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Physical Review A

Electric dipole interaction in quantum optics

L. Mandeļ

Published Oct 1, 1979• 38 citations
Publisher

Abstract

It is pointed out that the electromagnetic interaction $\ensuremath{-}\stackrel{^}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{E}$ between an atom and a field, in which the atom is modelled as a point dipole of moment $\stackrel{^}{\ensuremath{\mu}}$ interacting with the electric field $\stackrel{^}{E}$, is not strictly consistent with both quantum mechanics and Maxwell's equations. Quantum mechanics allows magnetic point dipoles but not electric point dipoles.

DOI
10.1103/physreva.20.1590

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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.

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Comment on "Electric dipole interaction in quantum optics"

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It is pointed out that the electromagnetic interaction \ensuremath{-}\stackrel{^}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{E} between an atom and a field, in which the atom is modelled as a point dipole of moment \stackrel{^}{\ensuremath{\mu}} interacting with the electric field \stackrel{^}{E}, is not strictly consistent with both quantum mechanics and Maxwell's equations. Quantum mechanics allows magnetic point dipoles but not electric point dipoles.
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.
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.
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.
The term of atom optics is due to the natural analogy with light optics or the optics of photons. Some elements of the optics of atoms and molecules, based on the interaction between spatially nonuniform static magnetic or electrical fields and the magnetic or electrical dipole moment of the particles, have long been known and used fairly successfully in experimental physics. In the presence of a magnetic or electric field, the quantum state of atom or molecule is shifted, the shift depending on the initial quantum state of the particle and the field strength. A particle in nonuniform magnetic field is acted upon by the force directed along the field strength gradient. The focusing properties of a magnetic lens depend on the magnetic sublevel of atoms.