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

Complex Light and Optical Forces

Investigating the degree of paraxiality in light–atom quadrupole interactions

Abdullah S. Alharbi, Kayn A. Forbes

Published Mar 5, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.3079712

Related papers

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

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.

Optics Express•Dec 4, 2023•4 citations

The paraxial approximation fails to describe the interaction of atoms with general vortex light fields

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

Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre-Gaussian and generalized Bessel beams. We demonstrate that all vortex beams here considered look the same close to the beam axis, where the atom is placed. From this, we conclude that the paraxial approximation fails for the large set of vortex beams, and that a full understanding of the interaction of optical vortices with atoms requires the inclusion of all vector components of the electric field.

arXiv:2004.00040•Mar 31, 2020•3 citations

Quadrupole interaction of Tricomi–Gaussian beams with atoms

Shuaishuai Ren, Mengyuan Han, Xinyi Cao +2 more

The interaction of light beams with atoms is one of the most important fields in physics, and the quadrupole transitions play an important role in such an interaction. In this work, we focus on the quadrupole interaction of Tricomi–Gaussian beams with atoms. The analytical expression of the paraxial Tricomi–Gaussian beams is derived using the angular spectrum representation. The formulas for describing the quadrupole interaction of Tricomi–Gaussian beams with two-level atoms are presented. The Cs atom is taken as an example, and the spatial distributions of the quadrupole Rabi frequency for such a two-level atom in Tricomi–Gaussian beams with different parameters are simulated and analyzed. It is revealed that the Rabi frequency of an atom in Tricomi–Gaussian beams can be flexibly controlled by choosing the asymmetric constants and order of the beams, which provides a pathway for tuning the repulsive potential and trapping potential acting on the atoms. This study is expected to provide useful insights into the interactions of Tricomi–Gaussian beams with atoms and their further applications.

Journal of the Optical Society of America B•Nov 25, 2024•4 citations

Interaction of a Laguerre–Gaussian beam with trapped Rydberg atoms

Koushik Mukherjee, Sonjoy Majumder, Pradip Kumar Mondal +1 more

Abstract Previous studies show that within paraxial limit and electric dipole approximations, the orbital angular momentum (OAM) of a Laguerre–Gaussian (LG) beam rotates the whole atom about the beam axis but does not affect the internal electronic motion. The contribution of the Gaussian part of the LG beam profile to the angular momentum exchange is not usually taken into account. In this paper, we develop a theory which shows that not only the OAM of a LG beam, but also the geometry of the beam cross section, plays an important role in the electronic lowest-order transitions in the interaction of the beam with a trapped Rydberg atom, due to the large span of its electronic wavefunctions. In this interaction, the standard dipole selection rules do not hold and our results show that the otherwise forbidden transitions can become sufficiently probable.

arXiv:1702.05969•Journal of Physics B Atomic Molecular and Optical Physics•Oct 4, 2017•14 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

Nonparaxial effects on laser-qubit operations

L. P. H. Gallagher, M. Mazzanti, Z. E. D. Ackerman +3 more

Tightly focused laser beams, or optical tweezers, are essential for analog and digital quantum simulation with neutral atoms and trapped ions. Despite this, most of the current intuition and theoretical treatment utilizes the paraxial approximation, which is an insufficient description at the focus of tightly focused optical tweezers. We develop an analytic model, which we use in tandem with numerical simulations, to quantify how nonparaxial effects will manifest in the next generation of scalable quantum hardware, where tightly focused beams are used for individual qubit control. In particular, we calculate the light potentials of Gaussian and Laguerre-Gaussian beams driving the quadrupole S 1 / 2 2 → D 5 / 2 2 transition in Ca + 40 . Longitudinal field components in the beam center cause spatially dependent Rabi frequencies and ac Stark shifts, leading to unexpected qubit-motion coupling. We characterize single- and two-qubit gate infidelities due to this effect with an analytic model and numerical simulation. We identify regimes where nonparaxial effects should be taken into account for high-precision quantum control, and highlight that the effects are potentially more severe for the case of neutral atom and molecule addressing.

arXiv:2502.19345•Physical Review Research•Dec 17, 2025•1 citation
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.
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.
Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre-Gaussian and generalized Bessel beams. We demonstrate that all vortex beams here considered look the same close to the beam axis, where the atom is placed. From this, we conclude that the paraxial approximation fails for the large set of vortex beams, and that a full understanding of the interaction of optical vortices with atoms requires the inclusion of all vector components of the electric field.
The interaction of light beams with atoms is one of the most important fields in physics, and the quadrupole transitions play an important role in such an interaction. In this work, we focus on the quadrupole interaction of Tricomi–Gaussian beams with atoms. The analytical expression of the paraxial Tricomi–Gaussian beams is derived using the angular spectrum representation. The formulas for describing the quadrupole interaction of Tricomi–Gaussian beams with two-level atoms are presented. The Cs atom is taken as an example, and the spatial distributions of the quadrupole Rabi frequency for such a two-level atom in Tricomi–Gaussian beams with different parameters are simulated and analyzed. It is revealed that the Rabi frequency of an atom in Tricomi–Gaussian beams can be flexibly controlled by choosing the asymmetric constants and order of the beams, which provides a pathway for tuning the repulsive potential and trapping potential acting on the atoms. This study is expected to provide useful insights into the interactions of Tricomi–Gaussian beams with atoms and their further applications.
Abstract Previous studies show that within paraxial limit and electric dipole approximations, the orbital angular momentum (OAM) of a Laguerre–Gaussian (LG) beam rotates the whole atom about the beam axis but does not affect the internal electronic motion. The contribution of the Gaussian part of the LG beam profile to the angular momentum exchange is not usually taken into account. In this paper, we develop a theory which shows that not only the OAM of a LG beam, but also the geometry of the beam cross section, plays an important role in the electronic lowest-order transitions in the interaction of the beam with a trapped Rydberg atom, due to the large span of its electronic wavefunctions. In this interaction, the standard dipole selection rules do not hold and our results show that the otherwise forbidden transitions can become sufficiently probable.
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.
Tightly focused laser beams, or optical tweezers, are essential for analog and digital quantum simulation with neutral atoms and trapped ions. Despite this, most of the current intuition and theoretical treatment utilizes the paraxial approximation, which is an insufficient description at the focus of tightly focused optical tweezers. We develop an analytic model, which we use in tandem with numerical simulations, to quantify how nonparaxial effects will manifest in the next generation of scalable quantum hardware, where tightly focused beams are used for individual qubit control. In particular, we calculate the light potentials of Gaussian and Laguerre-Gaussian beams driving the quadrupole S 1 / 2 2 → D 5 / 2 2 transition in Ca + 40 . Longitudinal field components in the beam center cause spatially dependent Rabi frequencies and ac Stark shifts, leading to unexpected qubit-motion coupling. We characterize single- and two-qubit gate infidelities due to this effect with an analytic model and numerical simulation. We identify regimes where nonparaxial effects should be taken into account for high-precision quantum control, and highlight that the effects are potentially more severe for the case of neutral atom and molecule addressing.