Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Optical Society of America B

Interaction Hamiltonian of quantum optics

Jay R. Ackerhalt, Peter W. Milonni

Published Mar 1, 1984• 49 citations
Publisher

Abstract

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.

DOI
10.1364/josab.1.000116

Related papers

Further remarks on the Hamiltonian of quantum optics

Edwin Albert Power, T. Thirunamachandran

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.

Journal of the Optical Society of America B•Jul 1, 1985•18 citations

Quantum versus semiclassical description of light interaction with atomic ensembles: Revision of the Maxwell-Bloch equations and single-photon superradiance

Anatoly A. Svidzinsky, Xiwen Zhang, Marlan O. Scully

Interaction of atoms with quantum states of light is a long-standing problem that apart from fundamental physics has potential applications for optical quantum-state storage, quantum communication, and quantum information. A fully quantum mechanical treatment of this problem is usually very complicated mathematically. Here we show, however, that quantum mechanical evolution equations describing single-photon emission (absorption) by atomic ensembles can be written in a form equivalent to the semiclassical Maxwell-Bloch equations. This connection allows us to find exact analytical solutions of the fully quantum mechanical problem. We also found that semiclassical Maxwell-Bloch equations should be written in a form different from those commonly used. Namely, the classical limit of the quantum problem gives a propagation equation with the Laplacian operator on the right-hand side rather than with the second-order time derivative.

Physical Review A•Jul 1, 2015•20 citations

Comment on "Electric dipole interaction in quantum optics"

Edwin Albert Power, T. Thirunamachandran

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.

Physical Review A•Dec 1, 1980•22 citations

QED-based Optical Bloch Equations without electric dipole approximation: A model for a two-level atom interacting with a monochromatic X-ray laser beam

Wenzhuo Zhang, Wu‐Ming Liu

We derive a set of optical Bloch equations (OBEs) directly from the minimal-coupling Hamiltonian density of the bound-state quantum electrodynamics (bound-state QED). Such optical Bloch equations are beyond the former widely-used ones due to that there is no electric dipole approximation (EDA) on the minimal-coupling Hamiltonian density of the bound-state QED. Then our optical Bloch equations can describe a two-level atom interacting with a monochromatic light of arbitrary wavelength, which are suitable to study the spectroscopy and the Rabi oscillations of two-level atoms in X-ray laser beams since that the wavelength of X-ray is close to an atom to make the electric dipole approximation (EDA) invalid.

arXiv:1206.3445•Jun 14, 2012•1 citation

Comment on "Electric dipole interaction in quantum optics"

W P Healy

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.

Physical Review A•Dec 1, 1980•29 citations

The interaction of light with neutral systems

Edwin Albert Power, R. Shail

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.

Mathematical Proceedings of the Cambridge Philosophical Society•Jan 1, 1959•12 citations
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.
Interaction of atoms with quantum states of light is a long-standing problem that apart from fundamental physics has potential applications for optical quantum-state storage, quantum communication, and quantum information. A fully quantum mechanical treatment of this problem is usually very complicated mathematically. Here we show, however, that quantum mechanical evolution equations describing single-photon emission (absorption) by atomic ensembles can be written in a form equivalent to the semiclassical Maxwell-Bloch equations. This connection allows us to find exact analytical solutions of the fully quantum mechanical problem. We also found that semiclassical Maxwell-Bloch equations should be written in a form different from those commonly used. Namely, the classical limit of the quantum problem gives a propagation equation with the Laplacian operator on the right-hand side rather than with the second-order time derivative.
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.
We derive a set of optical Bloch equations (OBEs) directly from the minimal-coupling Hamiltonian density of the bound-state quantum electrodynamics (bound-state QED). Such optical Bloch equations are beyond the former widely-used ones due to that there is no electric dipole approximation (EDA) on the minimal-coupling Hamiltonian density of the bound-state QED. Then our optical Bloch equations can describe a two-level atom interacting with a monochromatic light of arbitrary wavelength, which are suitable to study the spectroscopy and the Rabi oscillations of two-level atoms in X-ray laser beams since that the wavelength of X-ray is close to an atom to make the electric dipole approximation (EDA) invalid.
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.
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.