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arXiv:quant-ph/9902011

Comment on "Quantum Theory of Dispersive Electromagnetic Fields"

Gediminas Juzeliūnas

Published Feb 3, 1999• 0 citations
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Abstract

Recently Drummond and Hillery [Phys. Rev.A 59, 691(1999)] presented a quantum theory of dispersion based on the analysis of a coupled system of the electromagnetic field and atoms in the multipolar QED formulation. The theory has led to the explicit mode-expansions for various field-operators in a homogeneous medium characterized by an arbitrary number of resonant transitions with different frequencies. In this Comment, we drawn attention to a similar multipolar study by Juzeliunas [Phys. Rev.A 53, 3543 (1996); 55, 929 (1997)] on the field quantization in a discrete molecular (or atomic) medium. A comparative analysis of the two approaches is carried out, highlighting both common and distinctive features.

DOI
10.48550/arxiv.quant-ph/9902011

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Recently Drummond and Hillery [Phys. Rev.A 59, 691(1999)] presented a quantum theory of dispersion based on the analysis of a coupled system of the electromagnetic field and atoms in the multipolar QED formulation. The theory has led to the explicit mode-expansions for various field-operators in a homogeneous medium characterized by an arbitrary number of resonant transitions with different frequencies. In this Comment, we drawn attention to a similar multipolar study by Juzeliunas [Phys. Rev.A 53, 3543 (1996); 55, 929 (1997)] on the field quantization in a discrete molecular (or atomic) medium. A comparative analysis of the two approaches is carried out, highlighting both common and distinctive features.
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.
The microscopic operators have been investigated for radiation and polarization fields within a discrete molecular medium. The medium comprises atoms (or molecules) each containing an arbitrary number of energy levels. Explicit mode expansions have been derived for the quantized microfields in terms of normal Bose operators for polariton creation and annihilation. These microscopic operators have been demonstrated to yield the correct macroscopic and local field operators presented in part I [Phys. Rev. A 53, 3543 (1996)]. On the other hand, the commutation relations between the expanded microfields differ from the exact commutation relationships. This happens because the microfields have been described in terms of a continuous refractive index. Hence the expanded field operators do not extend to modes with extremely high frequencies ranging over the photon umklapp frequencies cG. In spite of that, the proper commutation relations hold between the resultant macroscopic (averaged) fields. This justifies separate quantization of the slowly modulated (macroscopic) part of the radiation field in linear dielectrics: such an approach is utilized in phenomenological quantization schemes. Finally, although the microscopic operators presented are not complete, the mode expansions adequately represent the quantized microfields associated with the optical modes in condensed molecular media.
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.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
The meaning of a recently proposed formalism for quantization of interacting fields is discussed by studying the consequences of the time-dependent unitary transformation which is essential for this approach. It turns out that non-relativistic quantum electrodynamics in dipole approximation may serve as a useful, although rather singular, example for this method. In the relativistic case a different point of view is suggested in order to avoid inconsistent interpretation. It is further possible to give arguments for a reasonable choice of the unitary transformation concerned