaneous emission directly by calculating the quantum flow from the emitting atom. Therefore, the method is free from divergences appearing in other approaches analysing the spontaneous decay in absorbing media.
We discuss the quantum fluctuations of the fields associated with a broad class of optical scattering and transmission problems by developing the quantum electrodynamics of an idealized linear, but nonuniform, dielectric medium. We present and compare two quantization schemes for this purpose. The first is based on the expansion of the field in terms of a set of single-frequency solutions of the Maxwell equations. The second involves expanding the field in the set of plane-wave solutions of the Maxwell equations in the vacuum. The relation between the two quantization schemes is discussed in the framework of the scattering theory that connects them. The methods presented are used to show that various field components within a dielectric medium may be either superfluctuant or subfluctuant relative to their fundamental uncertainties in the vacuum. These alterations of the fluctuation properties of the fields are shown to lead to changes in the spontaneous emission rates for both electric and magnetic dipole transitions of excited atoms within or near dielectric media. We also analyze the quantum properties of the transition radiation emitted by a fast charged particle in passing from one dielectric medium to another.
The electromagnetic field in a dispersive dielectric medium is quantized in a straightforward manner based on the classical expression for the field energy away from absorption resonances. Results for spontaneous and stimulated radiative rates, and absorption and gain coefficients, are obtained and compared with other expressions in the literature. Local field corrections, when they exist, are included. A level shift for an atom in a host dielectric medium is obtained and interpreted as the van der Waals interaction between that atom and its neighbours in the host medium.