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Physical Review A

Change in propagation constant and propagation velocity in a dielectric: A microscopic approach

P. R. Berman

Published Oct 9, 2007• 2 citations
Publisher

Abstract

The problem of a source atom radiating into a medium of dielectric atoms is considered using a microscopic approach. The source atom is excited by a pulsed, classical driving field, and the average field amplitude and intensity radiated by the source atom is calculated to first order in the dielectric density. It is shown how both the change in propagation vector and change in propagation speed of the radiation in the medium can be traced to scattering events in the medium.

DOI
10.1103/physreva.76.042106

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The problem of a source atom radiating into a medium of dielectric atoms is considered using a microscopic approach. The source atom is excited by a pulsed, classical driving field, and the average field amplitude and intensity radiated by the source atom is calculated to first order in the dielectric density. It is shown how both the change in propagation vector and change in propagation speed of the radiation in the medium can be traced to scattering events in the medium.
The propagation of an off-resonant, single-photon pulse in and into a medium of two-level atoms is considered. When the pulse is launched from within the medium, there are two propagation speeds, neither of which is the normal group velocity. For densities as small as 1011{}^{11}11 atoms/cm3{}^{3}3 and detunings of order 1011{}^{11}11 s${}^{\ensuremath{-}1}$, both propagation speeds approach one half the speed of light in vacuum. Moreover, rather remarkably, there are Rabi oscillations between the field and atomic excitation, even for this single-photon pulse. In contrast, for a pulse sent into the medium, the atom-field system remains adiabatically in a dressed state that propagates with the normal group velocity. In the limit that the index of refraction is approximately equal to unity, we obtain the Fresnel equations for the reflection and transmission coefficients based on this microscopic model. The transmission coefficient differs from the conventional result owing to the fact that we quantize the field in free space.
We derive the macroscopic electromagnetic-field and medium operators for a linear dispersive medium with a microscopic model. As an alternative to the previous treatments in the literature, we show that the canonical momentum for the macroscopic field can be chosen to be −∊0Ê instead of −Dˆ with the standard minimal-coupling Hamiltonian. We find that, despite the change in the field operator normalization constants, the equal-time commutators among the macroscopic electric-field, magnetic-field, and medium operators have the same values as their microscopic counterparts under a coarse-grained approximation. This preservation of the equal-time commutator is important from a fundamental standpoint, such as the preservation of micro-causality for macroscopic quantities. The existence of more than one normal frequency mode at each k vector in a realistic causal-response medium is shown to be responsible for the commutator preservation. The process of macroscopic averaging is discussed in our derivation. The macroscopic field operators we derive are valid for a wide range of frequencies below, above, and around resonances. Our derivation covers the lossless, slightly lossy, and dispersionless as well as dispersive regimes of the medium. The local-field correction is also included in the formalism by inclusion of dipole–dipole interactions. Comparisons are made with other derivations of the macroscopic field operators. Using our theory, we discuss the questions of field propagation across a dielectric boundary and the decay rate of an atom embedded in a dielectric medium. We also discuss the question of squeezing in a linear dielectric medium and the extension of our theory to the case of a nonuniform medium.
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.