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Physical Review A

Comment on "Electric dipole interaction in quantum optics"

Edwin Albert Power, T. Thirunamachandran

Published Dec 1, 1980• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.22.2894

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It is shown that both Hamiltonians, one with \ensuremath−\ensuremathμ\ensuremath→\ifmmode⋅\else\textperiodcentered\fiE\ensuremath→\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath−\ensuremathμ\ensuremath→​\ifmmode⋅\else\textperiodcentered\fiE\ensuremath→​ and the other with \ensuremath−\ensuremathμ\ensuremath→\ifmmode⋅\else\textperiodcentered\fiA\ensuremath→\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}\ensuremath−\ensuremathμ\ensuremath→​\ifmmode⋅\else\textperiodcentered\fiA\ensuremath→​ as interaction energy between an atom and the electromagnetic field, lead to Maxwell's equations. In contrast to the viewpoint in Mandel's paper, E\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}E\ensuremath→​ is considered to be proportional to the canonically conjugate field of the vector potential.
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 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.
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.
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.
Abstract The interaction between atoms or molecules and electromagnetic (EM) fields underlies all spectroscopic techniques and a great variety of desirable molecular properties. EM fields and EM waves are introduced via the famous Maxwell equations. The scalar and vector potential are defined, and the gauge freedom is outlined. The Coulomb gauge is adopted. The classical Hamiltonian for a charged particle in an EM field is derived, and from this the ‘minimal substitution’ rules for incorporating the fields in the quantum Hamiltonian are obtained. The operators describing the interaction of an electron with static electric and magnetic fields, including the magnetic fields from nuclear spins, are derived, followed by the derivation of the interaction between an electron and an EM wave.
Classical equations of motion that are first-order in time and conserve energy can only be quantized after their variables have been transformed to canonical ones, i.e., variables in which the energy is the system's Hamiltonian. The source-free version of Maxwell's equations is purely dynamical, first-order in time and has well-defined nonnegative conserved field energy but is decidedly noncanonical. That should long ago have made source-free Maxwell equation canonical Hamiltonization a research priority, and afterward, standard textbook fare, but textbooks seem unaware of the issue. The opposite parities of the electric and magnetic fields and consequent curl operations that typify Maxwell's equations are especially at odds with their being canonical fields. Transformation of the magnetic field into the transverse part of the vector potential helps but is not sufficient; further simple nonnegative symmetric integral transforms, which commute with all differential operators, are needed for both fields; such transforms also supplant the curls in the equations of motion. The canonical replacements of the source-free electromagnetic fields remain transverse-vector fields, but are more diffuse than their predecessors, albeit less diffuse than the transverse vector potential. Combined as the real and imaginary parts of a complex field, the canonical fields prove to be the transverse-vector wave function of a time-dependent Schrodinger equation whose Hamiltonian operator is the quantization of the free photon's square-root relativistic energy. Thus proper quantization of the source-free Maxwell equations is identical to second quantization of free photons that have normal square-root energy. There is no physical reason why first and second quantization of any relativistic free particle ought not to proceed in precise parallel, utilizing the square-root Hamiltonian operator. This natural procedure leaves no role for the completely artificial Klein-Gordon and Dirac equations, as accords with their grossly unphysical properties.