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Physical Review A

Hamiltonian formalism for charged-particle systems interacting with absorptive dielectrics

A. Tip

Published Jan 12, 2004• 5 citations
Publisher

Abstract

We use the auxiliary field method to obtain a quantized Hamilton formalism for charged particles interacting with absorptive dielectrics. We give various equivalent representations. For absorptive dielectrics extending over all space, such as absorptive photonic crystals, our formalism is equivalent to that of Dung et al. [Phys. Rev. A 65, 043813 (2002)]. We also show that, as for nonabsorptive photonic crystals, to leading order the excited states of embedded two-level atoms do not decay radiatively if their transition frequencies are in a band gap.

DOI
10.1103/physreva.69.013804

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We use the auxiliary field method to obtain a quantized Hamilton formalism for charged particles interacting with absorptive dielectrics. We give various equivalent representations. For absorptive dielectrics extending over all space, such as absorptive photonic crystals, our formalism is equivalent to that of Dung et al. [Phys. Rev. A 65, 043813 (2002)]. We also show that, as for nonabsorptive photonic crystals, to leading order the excited states of embedded two-level atoms do not decay radiatively if their transition frequencies are in a band gap.
The properties of atoms close to an absorptive dielectric are studied using the phenomenological Maxwell's equations. Although radiative decay has been considered by many authors, the coupling of atoms with longitudinal modes does not seem to have been treated in detail. Here we show that there are two main effects. The first is a change in the atomic interaction potential from the Coulomb one to a static potential, i.e., one that satisfies a Poisson equation featuring the static dielectric function ${\ensuremath{\epsilon}}_{\mathrm{stat}}=\ensuremath{\epsilon}(\ensuremath{\omega}){\ensuremath{\mid}}_{\ensuremath{\omega}=0}$. The second is the decay of excited atomic states through longitudinal field interactions. We find that the corresponding decay constant is nonzero only for atom-dielectric distances in the order of an atomic diameter and that it decreases exponentially fast on an atomic scale with increasing distance. We also show that the Hamiltonian used by the Jena group [Dung et al., Phys. Rev. A 65, 043813 (2002)], featuring the Coulomb potential, is unitarily equivalent to one containing the static potential.
Recently two methods have been developed for the quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics. The first is based upon the introduction of a quantum Langevin current in Maxwell's equations [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); Ho Trung Dung, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998); S. Scheel, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)], whereas the second makes use of a set of auxiliary fields, followed by a canonical quantization procedure [A. Tip, Phys. Rev. A 57, 4818 (1998)]. We show that both approaches are equivalent.
We present a formalism for the description of fluorescence from optically active materials embedded in a photonic crystal structure possessing a photonic band gap or pseudogap. An electromagnetic field expansion in terms of Bloch modes of the crystal is used to develop the equations for fluorescence in terms of the local density of photon modes available to the emitting atoms in either the high or low dielectric regions of the crystal. We then obtain expressions for fluorescence spectra and emission dynamics for luminescent materials in photonic crystals. The validity of our formalism is demonstrated through the calculation of relevant quantities for model photon densities of states. The connection of our calculations to the description of realistic systems is discussed. We also describe the consequences of these analyses on the accurate description of the interaction between radiative systems and the electromagnetic reservoir within photonic crystals.
For the description of spectral and radiative decay properties of atoms or molecules, placed in a photonic material, the electromagnetic field in the material must be quantized and active research is taking place in this area. Here a unified account is given of such quantization procedures. Led by the Maxwell example, we consider the canonical formalism and its quantization for a class of linear evolution equations ${\ensuremath{\partial}}_{t}F=NF\ensuremath{-}G$, obeying a conservation law for G=0G=0G=0. If NNN has a nonempty null space (zero is an eigenvalue with associated nonpropagating solutions), an abstract form of the gauge concept makes its appearance and generalizations of the familiar Coulomb and Lorentz gauges are obtained. A canonical formalism is set up and quantized. The application to spatially inhomogeneous nonconducting electrodynamical systems is immediate, including the interaction with matter. Next atomic decay in a medium is considered, in particular in the presence of band gaps. For a simple two-level model with transition frequency in the gap, single-photon decay is inhibited and also a different stable eigenvalue of the combined system is found. An open problem in connection with random dielectrics, showing Anderson localization, is discussed. Finally, a mass renormalization, by means of a Kramers transformation, is presented. In general, the renormalized mass is no longer a scalar quantity.
A model of the propagation of a quantized electromagnetic field in a one-dimensional photonic crystal that contains two-level atoms that interact with the field is proposed. The model separately considers the interaction of the quantized radiation with the quantum system and with the photonic crystal. The use of the model is exemplified by its application to the study of the decay of excited states of one- and two-level atoms placed in the photonic crystal. The possibility of the transformation of the front of a single photon by a finite dielectric is demonstrated.
We derive an effective Hamiltonian for the nonlinear process of parametric downconversion in the presence of absorption. Based upon the Green-function method for quantizing the electromagnetic field, we first set up Heisenberg's equations of motion for a single atom driven by an external electric field and in the presence of an absorbing dielectric material. The equations of motion are then solved to second order in perturbation theory which, in rotating-wave approximation, yields the standard effective interaction Hamiltonian known from free-space nonlinear optics. In a second step, we derive the local-field-corrected Hamiltonian for an atom embedded in a dielectric host medium, that is, a nonlinear crystal. Here we show that the resulting effective Hamiltonian is found to be trilinear in the electric and noise-polarization fields and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption, and hence the noise-polarization field, vanishes.