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Physical Review A

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

G. Fields, R. Brienza, F. B. Dunning, S. Yoshida, Joachim Burgdörfer

Published Sep 14, 2021• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.104.032817

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Studies of high-n,120\ensuremath≲n\ensuremath≲160n, 120\ensuremath{\lesssim}n\ensuremath{\lesssim}160n,120\ensuremath≲n\ensuremath≲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-\ensuremathℓ\ensuremath{\ell}\ensuremathℓ states in the high-nnn limit. With increasing \ensuremathℓ\ensuremath{\ell}\ensuremathℓ 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-\ensuremathℓ\ensuremath{\ell}\ensuremathℓ states the quantum defects are determined predominantly by the polarizability of the inner electrons.
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.
Rydberg atoms have a long history of scientific research, but they have recently seen a resurgence in popularity due to their advantages when dealing with coupled internal Rydberg states, or externally-coupled Rydberg atoms. The use of convenient RF and/or microwave generation techniques allows for straightforward coupling of many nearby Rydberg states, enabling quantum simulation studies, such as the generation of a synthetic lattice. Strontium is particularly attractive for Rydberg atom studies due to its optically active core, which allows for two-electron excited states and its conveniently accessible transition frequencies that match those available using commercial diode lasers. Before these experiments and applications can be realized however, precise knowledge of Rydberg state energies and quantum defects is required. Therefore, this work undertakes an extensive study of the quantum defects of both singly and doubly-excited strontium Rydberg atoms for a wide range of n and l. Precision microwave spectroscopy between nearby Rydberg states produces an improved self-consistent set of quantum defects for singly-excited 1S0, 1P1, 1D2, and 1F3 states, described by their corresponding Rydberg-Ritz parameters. Quantum defects and lifetimes for autoionizing two-electron-states are also presented. These elucidate the effects of core penetration by the “outer” excited Rydberg electron on transitions within the core ion and on the Rydberg level itself.
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.
The authors use microwave spectroscopy to measure the quantum defects for lower-lll singlet strontium Rydberg states at intermediate nnn. The results are used to develop a revised set of Rydberg-Ritz quantum defect parameters that could help make accurate predictions of the energy intervals between other high-lying states.
The complex Rydberg structure of ytterbium atoms is shaped by multiple low-lying ion-core-excited states and strong channel interactions, which presents both opportunities and challenges for quantum information processing and precision metrology. In this work, we extend high-resolution microwave spectroscopy and multichannel quantum defect theory (MQDT) modeling of singly excited $6sn\ensuremath{\ell}$ Rydberg states in 174Yb^{174}\mathrm{Yb}174Yb and 171Yb^{171}\mathrm{Yb}171Yb to include the $\ensuremath{\ell}=3$ (fff) and $\ensuremath{\ell}=4$ (ggg) series. Our measurements reveal $p\text{\ensuremath{-}}f$ mixing in odd-parity Rydberg states of 171Yb^{171}\mathrm{Yb}171Yb, which we incorporate by combined MQDT models for 6snp6snp6snp and 6snf6snf6snf series. Additionally, we observe that for $\ensuremath{\ell}=4$ the spin-orbit interaction dominates over the exchange interaction, such that the 6sng6sng6sng states are more accurately described in a jjjjjj-coupled basis. We validate our models by comparing the predicted Land\'e ggg factors and static dipole polarizabilities with experimental measurements, finding excellent agreement. These results provide important input for designing high-fidelity entangling gates with ytterbium atoms.
The use of autoionization as a tool to quantitatively analyze transitions between high-n strontium Rydberg states with low-to-intermediate values of L is described and is demonstrated through observations of Rabi oscillations when driving 5snf 1F3 → 5s(n + 1)g 1G4, 5snf 1F3 → 5s(n + 2)h 1H5, and 5snp 1P1 → 5s(n + 1)d 1D2 transitions using microwave fields. The technique is shown to offer advantages when compared to selective field ionization and can be used with other alkaline-earth and alkaline-earth-like metals.