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

Molecular Physics

On the hamiltonians for magneto-optical intensities

Aage E. Hansen, Erik Nørby Svendsen

Published Oct 1, 1974• 9 citations
Publisher

Abstract

A multipolar semi-classical hamiltonian for magneto-optical intensities is derived by a canonical transformation taking Feynman's formulation of the conventional Pauli hamiltonian as the point of departure. Only interactions up to the order of electric quadrupole and magnetic dipole terms are retained. The resulting hamiltonians are divided into their respective time-independent, single photon and double photon parts, and the difference in the appearance of magnetic field dependent contributions in the perturbation terms is analysed. On the basis of this analysis it is argued that the magnetic field dependent term in the magnetic dipole transition moment operator may lead to a unique single photon process with an amplitude proportional to the static magnetic field. This process leads, amongst others, to transitions between totally symmetric atomic states, in contrast to the standard multipolar selection rules of ordinary spectroscopy.

DOI
10.1080/00268977400102371

Related papers

Gauge and unitary transformations in multipolar quantum optics

M. Babiker

Multipolar quantum optics deals with the interaction of light with matter as a many-body bound system of charged particles where the coupling to electromagnetic fields is in terms of the multipolar electric polarization and magnetization. We describe two transformations applied to the conventional non-relativistic formalism, namely a gauge transformation applied directly to the fields at the Lagrangian stage and a unitary transformation applied to the old Hamiltonian. We show how such transformations lead to the same Power-Zienau-Woolley (PZW) formulation of the quantum electrodynamics (QED) of an overall electrically neutral many-body bound system of charges, including the internal motion as well as the gross dynamics of the centre of mass. Besides highlighting the utility of the multipolar formalism as a reliable and convenient platform in dealing with optical processes in atomic and molecular physics, it is shown how the analysis can also lead to the identification of the Röntgen effect arising from the gross motion of an electric dipole moment in a magnetic field and the Aharonov-Casher effect due to the motion of a magnetic dipole moment in an electric field. The importance of the two effects is pointed out in both experimental and theoretical contexts.This article is part of the theme issue 'The quantum theory of light'.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 24, 2024•0 citations

Field quantization in dielectric media and the generalised multipolar hamiltonian

Bryan J. Dalton, S. M. Dutra, E. S. Guerra +1 more

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.

European Quantum Electronics Conference (EQEC)•Aug 24, 2005•0 citations

Electronic structure of atoms in magnetic quadrupole traps

Igor Lesanovsky, Jörg Schmiedmayer, Peter Schmelcher

We investigate the electronic structure and properties of atoms exposed to a magnetic quadrupole field. The spin spatial as well as generalized time-reversal symmetries are established and shown to lead to a twofold degeneracy of the electronic states in the presence of the field. Low lying as well as highly excited Rydberg states are computed and analyzed for a broad regime of field gradients. The delicate interplay between the Coulomb and various magnetic interactions leads to complex patterns of the spatial spin polarization of individual excited states. Electromagnetic transitions in the quadrupole field are studied in detail thereby providing the selection rules and, in particular, the transition wavelengths and corresponding dipole strengths. The peculiar property that the quadrupole magnetic field induces permanent electric dipole moments of the atoms is derived and discussed.

arXiv:physics/0402109•Physical Review A•May 12, 2004•14 citations

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

Time dependence of operators in minimal and multipolar nonrelativistic quantum electrodynamics. I. Electromagnetic fields in the neighborhood of an atom

Edwin Albert Power, T. Thirunamachandran

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.

Physical Review A•Dec 1, 1999•21 citations

Field quantization in dielectric media and the generalized multipolar Hamiltonian

Bryan J. Dalton, E. S. Guerra, P. L. Knight

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.

Physical Review A•Sep 1, 1996•75 citations
A multipolar semi-classical hamiltonian for magneto-optical intensities is derived by a canonical transformation taking Feynman's formulation of the conventional Pauli hamiltonian as the point of departure. Only interactions up to the order of electric quadrupole and magnetic dipole terms are retained. The resulting hamiltonians are divided into their respective time-independent, single photon and double photon parts, and the difference in the appearance of magnetic field dependent contributions in the perturbation terms is analysed. On the basis of this analysis it is argued that the magnetic field dependent term in the magnetic dipole transition moment operator may lead to a unique single photon process with an amplitude proportional to the static magnetic field. This process leads, amongst others, to transitions between totally symmetric atomic states, in contrast to the standard multipolar selection rules of ordinary spectroscopy.
Multipolar quantum optics deals with the interaction of light with matter as a many-body bound system of charged particles where the coupling to electromagnetic fields is in terms of the multipolar electric polarization and magnetization. We describe two transformations applied to the conventional non-relativistic formalism, namely a gauge transformation applied directly to the fields at the Lagrangian stage and a unitary transformation applied to the old Hamiltonian. We show how such transformations lead to the same Power-Zienau-Woolley (PZW) formulation of the quantum electrodynamics (QED) of an overall electrically neutral many-body bound system of charges, including the internal motion as well as the gross dynamics of the centre of mass. Besides highlighting the utility of the multipolar formalism as a reliable and convenient platform in dealing with optical processes in atomic and molecular physics, it is shown how the analysis can also lead to the identification of the Röntgen effect arising from the gross motion of an electric dipole moment in a magnetic field and the Aharonov-Casher effect due to the motion of a magnetic dipole moment in an electric field. The importance of the two effects is pointed out in both experimental and theoretical contexts.This article is part of the theme issue 'The quantum theory of light'.
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.
We investigate the electronic structure and properties of atoms exposed to a magnetic quadrupole field. The spin spatial as well as generalized time-reversal symmetries are established and shown to lead to a twofold degeneracy of the electronic states in the presence of the field. Low lying as well as highly excited Rydberg states are computed and analyzed for a broad regime of field gradients. The delicate interplay between the Coulomb and various magnetic interactions leads to complex patterns of the spatial spin polarization of individual excited states. Electromagnetic transitions in the quadrupole field are studied in detail thereby providing the selection rules and, in particular, the transition wavelengths and corresponding dipole strengths. The peculiar property that the quadrupole magnetic field induces permanent electric dipole moments of the atoms is derived and discussed.
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.
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.
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.