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Scientific Reports

The quantum-optics Hamiltonian in the Multipolar gauge

Emmanuel Rousseau, Didier Felbacq

Published Sep 5, 2017• 15 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-017-11076-5

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

Reply to ”The gauge-invariant lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics.” by A. Vuckis et al

Emmanuel Rousseau, Didier Felbacq

This note is a reply to the paper (arXiv:1801.0559v2): " The gauge-invariant Lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics" by G. Kónya, et al.See also our first reply (arXiv:1804.07472v1) in response to their initial comment "The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " ( arXiv:1801.05590v1) ————————————– In a recent paper, we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. Kónya and al. They first claim an error [arXiv:1801.05590v1] followed by a subsequent allegation that the Power-Zienau-Woolley hamiltonian can be derived starting from the electromagnetic action.We do not share their conclusions and show in two back-to-back replies [arXiv:1804.07472v1 followed by this note] that their successive statements are not correct.

•Apr 6, 2019•1 citation

A study of basic interaction Hamiltonians for quantum optics via an exactly solvable model

Marek Żukowski

The two basic interaction Hamiltonians for the quantum optical problems, namely the so-called minimal coupling one (p.A) and the multipolar one (d.E) are studied in the dipole approximation. The rigorous and exactly solvable model of an oscillator interacting with the electromagnetic field is employed. The Power-Zienau transformation is discussed. It is concluded that both Hamiltonians lead to the same S-matrix. The existence of an example like the one studied in this paper can be treated as support for the formal proofs given in the literature, which show that both Hamiltonians lead to the same predictions in the case of the more realistic description of the interaction of atoms with photons.

Quantum Optics Journal of the European Optical Society Part B•Dec 1, 1989•0 citations

Reply to " The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " by G. Kónya, et al.

Emmanuel Rousseau, Didier Felbacq

This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{\'e} gauge from the very first principles" by G. K{\'o}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{\'o}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{\'e} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{\'o}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{\'o}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{\'e} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{\'e} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{\'o}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum $\pi$(x,t) conjugated to the vector potential A(x,t) is given by $\pi$(x,t) = --$\epsilon$\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to $\partial$\_t $\phi$ (x,t) where $\phi$ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{\'o}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.

•Apr 6, 2018•0 citations

Reply to "The equivalence of the Power-Zineau-Woolley picture and the Poincar{é} gauge from the very first principles" by G. K{ó}nya, et al

Emmanuel Rousseau, Didier Felbacq

This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{é} gauge from the very first principles" by G. K{ó}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{ó}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{é} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{ó}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{ó}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{é} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{é} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{ó}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum $π$(x,t) conjugated to the vector potential A(x,t) is given by $π$(x,t) = --$ε$\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to $\partial$\_t $ϕ$ (x,t) where $ϕ$ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{ó}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.

arXiv:1804.07472•Apr 20, 2018•0 citations

Gauge invariant wave mechanics and the Power-Zienau-Woolley transformation

R. Guy Woolley

The wave mechanical formulation of quantum electrodynamics is investigated in an explicitly gauge invariant form and this leads to the connection between manifest charge conservation and the Power-Zienau-Woolley transformation.

Journal of Physics A Mathematical and General•Aug 1, 1980•26 citations
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.
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'.
This note is a reply to the paper (arXiv:1801.0559v2): " The gauge-invariant Lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics" by G. Kónya, et al.See also our first reply (arXiv:1804.07472v1) in response to their initial comment "The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " ( arXiv:1801.05590v1) ————————————– In a recent paper, we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. Kónya and al. They first claim an error [arXiv:1801.05590v1] followed by a subsequent allegation that the Power-Zienau-Woolley hamiltonian can be derived starting from the electromagnetic action.We do not share their conclusions and show in two back-to-back replies [arXiv:1804.07472v1 followed by this note] that their successive statements are not correct.
The two basic interaction Hamiltonians for the quantum optical problems, namely the so-called minimal coupling one (p.A) and the multipolar one (d.E) are studied in the dipole approximation. The rigorous and exactly solvable model of an oscillator interacting with the electromagnetic field is employed. The Power-Zienau transformation is discussed. It is concluded that both Hamiltonians lead to the same S-matrix. The existence of an example like the one studied in this paper can be treated as support for the formal proofs given in the literature, which show that both Hamiltonians lead to the same predictions in the case of the more realistic description of the interaction of atoms with photons.
This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{\'e} gauge from the very first principles" by G. K{\'o}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{\'o}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{\'e} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{\'o}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{\'o}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{\'e} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{\'e} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{\'o}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum π\piπ(x,t) conjugated to the vector potential A(x,t) is given by π\piπ(x,t) = --ϵ\epsilonϵ\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to ∂\partial∂\_t ϕ\phiϕ (x,t) where ϕ\phiϕ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{\'o}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.
This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{é} gauge from the very first principles" by G. K{ó}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{ó}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{é} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{ó}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{ó}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{é} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{é} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{ó}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum πππ(x,t) conjugated to the vector potential A(x,t) is given by πππ(x,t) = --εεε\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to ∂\partial∂\_t ϕϕϕ (x,t) where ϕϕϕ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{ó}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.