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

Optics Express

Significance of the longitudinal component of paraxial light in position-dependent selection rules for quadrupole atomic transitions

Abdullah F. Alharbi, A. Lyras, V. E. Lembessis

Published Dec 4, 2023• 4 citations
Publisher

Abstract

It is well established that the longitudinal component of paraxial optical vortices has a key role in specific atomic quadrupole transitions near the beam axis when the spin and orbital angular momenta are antiparallel. By deriving analytical expressions for the position-dependent selection rules, this work shows that the significant role of the longitudinal fields is not limited to this case, but rather is a more general feature for any paraxial light including, for example, optical vortices with parallel spin and orbital angular momenta as well as Guassian beams. Numerically, the transition strengths induced by the weaker longitudinal component can be twice as high as those by the stronger transverse component. We also show that there are transitions that can be induced exclusively by the longitudinal component for light carrying two quanta of orbital angular momentum.

DOI
10.1364/oe.510120

Related papers

On the importance of the longitudinal component of paraxial optical vortices in the interaction with atoms

G. F. Quinteiro

Experimental evidence and theory on the head-on excitation of atoms by paraxial Laguerre–Gaussian beams revealed that the longitudinal component of the field has to be taken into account. Optical vortices are in fact a large family of fields, Laguerre–Gaussian being only one particular case. Here, we extend the previous study to a broader set of vortex fields. We demonstrate that, in general, paraxial optical vortices that have opposite orbital and spin angular momenta exhibit a longitudinal component that cannot be disregarded in the light–matter interaction.

Journal of the Optical Society of America B•Feb 8, 2023•6 citations

Twisted-Light–Ion Interaction: The Role of Longitudinal Fields

G. F. Quinteiro, Ferdinand Schmidt‐Kaler, Christian Tomás Schmiegelow

The propagation of light beams is well described using the paraxial approximation, where field components along the propagation direction are usually neglected. For strongly inhomogeneous or shaped light fields, however, this approximation may fail, leading to intriguing variations of the light-matter interaction. This is the case of twisted light having opposite orbital and spin angular momenta. We compare experimental data for the excitation of a quadrupole transition in a single trapped ^{40}Ca^{+} ion from Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)NCAOBW2041-172310.1038/ncomms12998] with a complete model where longitudinal components of the electric field are taken into account. Our model matches the experimental data and excludes by 11 standard deviations the approximation of a complete transverse field. This demonstrates the relevance of all field components for the interaction of twisted light with matter.

arXiv:1707.04776•Physical Review Letters•Dec 20, 2017•91 citations

Excitation of an Atomic Transition with a Vortex Laser Beam

Christian Tomás Schmiegelow, J. Schulz, Henning Kaufmann +3 more

Photons carry one unit of angular momentum associated with their spin~\cite{Beth1936}. Structured vortex beams carry additional orbital angular momentum which can also be transferred to matter~\cite{Allen1992}. This extra twist has been used for example to drive motion of microscopic particles in optical tweezers as well as to create vortices in degenerate quantum gases~\cite{He1995,Andersen2006}. Here we demonstrate the transfer of optical orbital angular momentum from the transverse spatial structure of the beam to the internal (electronic) degrees of freedom of an atom. Probing a quadrupole transition of a single trapped $^{40}$Ca$^+$ ion localized at the center of the vortex, we observe strongly modified selection rules, accounting for both the photon spin and the vorticity of the field. In particular, we show that an atom can absorb two quanta of angular momentum from a single photon even when rotational symmetry is conserved. In contrast to previous findings~\cite{Araoka2005,Loeffler2011a,Mathevet2013}, our experiment allows for conditions where the vorticity of the laser beam determines the optical excitation, contributing to the long-standing discussion on whether the orbital angular momentum of photons can be transferred to atomic internal degrees of freedom~\cite{VanEnk1994,Babiker2002,Jauregui2004, Schmiegelow2012, Mondal2014, Scholz-Marggraf2014} and paves the way for its use to tailor light-matter interactions.

arXiv:1511.07206•Nov 23, 2015•6 citations

Transversality Locks Longitudinal Gradients in Structured-Light Quadrupole Transitions

Kayn A. Forbes, Dale Green, Abdullah F. Alharbi +1 more

Electric quadrupole absorption is driven by optical field gradients, not by field amplitudes alone. This distinction is crucial for structured light, where a paraxially small longitudinal field can generate a leading-order longitudinal gradient. For fully vectorial Laguerre-Gaussian modes, Maxwell transversality locks this longitudinal gradient to the transverse structured gradient of the leading paraxial field. This locking is therefore a general electromagnetic constraint, not a focusing-dependent correction that can be removed independently. Decomposing the symmetric-traceless optical field-gradient tensor into spherical components reveals a strongly channel-selective response: the $Δm=0$ transition requires the longitudinal gradient at the retained order, the $Δm=\pm1$ channels receive scalar and vectorial nonparaxial corrections on top of the carrier-scale longitudinal derivative of the transverse field, and the $Δm=\pm2$ channels remain transverse at the same order. These results show that longitudinal optical structure can enter structured-light quadrupole transitions as an indispensable leading-gradient contribution, even when the longitudinal field amplitude itself is paraxially small.

arXiv:2608.19054•Aug 19, 2026•0 citations

Investigating the degree of paraxiality in light–atom quadrupole interactions

Abdullah S. Alharbi, Kayn A. Forbes

We examine the applicability and limitations of several established measures of paraxiality in the context of electric quadrupole interactions between light and trapped atoms. Focusing on Laguerre–Gaussian beams, we compare analytical predictions—particularly transition amplitudes and selection rules—derived under the paraxial assumption with full numerical solutions obtained via diffraction integrals that account for the complete vectorial field structure. This comparison allows us to assess the validity and relevance of these measures across a range of beam parameters. We also investigate the role of the power-series expansion in the paraxial parameter in bridging paraxial and nonparaxial regimes.

Complex Light and Optical Forces•Mar 5, 2026•0 citations

Quadrupole absorption rate for atoms in circularly-polarized optical vortices

Smail B. Bougouffa

Twisted light beams, or optical vortices, have been used to drive the circular motion of microscopic particles in optical tweezers and have been shown to generate vortices in quantum gases. Recent studies have established that electric quadrupole interactions can mediate an orbital angular momentum exchange between twisted light and the electronic degrees of freedom of atoms. Here we consider a quadrupole atomic transition mediated by a circularly-polarized optical vortex. We evaluate the transfer rate of the optical angular momentum to a Ca+ ion involving the 42S1∕2→32D5∕2 quadrupole transition and explain how the polarization state and the topological charge of the vortex beam determine the selection rules.

arXiv:2101.07294•Results in Physics•Jul 15, 2021•5 citations
It is well established that the longitudinal component of paraxial optical vortices has a key role in specific atomic quadrupole transitions near the beam axis when the spin and orbital angular momenta are antiparallel. By deriving analytical expressions for the position-dependent selection rules, this work shows that the significant role of the longitudinal fields is not limited to this case, but rather is a more general feature for any paraxial light including, for example, optical vortices with parallel spin and orbital angular momenta as well as Guassian beams. Numerically, the transition strengths induced by the weaker longitudinal component can be twice as high as those by the stronger transverse component. We also show that there are transitions that can be induced exclusively by the longitudinal component for light carrying two quanta of orbital angular momentum.
Experimental evidence and theory on the head-on excitation of atoms by paraxial Laguerre–Gaussian beams revealed that the longitudinal component of the field has to be taken into account. Optical vortices are in fact a large family of fields, Laguerre–Gaussian being only one particular case. Here, we extend the previous study to a broader set of vortex fields. We demonstrate that, in general, paraxial optical vortices that have opposite orbital and spin angular momenta exhibit a longitudinal component that cannot be disregarded in the light–matter interaction.
The propagation of light beams is well described using the paraxial approximation, where field components along the propagation direction are usually neglected. For strongly inhomogeneous or shaped light fields, however, this approximation may fail, leading to intriguing variations of the light-matter interaction. This is the case of twisted light having opposite orbital and spin angular momenta. We compare experimental data for the excitation of a quadrupole transition in a single trapped ^{40}Ca^{+} ion from Schmiegelow et al. [Nat. Commun. 7, 12998 (2016)NCAOBW2041-172310.1038/ncomms12998] with a complete model where longitudinal components of the electric field are taken into account. Our model matches the experimental data and excludes by 11 standard deviations the approximation of a complete transverse field. This demonstrates the relevance of all field components for the interaction of twisted light with matter.
Photons carry one unit of angular momentum associated with their spin~\cite{Beth1936}. Structured vortex beams carry additional orbital angular momentum which can also be transferred to matter~\cite{Allen1992}. This extra twist has been used for example to drive motion of microscopic particles in optical tweezers as well as to create vortices in degenerate quantum gases~\cite{He1995,Andersen2006}. Here we demonstrate the transfer of optical orbital angular momentum from the transverse spatial structure of the beam to the internal (electronic) degrees of freedom of an atom. Probing a quadrupole transition of a single trapped 40^{40}40Ca+^++ ion localized at the center of the vortex, we observe strongly modified selection rules, accounting for both the photon spin and the vorticity of the field. In particular, we show that an atom can absorb two quanta of angular momentum from a single photon even when rotational symmetry is conserved. In contrast to previous findings~\cite{Araoka2005,Loeffler2011a,Mathevet2013}, our experiment allows for conditions where the vorticity of the laser beam determines the optical excitation, contributing to the long-standing discussion on whether the orbital angular momentum of photons can be transferred to atomic internal degrees of freedom~\cite{VanEnk1994,Babiker2002,Jauregui2004, Schmiegelow2012, Mondal2014, Scholz-Marggraf2014} and paves the way for its use to tailor light-matter interactions.
Electric quadrupole absorption is driven by optical field gradients, not by field amplitudes alone. This distinction is crucial for structured light, where a paraxially small longitudinal field can generate a leading-order longitudinal gradient. For fully vectorial Laguerre-Gaussian modes, Maxwell transversality locks this longitudinal gradient to the transverse structured gradient of the leading paraxial field. This locking is therefore a general electromagnetic constraint, not a focusing-dependent correction that can be removed independently. Decomposing the symmetric-traceless optical field-gradient tensor into spherical components reveals a strongly channel-selective response: the Δm=0Δm=0Δm=0 transition requires the longitudinal gradient at the retained order, the Δm=±1Δm=\pm1Δm=±1 channels receive scalar and vectorial nonparaxial corrections on top of the carrier-scale longitudinal derivative of the transverse field, and the Δm=±2Δm=\pm2Δm=±2 channels remain transverse at the same order. These results show that longitudinal optical structure can enter structured-light quadrupole transitions as an indispensable leading-gradient contribution, even when the longitudinal field amplitude itself is paraxially small.
We examine the applicability and limitations of several established measures of paraxiality in the context of electric quadrupole interactions between light and trapped atoms. Focusing on Laguerre–Gaussian beams, we compare analytical predictions—particularly transition amplitudes and selection rules—derived under the paraxial assumption with full numerical solutions obtained via diffraction integrals that account for the complete vectorial field structure. This comparison allows us to assess the validity and relevance of these measures across a range of beam parameters. We also investigate the role of the power-series expansion in the paraxial parameter in bridging paraxial and nonparaxial regimes.
Twisted light beams, or optical vortices, have been used to drive the circular motion of microscopic particles in optical tweezers and have been shown to generate vortices in quantum gases. Recent studies have established that electric quadrupole interactions can mediate an orbital angular momentum exchange between twisted light and the electronic degrees of freedom of atoms. Here we consider a quadrupole atomic transition mediated by a circularly-polarized optical vortex. We evaluate the transfer rate of the optical angular momentum to a Ca+ ion involving the 42S1∕2→32D5∕2 quadrupole transition and explain how the polarization state and the topological charge of the vortex beam determine the selection rules.