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Angular Geometry of Atomic Multipole Transitions

Wesley C. Campbell

TL;DR

This work presents a geometry-centric framework for driving atomic and nuclear multipole transitions by expressing the light–matter coupling as a dot product between the laser polarization and a vector spherical harmonic that encodes the transition’s angular structure. By recasting the interaction in terms of vector spherical harmonics and the helicity frame, the method yields a compact, Wigner-Eckart–based expression for the resonant Rabi frequency, $\Omega_{\mathrm{eg}}$, applicable to arbitrary rank $K$ and adaptable to hyperfine structure. The paper carefully demonstrates how beam geometry (Gaussian, Hermite–Gauss, Laguerre–Gauss, and vector modes) and multi-beam configurations shape selection rules and coupling strengths, providing both perturbative corrections and nontrivial, finite couplings in experimentally relevant settings. The approach offers intuitive visualization of the interaction, systematic paths to optimize coupling, and straightforward extensions to more complex beam geometries and vector fields with potential impact on precision measurements and quantum information processing.

Abstract

A simple way to calculate Rabi frequencies is outlined for interactions of atomic or nuclear multipole moments with laser fields that focuses on their relative geometry. The resulting expression takes the form of a dot product between the laser polarization and a vector spherical harmonic, thereby naturally connecting to the multipole's far-field spontaneous-emission pattern and providing a way to visualize the interaction. Since the vector spherical harmonics are not yet a standard tool in quantum science, their relevant properties are reviewed. This approach is illustrated in the calculation of a variety of beam effects, yielding both perturbative corrections and some nontrivial cases with non-vanishing coupling.

Angular Geometry of Atomic Multipole Transitions

TL;DR

This work presents a geometry-centric framework for driving atomic and nuclear multipole transitions by expressing the light–matter coupling as a dot product between the laser polarization and a vector spherical harmonic that encodes the transition’s angular structure. By recasting the interaction in terms of vector spherical harmonics and the helicity frame, the method yields a compact, Wigner-Eckart–based expression for the resonant Rabi frequency, , applicable to arbitrary rank and adaptable to hyperfine structure. The paper carefully demonstrates how beam geometry (Gaussian, Hermite–Gauss, Laguerre–Gauss, and vector modes) and multi-beam configurations shape selection rules and coupling strengths, providing both perturbative corrections and nontrivial, finite couplings in experimentally relevant settings. The approach offers intuitive visualization of the interaction, systematic paths to optimize coupling, and straightforward extensions to more complex beam geometries and vector fields with potential impact on precision measurements and quantum information processing.

Abstract

A simple way to calculate Rabi frequencies is outlined for interactions of atomic or nuclear multipole moments with laser fields that focuses on their relative geometry. The resulting expression takes the form of a dot product between the laser polarization and a vector spherical harmonic, thereby naturally connecting to the multipole's far-field spontaneous-emission pattern and providing a way to visualize the interaction. Since the vector spherical harmonics are not yet a standard tool in quantum science, their relevant properties are reviewed. This approach is illustrated in the calculation of a variety of beam effects, yielding both perturbative corrections and some nontrivial cases with non-vanishing coupling.

Paper Structure

This paper contains 24 sections, 3 theorems, 110 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Multipole expansion
  3. Calculation of the Rabi Frequency
  4. Applications
  5. Elementary examples
  6. $E1$ transitions
  7. $E2$ transitions
  8. Single-photon transitions with multiple, coherent plane waves
  9. Beam effects
  10. Centered Gaussian beams
  11. Off-center Gaussian beams
  12. Gouy phase
  13. Hermite-Gauss beams
  14. Helical Laguerre-Gauss beams
  15. Classical, nonseparable (vector-mode) beams
  16. ...and 9 more sections

Key Result

Lemma G.1

A stretched, nested, rank-$K$ irreducible tensor product $T^{(K)}[\mathbf{a},\mathbf{a}, \ldots , \mathbf{a}]$ of a vector $\mathbf{a}$ with itself $K$ times is proportional to a solid harmonic and given by where $Y_{(K)}(\mathbf{\hat{a}})$ is the rank-$K$ irreducible tensor whose $p$ component is given by the spherical harmonic $Y_{K,p}(\vartheta_a,\varphi_a)$.

Figures (5)

  • Figure 2.1: Rank-1 vector spherical harmonics of type $\lambda = +1$, depicted on the unit sphere. The color scale depicts the vector magnitude of each part, while the arrows, which are all the same length, only indicate direction. It is recommended to disable the Enhance thin lines option in Preferences$\rightarrow$Page Display to see the details in these plots.
  • Figure 2.2: Rank-2 vector spherical harmonics of type $\lambda = +1$. The rank-3 vector spherical harmonics are plotted in Appendix \ref{['app:VectorSphericalHarmonics']}.
  • Figure A.1: Rank-3 vector spherical harmonics of type $\lambda = +1$.
  • Figure A.2: Rank-1 vector spherical harmonics of type $\lambda = 0$, which can be used to describe the quadrature field to the field interacting with a multipole moment.
  • Figure B.1: Basis vectors for describing transverse vector fields in the far-field limit. The atom is at the origin. (a) The $\mathbf{k}$ vector of the light points in the radial direction $\mathbf{\hat{k}} = \mathbf{\hat{e}}'_{z'} \doteq (\vartheta_k,\varphi_k)$. I choose the helicity-frame basis vectors such that: (b) $\mathbf{\hat{e}}'_{x'} = {\hbox{\boldmath${\hat{\vartheta}}$}}$ and (c) $\mathbf{\hat{e}}'_{y'} = {\hbox{\boldmath${\hat{\varphi}}$}}$.

Theorems & Definitions (3)

  • Lemma G.1
  • Lemma G.2
  • Theorem G.3