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Physical Review A

Quantization of evanescent electromagnetic waves based on detector modes

Tetsuya Inoue, Hirokazu Hori

Published May 9, 2001• 31 citations
Publisher

Abstract

The problem of field quantization in a half space is discussed based on the detector modes, which provide a useful basis for the study of atomic or molecular interaction with radiation modes involving evanescent electromagnetic waves. The detector modes, each of which involves a single outgoing wave, are introduced in terms of the time-reversal and spatial-rotation transforms of the widely used triplet modes given by Carniglia and Mandel. The derivation of the orthogonal relation for the detector modes is therefore straightforward. We represent the creation and annihilation operators for the detector modes in terms of those for the triplet modes, and obtain the field operator, Hamiltonian, number operator, and pseudomomentum operator. Based on this formalism, we evaluate the differential radiation probability for atomic dipole radiation near a dielectric surface. The evaluation of the final-state mode density is straightforward for the detector modes involving single outgoing waves. With respect to the image dipole picture, the quantum-to-classical correspondence is clear in the detector-mode formalism.

DOI
10.1103/physreva.63.063805

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The problem of field quantization in a half space is discussed based on the detector modes, which provide a useful basis for the study of atomic or molecular interaction with radiation modes involving evanescent electromagnetic waves. The detector modes, each of which involves a single outgoing wave, are introduced in terms of the time-reversal and spatial-rotation transforms of the widely used triplet modes given by Carniglia and Mandel. The derivation of the orthogonal relation for the detector modes is therefore straightforward. We represent the creation and annihilation operators for the detector modes in terms of those for the triplet modes, and obtain the field operator, Hamiltonian, number operator, and pseudomomentum operator. Based on this formalism, we evaluate the differential radiation probability for atomic dipole radiation near a dielectric surface. The evaluation of the final-state mode density is straightforward for the detector modes involving single outgoing waves. With respect to the image dipole picture, the quantum-to-classical correspondence is clear in the detector-mode formalism.
The problem of the quantization of evanescent waves, which appear in the angular spectrum representation of the electromagnetic field in a half-space, is discussed. Although evanescent waves are associated with material sources, scatterers, etc., we are able to treat the electromagnetic field, including the evanescent waves, effectively as a free field, by making use of the idea of the refractive index of a passive, macroscopically continuous medium. We consider a space which is filled with a homogeneous dielectric to the left of the plane z=0z=0z=0, and is empty to the right of the plane. Triplets of incident, reflected, and transmitted waves at the interface form the fundamental orthogonal modes of the space. By expanding the field in terms of these triplet modes, we show that the field Hamiltonian reduces to the sum of independent harmonic-oscillator Hamiltonians. The quantization is therefore straightforward. We introduce the creation and annihilation operators for the triplet wave modes, and encounter Fock states, coherent states, etc., for a field having evanescent wave components. The field commutator at two space-time points in the right half-space is shown to have an explicit contribution from evanescent waves, characterized by an exponential decay to the right and a propagation parallel to the interface. We also examine the problem of atomic excitation by quantized evanescent waves, and show that the results are of the form given by semiclassical treatments.
The problem of the momentum of the electromagnetic field near a homogeneous dielectric plane surface is studied. In the particular case of evanescent waves, our calculations are consistent with the experimental result of S. Huard and C. Imbert which points out that, during an interaction between a moving atom and a surface wave of pulsation $\ensuremath{\omega}$, the exchanged momentum is greater than $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{c}$. This work leads at first to a straightforward procedure for quantizing the momentum parallel to the diopter of the field. The authors proceed to investigate the form of the component perpendicular to the interface which is transmitted by the field to an atom during an interaction. (In the case of an evanescent wave, this problem involves the interpretation of the imaginary part of the wave vector.) A general theory shows that the presence of a medium widens and shifts the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ levels of the field. This result agrees with the uncertainty principle.
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.
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 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.
ABSTRACT A theoretical framework is introduced for deriving expressions for electromagnetic (EM) potentials and fields of single atomic or molecular emitters modeled as oscillating dipoles, which follows a recently developed method to solve inhomogeneous wave equations for time‐dependent distributions of charge. This framework is first used to evaluate the physical implications of the simplifying assumptions made in the standard approach to the quantization of EM fields and the impact of such assumptions on the predicted energy and momentum quantization. Then, the exact relationships between the potentials and the oscillating (transition) dipole properties derived under the present framework are used to quantize EM fields radiated by single emitters and to achieve agreement with the classical dipole radiation pattern, while maintaining the quantum mechanical description of electromagnetic radiation in terms of the probability distribution of quantum modes. Contributions of the present analysis to the understanding of photon emission from excited atoms or molecules stimulated by light or vacuum field fluctuations are highlighted, and possible experimental tests and practical applications are proposed.