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Physical Review A

Quantum electrodynamics with nonrelativistic sources. IV. Poynting vector, energy densities, and other quadratic operators of the electromagnetic field

Edwin Albert Power, T. Thirunamachandran

Published Jan 1, 1992• 50 citations
Publisher

Abstract

The quantized electromagnetic field in interaction with nonrelativistic sources, i.e., atoms and molcules, has been the subject of study in the previous papers in this series. The Heisenberg formalism is used to calculate the electric- and magnetic-field operators arising from charges and currents expressed in multipolar form. In this paper, quadratic operators in these fields such as the Thompson energy density and the Poynting vector and their expectation values for specified states are found. The expectation values are, in general, time dependent. For initial conditions given at t=0 it is shown that the fields are causal, i.e., for tr/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting's theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules. Similarly the magnetic-energy density is derived and is used in the calculation of energy shifts. Quadratic operators referring to different field points are also discussed.

DOI
10.1103/physreva.45.54

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The quantized electromagnetic field in interaction with nonrelativistic sources, i.e., atoms and molcules, has been the subject of study in the previous papers in this series. The Heisenberg formalism is used to calculate the electric- and magnetic-field operators arising from charges and currents expressed in multipolar form. In this paper, quadratic operators in these fields such as the Thompson energy density and the Poynting vector and their expectation values for specified states are found. The expectation values are, in general, time dependent. For initial conditions given at t=0 it is shown that the fields are causal, i.e., for tr/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting's theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules. Similarly the magnetic-energy density is derived and is used in the calculation of energy shifts. Quadratic operators referring to different field points are also discussed.
The electromagnetic field operators and the electron field operators for the coupled system governed by the multipolar Hamiltonian are obtained within the Heisenberg picture. Their causal behavior and their relationship to the minimal-coupling forms are discussed. The basic fields of the multipolar theory, namely, the displacement vector and the magnetic field, are calculated up to terms quadratic in the multipole moment sources. The terms linear in the transition moments are the quantum counterparts of the classical fields and do not change the photon occupation number. The quadratic terms have no classical analogs: they act in both the photon and electron occupation-number spaces. It is shown that it is necessary to include these second-order terms in the calculation of the Poynting vector for an emitting dipole, thus demonstrating their role in the transport of radiative energy.
The electromagnetic field operators for a chiral (optically active) molecule are calculated in the Heisenberg picture, using the multipolar Hamiltonian. Interactions involving electric-dipole, magnetic-dipole, and electric-quadrupole moments are taken into account. The operators are expanded in powers of moments, and expressions correct to quadratic powers in the moments are given. Comparable contributions from diamagnetic interactions are also presented. The properties of these operators are discussed. Terms that are linear in the moments are found to act in the electron space alone; those quadratic in the moments act in both the electron and photon spaces. The operators are used to calculate the Poynting vector, from which the spontaneous emission rates for magnetic-dipole and electric-quadrupole transitions are obtained. Attention is drawn to the inverse square dependence of the Poynting vector on the distance of the field point from the source. This behavior is shown to be consistent with the requirement of local conservation of energy. The operators are used in the following paper [J. K. Jenkins, A. Salam, and T. Thirunamachandran, Phys. Rev. A 50, 4767 (1994)] to calculate dispersion energies between two optically active molecules.
The electromagnetic field operators in the Heisenberg picture are found in both the minimal and multipolar forms of nonrelativistic quantum electrodynamics within the electric dipole approximation. These are made up of free- and source-field terms. The free electric- and magnetic-field operators in the two formalisms differ since the creation/annihilation operators are manifestly distinct. Their explicit forms in the Heisenberg picture are given. However, the source fields are formally the same when expressed in terms of dipole moment operators at positive retarded time. The cancellations that lead to this equivalence are presented. In many applications it is necessary to express the operators at retarded time in terms of creation and annihilation operators at the initial time. Explicit expressions for the fields up to second order in terms of the transition dipole moments are given. These enable the formulas for induced dipole moments to be derived within the context of quantum electrodynamics whereas previous derivations have been based on semiclassical electrodynamics.
The relationship between the multipolar and minimal Hamiltonians through a canonical transformation is used to analyze the time dependence of quantum operators in the two formalisms. It is shown that operators dependent on particle position, the vector potential, and their time derivatives have the same time dependence. However, operators such as the photon number and the atomic population number evolve differently in the two cases. Expressions correct to second order in the electric-dipole moments for these are given. The expectation values of the photon number operator are calculated in the two formalisms and are used to predict natural line shapes. These theoretical shapes differ. The observed shape will depend on the particular setup of the experiment involving the radiative decay of an excited atom. A comparison with the theoretical predictions will determine which of the two frameworks is most appropriate to describe the decay. Finally, the energy density of the electromagnetic field in the neighborhood of an atom is calculated within the two formalisms.
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.