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arXiv:2610.22527

Nonrelativistic quantum theory of non-dipole effects in a multielectron atom: radiation transition operator

A. N. Hopersky, A. M. Nadolinsky, Конеев Рустам Викторович

Published Oct 15, 2026• 0 citations
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Abstract

In the work of Hopersky A.N. et al., arXiv: 2609.33494v1 [physics. atom-ph] in the Loudon integral representation, the general analytical structure of the complete operator of a multipole radiation transition taking into account non-dipole effects for a multielectron atom is established. In this Letter this structure is converted to the sum only for all multipoles.

DOI
10.48550/arxiv.2610.22527

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In the work of Hopersky A.N. et al., arXiv: 2609.33494v1 [physics. atom-ph] in the Loudon integral representation, the general analytical structure of the complete operator of a multipole radiation transition taking into account non-dipole effects for a multielectron atom is established. In this Letter this structure is converted to the sum only for all multipoles.
This Letter establishes the analytical structure of the dipole radiation transition operator, taking into account non-dipole effects for a multielectron atom. The results summarize those of Komninos Y. et al., Phys. Rev. A 86, 023420 (2012), obtained in an approximation of a "split" electron of a continuous spectrum. The result for the complete operator of a multipole radiation transition [see formula (10)] can be specified within the framework of the methods of the theory of irreducible tensor operators and for higher multipoles.
In this paper, we extend to three-photon radiative transitions in hydrogenic ions our fully relativistic multipole approach [Z. Bona et al., Phys. Rev. A 89, 022514 (2014)], which has proven so far to yield extremely accurate results for two-photon emission processes. Closed-form formulas of double and single differential frequency distributions as well as total emission probabilities are derived for arbitrary multipole channels by using the Dirac-Coulomb Sturmian functions of the first order. Two cases of atomic systems are considered in the study, namely, ions with a spinless nucleus and those with a nonzero nuclear spin to check the validity of the Bose-Einstein statistics for three-photon emission investigated by Zalialiutdinov et al. [J. Phys. B: At. Mol. Opt. Phys. 49, 055001 (2016)]. In an effort to assess relativistic effects and the influence of the negative spectrum of the set of Dirac-Coulomb Sturmians, we formulate two nonrelativistic schemes that involve Schr\"odinger-Coulomb Sturmians, a transition operator with retardation, on one hand, and a transition operator in the long wavelength approximation, on the other hand. An application of these theories is made for the relativistic and nonrelativistic atomic transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{3/2}\ensuremath{\rightarrow}2{s}_{1/2}$ and $2s\ensuremath{\rightarrow}1s, 2p\ensuremath{\rightarrow}1s$, respectively, with nuclear charge ranging from 1 to 100. Some of our numerical values are compared with the scarce data available in the literature and they enable us to draw inferences as to the improvements obtained with our approach, while the others are additional results that we provide as a further step for more investigations of three-photon decays.
The author reviews investigations of the photon localization and quantum phase problems based on the use of the representation of multipole photons. Section II presents a general consideration of the field quantization, including a comparison of zero-point oscillations of the plane and multipole waves of photons. Section III discusses the atom-field interaction leading to the multipole radiation. The SU(2) quantum phase representation of atomic variables is considered, as is a connection between the SU(2) quantum phase states and entanglement phenomenon. In Section IV the author describes the quantum phase of multipole radiation caused by the angular momentum conservation in the process of radiation. This approach is compared with the Pegg–Barnett formalism and with Mandel's operational approach. Section V covers the quantum polarization properties of multipole radiation. Photon localization, quantum measurements, and causality are considered in Section VI.
The electromagnetic field operators and the electron field operators for the coupled system governed by the multipolar Hamiltonian are obtained within the Heisenberg picture. Their causal behavior and their relationship to the minimal-coupling forms are discussed. The basic fields of the multipolar theory, namely, the displacement vector and the magnetic field, are calculated up to terms quadratic in the multipole moment sources. The terms linear in the transition moments are the quantum counterparts of the classical fields and do not change the photon occupation number. The quadratic terms have no classical analogs: they act in both the photon and electron occupation-number spaces. It is shown that it is necessary to include these second-order terms in the calculation of the Poynting vector for an emitting dipole, thus demonstrating their role in the transport of radiative energy.
The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.
We investigate the interaction between the hydrogen atomic orbitals and the quantized modes of the electromagnetic field within the domain of nonrelativistic quantum electrodynamics in the Coulomb gauge. Contrary to the conventional dipole approximation and higher-order multipole approximations, which hold in the long-wavelength regime, we make no such approximations and investigate the rigorous wavelength dependence of the interaction matrix elements. Using analytical as well as numerical considerations, we establish the short-wavelength behaviour in the domain that is beyond the multipole approximation but still within the nonrelativistic regime. Our results differ significantly from the predictions of the dipole approximation. These features are graphically demonstrated in the case of the radiative 1 s -2 p z coupling in the hydrogen atom. We report a previously unnoticed short-wavelength contribution to the interaction that vanishes in the multipole expansion. We note that the convergence properties of radiative energy shifts are incorrectly predicted by the dipole approximation and that the unapproximated result is in fact quickly convergent.