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

Physical Review A

Indirect spin-orbit interaction in high- L Rydberg states with 2S1/2 cores

Erica L. Snow, R. A. Komara, M. A. Gearba, S. R. Lundeen

Published Aug 29, 2003• 16 citations
Publisher

Abstract

In high-$L$ Rydberg states, where exchange interactions are of negligible size, spin-orbit interactions are expected to determine the energy differences between states of different spin orientations. This is well studied in the case of helium, where it results in a fourfold splitting of each high-$L$ term, coming from the two electrons' spin-orbit interactions. A similar structure might be expected in heavier Rydberg states, whose core ion is a ${}^{2}{S}_{1/2}$ state. However, observations in barium Rydberg states and recent observations in Rydberg states of ${\mathrm{Si}}^{2+}$ reveal significant deviations from this expectation. Both can be explained as an indirect effect of spin-orbit interactions in virtual excitations of the core ion. A perturbative treatment predicts an indirect spin-orbit interaction for the Rydberg state proportional to the inverse sixth power of the Rydberg electron's radial coordinate. The significance of this term, relative to the direct magnetic interactions, varies by more than six orders of magnitude between the three systems studied, helium, ${\mathrm{Si}}^{2+},$ and barium.

DOI
10.1103/physreva.68.022510

Related papers

Rydberg states of helium in electric and magnetic fields of arbitrary relative orientation

Ondřej Tkáč, Matija Žeško, Josef A. Agner +2 more

A spectroscopic study of Rydberg states of helium (n = 30 and 45) in magnetic, electric and combined magnetic and electric fields with arbitrary relative orientations of the field vectors is presented. The emphasis is on two special cases where (i) the diamagnetic term is negligible and both paramagnetic Zeeman and Stark effects are linear (n = 30, B ≤ 120 mT and F = 0-78 V cm⁻¹), and (ii) the diamagnetic term is dominant and the Stark effect is linear (n = 45, B = 277 mT and F = 0-8 V cm⁻¹). Both cases correspond to regimes where the interactions induced by the electric and magnetic fields are much weaker than the Coulomb interaction, but much stronger than the spin-orbit interaction. The experimental spectra are compared to spectra calculated by determining the eigenvalues of the Hamiltonian matrix describing helium Rydberg states in the external fields. The spectra and the calculated energy-level diagrams in external fields reveal avoided crossings between levels of different m ₗ values and pronounced m ₁-mixing effects at all angles between the electric- and magnetic-field vectors other than 0. These observations are discussed in the context of the development of a method to generate dense samples of cold atoms and molecules in a magnetic trap following Rydberg-Stark deceleration.

•Apr 1, 2016•5 citations

Indirect spin-orbit K splittings in strontium

Jirakan Nunkaew, E. S. Shuman, Thomas F. Gallagher

We use delayed field ionization to observe the microwave resonance transitions of Sr from $5snf$ to $5sng$, $5snh$, and $5sni$. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}>3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the $K$ splittings. From the $K$ splittings we extract the ionic dipole and quadrupole matrix elements, $⟨5s|r|5p⟩=3.65(25){a}_{0}$ and $⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}$. From the dipole matrix element we obtain the dipole polarizability of ${\text{Sr}}^{+}$. With the dipole polarizability and the quantum defect differences of adjacent $l$ states, we can determine the ${\text{Sr}}^{+}$ quadrupole polarizability. The dipole and quadrupole polarizabilities of ${\text{Sr}}^{+}$ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.

Physical Review A•May 4, 2009•14 citations

Indirect spin-orbit K splittings in strontium

Jirakan Nunkaew, E. S. Shuman, Thomas F. Gallagher

We use delayed field ionization to observe the microwave resonance transitions of Sr from $5snf$ to $5sng$, $5snh$, and $5sni$. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}g3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the $K$ splittings. From the $K$ splittings we extract the ionic dipole and quadrupole matrix elements, $⟨5s|r|5p⟩=3.65(25){a}_{0}$ and $⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}$. From the dipole matrix element we obtain the dipole polarizability of ${\text{Sr}}^{+}$. With the dipole polarizability and the quantum defect differences of adjacent $l$ states, we can determine the ${\text{Sr}}^{+}$ quadrupole polarizability. The dipole and quadrupole polarizabilities of ${\text{Sr}}^{+}$ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.

Bulletin of the American Physical Society•May 21, 2009•1 citation

Autoionization of Rydberg states with large angular momentum: Application to alkaline-earth atoms

Michel Poirier

This paper is devoted to the autoionization process of large-angular-momentum Rydberg states. After a brief account of autoionization formalism in nonrelativistic theory and single-configuration approximation, we apply it to atomic states with one electron of large angular momentum. The dominant contribution is the direct one and the exchange contribution is ignored. In this framework, the autoionization width varies with the principal quantum number as ${n}^{\mathrm{\ensuremath{-}}3}$ and exponentially with the angular momentum. When two excited electrons interact with a spherically symmetric core, it is demonstrated that the interaction between the Rydberg electron and the core polarized by the valence electron is proportional to the interaction between the two active electrons. Using newly derived radial matrix elements, we obtain a generally good agreement between our results and experiments on barium and strontium. Large-l Rydberg states are shown to be stable versus autoionization as soon as l is about 7. Possible refinements of our method are reviewed.

Physical Review A•Oct 1, 1988•51 citations

High- n limit of quantum defects for high- ℓ strontium Rydberg states

G. Fields, R. Brienza, F. B. Dunning +2 more

Studies of high-$n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160$, strontium $5sng, 5snh$, and $5sni$ Rydberg states are reported. Measurements using microwave spectroscopy provide the quantum defects for these states and furnish a valuable test of quantum defects reported for lower-$\ensuremath{\ell}$ states in the high-$n$ limit. With increasing $\ensuremath{\ell}$ the relativistic spin-orbit interaction becomes comparable to the electron-electron interaction resulting in the breakdown of the standard $LS$-coupling scheme. Heavier alkaline-earth quasi-two-electron systems differ significantly from a true two-electron system such as helium because indirect spin-orbit effects are dramatically enhanced. We explore the high-$n$ limit of this transition between $LS$ and $jj$ coupling. For high-$\ensuremath{\ell}$ states the quantum defects are determined predominantly by the polarizability of the inner electrons.

Physical Review A•Sep 14, 2021•4 citations

Rydberg states of helium in electric and magnetic fields of arbitrary relative orientation

Ondřej Tkáč, Matija Žeško, Josef A. Agner +2 more

Abstract A spectroscopic study of Rydberg states of helium ( n = 30 and 45) in magnetic, electric and combined magnetic and electric fields with arbitrary relative orientations of the field vectors is presented. The emphasis is on two special cases where (i) the diamagnetic term is negligible and both paramagnetic Zeeman and Stark effects are linear ( n = 30, B ≤ 120 mT and F = 0–78 V cm −1 ), and (ii) the diamagnetic term is dominant and the Stark effect is linear ( n = 45, B = 277 mT and F = 0–8 V cm −1 ). Both cases correspond to regimes where the interactions induced by the electric and magnetic fields are much weaker than the Coulomb interaction, but much stronger than the spin–orbit interaction. The experimental spectra are compared to spectra calculated by determining the eigenvalues of the Hamiltonian matrix describing helium Rydberg states in the external fields. The spectra and the calculated energy-level diagrams in external fields reveal avoided crossings between levels of different m l values and pronounced m l -mixing effects at all angles between the electric- and magnetic-field vectors other than 0. These observations are discussed in the context of the development of a method to generate dense samples of cold atoms and molecules in a magnetic trap following Rydberg–Stark deceleration.

arXiv:1701.01638•Journal of Physics B Atomic Molecular and Optical Physics•Apr 22, 2016•6 citations
In high-LLL Rydberg states, where exchange interactions are of negligible size, spin-orbit interactions are expected to determine the energy differences between states of different spin orientations. This is well studied in the case of helium, where it results in a fourfold splitting of each high-LLL term, coming from the two electrons' spin-orbit interactions. A similar structure might be expected in heavier Rydberg states, whose core ion is a 2S1/2{}^{2}{S}_{1/2}2S1/2​ state. However, observations in barium Rydberg states and recent observations in Rydberg states of Si2+{\mathrm{Si}}^{2+}Si2+ reveal significant deviations from this expectation. Both can be explained as an indirect effect of spin-orbit interactions in virtual excitations of the core ion. A perturbative treatment predicts an indirect spin-orbit interaction for the Rydberg state proportional to the inverse sixth power of the Rydberg electron's radial coordinate. The significance of this term, relative to the direct magnetic interactions, varies by more than six orders of magnitude between the three systems studied, helium, Si2+,{\mathrm{Si}}^{2+},Si2+, and barium.
A spectroscopic study of Rydberg states of helium (n = 30 and 45) in magnetic, electric and combined magnetic and electric fields with arbitrary relative orientations of the field vectors is presented. The emphasis is on two special cases where (i) the diamagnetic term is negligible and both paramagnetic Zeeman and Stark effects are linear (n = 30, B ≤ 120 mT and F = 0-78 V cm⁻¹), and (ii) the diamagnetic term is dominant and the Stark effect is linear (n = 45, B = 277 mT and F = 0-8 V cm⁻¹). Both cases correspond to regimes where the interactions induced by the electric and magnetic fields are much weaker than the Coulomb interaction, but much stronger than the spin-orbit interaction. The experimental spectra are compared to spectra calculated by determining the eigenvalues of the Hamiltonian matrix describing helium Rydberg states in the external fields. The spectra and the calculated energy-level diagrams in external fields reveal avoided crossings between levels of different m ₗ values and pronounced m ₁-mixing effects at all angles between the electric- and magnetic-field vectors other than 0. These observations are discussed in the context of the development of a method to generate dense samples of cold atoms and molecules in a magnetic trap following Rydberg-Stark deceleration.
We use delayed field ionization to observe the microwave resonance transitions of Sr from 5snf5snf5snf to 5sng5sng5sng, 5snh5snh5snh, and 5sni5sni5sni. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}>3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the KKK splittings. From the KKK splittings we extract the ionic dipole and quadrupole matrix elements, ⟨5s∣r∣5p⟩=3.65(25)a0⟨5s|r|5p⟩=3.65(25){a}_{0}⟨5s∣r∣5p⟩=3.65(25)a0​ and ⟨5s∣r2∣4d⟩=12(14)a02⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}⟨5s∣r2∣4d⟩=12(14)a02​. From the dipole matrix element we obtain the dipole polarizability of Sr+{\text{Sr}}^{+}Sr+. With the dipole polarizability and the quantum defect differences of adjacent lll states, we can determine the Sr+{\text{Sr}}^{+}Sr+ quadrupole polarizability. The dipole and quadrupole polarizabilities of Sr+{\text{Sr}}^{+}Sr+ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.
We use delayed field ionization to observe the microwave resonance transitions of Sr from 5snf5snf5snf to 5sng5sng5sng, 5snh5snh5snh, and 5sni5sni5sni. The energies of the $5sn\ensuremath{\ell}$, $\ensuremath{\ell}g3$ states are split by the indirect spin-orbit coupling of the Sr core to the Rydberg electron, producing the KKK splittings. From the KKK splittings we extract the ionic dipole and quadrupole matrix elements, ⟨5s∣r∣5p⟩=3.65(25)a0⟨5s|r|5p⟩=3.65(25){a}_{0}⟨5s∣r∣5p⟩=3.65(25)a0​ and ⟨5s∣r2∣4d⟩=12(14)a02⟨5s|{r}^{2}|4d⟩=12(14){a}_{0}^{2}⟨5s∣r2∣4d⟩=12(14)a02​. From the dipole matrix element we obtain the dipole polarizability of Sr+{\text{Sr}}^{+}Sr+. With the dipole polarizability and the quantum defect differences of adjacent lll states, we can determine the Sr+{\text{Sr}}^{+}Sr+ quadrupole polarizability. The dipole and quadrupole polarizabilities of Sr+{\text{Sr}}^{+}Sr+ are ${\ensuremath{\alpha}}_{d}=86(11){a}_{0}^{3}$ and ${\ensuremath{\alpha}}_{q}=1.1\ifmmode\times\else\texttimes\fi{}{10}^{3}(10){a}_{0}^{5}$, respectively.
This paper is devoted to the autoionization process of large-angular-momentum Rydberg states. After a brief account of autoionization formalism in nonrelativistic theory and single-configuration approximation, we apply it to atomic states with one electron of large angular momentum. The dominant contribution is the direct one and the exchange contribution is ignored. In this framework, the autoionization width varies with the principal quantum number as ${n}^{\mathrm{\ensuremath{-}}3}$ and exponentially with the angular momentum. When two excited electrons interact with a spherically symmetric core, it is demonstrated that the interaction between the Rydberg electron and the core polarized by the valence electron is proportional to the interaction between the two active electrons. Using newly derived radial matrix elements, we obtain a generally good agreement between our results and experiments on barium and strontium. Large-l Rydberg states are shown to be stable versus autoionization as soon as l is about 7. Possible refinements of our method are reviewed.
Studies of high-$n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160$, strontium 5sng,5snh5sng, 5snh5sng,5snh, and 5sni5sni5sni Rydberg states are reported. Measurements using microwave spectroscopy provide the quantum defects for these states and furnish a valuable test of quantum defects reported for lower-$\ensuremath{\ell}$ states in the high-nnn limit. With increasing $\ensuremath{\ell}$ the relativistic spin-orbit interaction becomes comparable to the electron-electron interaction resulting in the breakdown of the standard LSLSLS-coupling scheme. Heavier alkaline-earth quasi-two-electron systems differ significantly from a true two-electron system such as helium because indirect spin-orbit effects are dramatically enhanced. We explore the high-nnn limit of this transition between LSLSLS and jjjjjj coupling. For high-$\ensuremath{\ell}$ states the quantum defects are determined predominantly by the polarizability of the inner electrons.
Abstract A spectroscopic study of Rydberg states of helium ( n = 30 and 45) in magnetic, electric and combined magnetic and electric fields with arbitrary relative orientations of the field vectors is presented. The emphasis is on two special cases where (i) the diamagnetic term is negligible and both paramagnetic Zeeman and Stark effects are linear ( n = 30, B ≤ 120 mT and F = 0–78 V cm −1 ), and (ii) the diamagnetic term is dominant and the Stark effect is linear ( n = 45, B = 277 mT and F = 0–8 V cm −1 ). Both cases correspond to regimes where the interactions induced by the electric and magnetic fields are much weaker than the Coulomb interaction, but much stronger than the spin–orbit interaction. The experimental spectra are compared to spectra calculated by determining the eigenvalues of the Hamiltonian matrix describing helium Rydberg states in the external fields. The spectra and the calculated energy-level diagrams in external fields reveal avoided crossings between levels of different m l values and pronounced m l -mixing effects at all angles between the electric- and magnetic-field vectors other than 0. These observations are discussed in the context of the development of a method to generate dense samples of cold atoms and molecules in a magnetic trap following Rydberg–Stark deceleration.