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Physical review. B, Condensed matter

Phenomenological quantum electrodynamics in periodic dielectric media

Ariel Caticha, Nestor Caticha

Published Jul 1, 1992• 5 citations
Publisher

Abstract

The quantization of the x-ray field in a crystalline medium phenomenologically described by a periodic dielectric susceptibility \ensuremath{\chi}(r) is carried out exactly, that is, the usual approximation of the dynamical theory of x-ray diffraction of neglecting terms that are of second and higher orders in \ensuremath{\chi} is not employed. This can be achieved after exact orthogonality relations for the classical electromagnetic Bloch waves are obtained. This formalism provides a simple framework for the study of a number of QED processes involving x-ray photons undergoing diffraction in a crystal. As a brief application, we show that the interaction of the quantized radiation field with an arbitrary classical current, i.e., the radiation of coherent states of Bloch photons can be studied along lines similar to the analogous situation in vacuum. The emission of such x-ray Bloch photons is governed by Poisson distributions.

DOI
10.1103/physrevb.46.479

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The quantization of the x-ray field in a crystalline medium phenomenologically described by a periodic dielectric susceptibility \ensuremath{\chi}(r) is carried out exactly, that is, the usual approximation of the dynamical theory of x-ray diffraction of neglecting terms that are of second and higher orders in \ensuremath{\chi} is not employed. This can be achieved after exact orthogonality relations for the classical electromagnetic Bloch waves are obtained. This formalism provides a simple framework for the study of a number of QED processes involving x-ray photons undergoing diffraction in a crystal. As a brief application, we show that the interaction of the quantized radiation field with an arbitrary classical current, i.e., the radiation of coherent states of Bloch photons can be studied along lines similar to the analogous situation in vacuum. The emission of such x-ray Bloch photons is governed by Poisson distributions.
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.
Introduction The propagation of light in periodically ordered photonic crystals bears a strong analogy to the wave propagation of conduction electrons in atomic crystals. The development of Bloch modes and dispersion are determined by the interference of waves that are diffracted by lattice planes [19, 52]. The periodicity commensurate with the wavelength (a ≈ λ/2) gives rise to Bragg diffraction that is associated with frequency windows for which waves are forbidden to propagate in a particular direction. Such stop gaps have long been known to arise for electromagnetic waves, notably for X-rays in atomic crystals [214]. At optical frequencies, the stop gaps that occur in one-dimensional periodic structures, known as Bragg stacks, are widely used in optical laboratories as broadband highly reflecting dielectric mirrors [556]. The distinguishing feature of three-dimensional (3D) photonic crystals is that a common stop gap can be achieved for all directions and for all polarizations simultaneously: the widely pursued 3D photonic band gap . Historically a 3D band gap has never been considered for the propagation of X-rays in periodic media, since a gap requires a high refractive index contrast of order unity, whereas the refractive index varies by less then 10 −4 in the X-ray range. At frequencies inside the 3D photonic band gap, the density of optical states (DOS) completely vanishes. As the density of states can also be interpreted as the density of vacuum fluctuations [334], a 3D band gap thus serves as an effective shield to these fluctuations. The total absence of optical modes in a photonic band gap has implications beyond classical optics that break the analogy between the behavior of light in photonic crystals and the behavior of electrons in atomic crystals. 3D photonic band gap crystals play an important role in cavity quantum electrodynamics (cQED) [64, 549], where they offer at least five prospects for new physics. Firstly, probably the most eagerly pursued phenomenon is the complete inhibition of spontaneous emission: an excited quantum emitter – such as an atom, molecule, or quantum dot – embedded in a crystal with its transition frequency tuned to within the 3D band gap remains forever excited since it cannot decay to the ground state by emitting a photon.
After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.
An introduction to the quantum theory of nonlinear optics This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.
In this work the electromagnetic field in a dielectric medium is quantized in a simple and straightforward way based on the classical expression for the energy density for the case of weak absorption. The quantized field leads to expressions for spontaneous and stimulated emission rates, and gain and absorption coefficients, for the general case of a dispersive dielectric host medium. Local field corrections, when they apply, arise in the same manner as in classical electromagnetic theory. The theory also leads to a radiative level shift dependent on the dispersion of the dielectric, and this shift may be interpreted as a van der Waals interaction between the guest atom and the atoms of the host medium.>
A theoretical model that takes into full account the vectorial nature of electromagnetic (EM) fields is developed to investigate various types of quantum electrodynamics (QED) in a three-dimensional (3D) photonic crystal. The EM fields are quantized via solving the eigenproblem of photonic crystals with the use of a plane-wave expansion method. It is found that the light-atom coupling coefficients strongly depend on the Bloch states, and the key physical function concerning the atomic QED is the photon local density of states (LDOS) instead of the DOS. Both the DOS and the LDOS vary slowly near the band edge and no singularity takes place. This vectorial model show that the spontaneous emission from a two-level atom can be solved via the conventional Weisskopf-Wigner approximation theory, which exhibits a pure exponential decay behavior with a rate proportional to the LDOS. The quantum interference effect from a three-level atom is greatly weakened at the band edge, instead of enhanced. Neglecting this vectorial nature will lead to many discrepancies in understanding quantum optics in 3D photonic crystals and other inhomogeneous media.