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Physical Review A

Comment on "Electric dipole interaction in quantum optics"

W P Healy

Published Dec 1, 1980• 29 citations
Publisher

Abstract

It is shown that Mandel's argument that the electric dipole interaction in quantum electrodynamics is inconsistent with Maxwell's equations relies on the use of the wrong canonical momenta in the multipolar form of the Hamiltonian; the field conjugate to the vector potential in the Coulomb gauge is proportional to the transverse displacement vector and not, as in the minimal-coupling form of the Hamiltonian, to the transverse electric field, and similarly the canonical momenta of the charged particles differ in the two cases. Some consequences of this for time-dependent perturbation theory are discussed.

DOI
10.1103/physreva.22.2891

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It is shown that Mandel's argument that the electric dipole interaction in quantum electrodynamics is inconsistent with Maxwell's equations relies on the use of the wrong canonical momenta in the multipolar form of the Hamiltonian; the field conjugate to the vector potential in the Coulomb gauge is proportional to the transverse displacement vector and not, as in the minimal-coupling form of the Hamiltonian, to the transverse electric field, and similarly the canonical momenta of the charged particles differ in the two cases. Some consequences of this for time-dependent perturbation theory are discussed.
It is shown that both Hamiltonians, one with $\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}$ and the other with $\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}$ as interaction energy between an atom and the electromagnetic field, lead to Maxwell's equations. In contrast to the viewpoint in Mandel's paper, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}$ is considered to be proportional to the canonically conjugate field of the vector potential.
The equations of motion and the class of Hamiltonians for electrodynamics, both classical and quantal, are considered for the interaction of radiation with bound charges within the electric-dipole approximation. It is emphasized that, despite the uniqueness of the equations of motion for physical variables, the form of the Hamiltonian is not unique and that a physical system is associated with an equivalence class of Hamiltonians. Two of the members of this class are the minimal-coupling and multipolar forms extensively used in quantum optics. It is shown that although different choices of Hamiltonians lead to different first-order equations of motion that connect canonical coordinates and momenta, they lead to the same second-order equations of motion for the physical variables and physical electromagnetic fields. In the quantum theory of electrodynamics the role played by the choice of representation for the operators and for the states is analyzed with care. The choice of representation leads to a freedom additional to that provided by canonical transformations in classical electrodyamics, although the two lead to the same operator equations of motion.
It is shown that Haller's interpretation of the unitary transformation from the minimalcoupling to the multipolar form of the Hamiltonian in nonrelativistic quantum electrodynamics not only conflicts with the Maxwell-Lorentz equations but also ascribes a special status to the minimal-coupling Hamiltonian which is not warranted by the general method of canonical quantization. The correct interpretation of the transformation as the quantum analog of a classical canonical transformation is reiterated. It is also shown that the transformation is equivalent to a gauge transformation of the electromagnetic potentials.
This article deals with the fundamental problem of light-matter interaction in the quantum theory. Although it is described through the vector potential in quantum electrodynamics, it is believed by some that a hamiltonian involving only the electric and the magnetic fields is preferable. In the literature this hamiltonian is known as the Power-Zienau-Woolley hamiltonian. We question its validity and show that it is not equivalent to the minimal-coupling hamiltonian. In this article, we show that these two hamiltonians are not connected through a gauge transformation. We find that the gauge is not fixed in the Power-Zienau-Woolley hamiltonian. The interaction term is written in one gauge whereas the rest of the hamiltonian is written in another gauge. The Power-Zienau-Woolley hamiltonian and the minimal-coupling one are related through a unitary transformation that does not fulfill the gauge fixing constraints. Consequently, they predict different physical results. In this letter, we provide the correct quantum theory in the multipolar gauge with a hamiltonian involving only the physical fields.
A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.
We consider the fully quantum-mechanical Hamiltonian for the interaction of light with bound electrons. There are, roughly speaking, two different viewpoints relating the minimal-coupling and electric-dipole forms of the Hamiltonian. We attempt to clarify the situation by showing that either viewpoint is justified. As an example we discuss the Maxwell–Bloch equations and show that the usual derivation of these equations is somewhat misleading. In addition, we find that the Maxwell–Bloch equations appearing in standard texts are inconsistent.