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Microwave spectroscopy and multi-channel quantum defect analysis of ytterbium Rydberg states

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Ytterbium's 6snp and 6snf Rydberg series are coupled by a small p-f mixing angle, and its 6sng series are jj-coupled, as shown by models that reproduce measured energies, g-factors, and polarizabilities.

desk verdict Strong new data and honest modeling; the f-channel model's unidentified fifth perturber deserves scrutiny before gate-design use, but the paper deserves a serious referee. read the letter →

arxiv 2507.11487 v1 pith:B4DLUDU3 submitted 2025-07-15 physics.atom-ph quant-ph

Rin Kuroda , Vernon M. Hughes , Martin Poitrinal , Michael Peper , Jeff D. Thompson This is my paper · ORCID
classification physics.atom-phquant-ph
keywords ytterbiumRydbergstatesmultichannelquantumdefecttheorymicrowavespectroscopyp-fmixingjjcouplingLandég-factorsstaticdipolepolarizabilityhyperfinestructure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends precision microwave spectroscopy of ytterbium Rydberg states to the $\ell=3$ ($f$) and $\ell=4$ ($g$) series in both $^{174}$Yb and $^{171}$Yb, and builds multichannel quantum defect theory (MQDT) models for them. The central finding is that in $^{171}$Yb the $6snp$ and $6snf$ series with total angular momentum $F=3/2$ and $F=5/2$ are mixed by configuration interaction; a single mixing angle between the $6s_{1/2}np_{3/2}$ and $6s_{1/2}nf_{5/2}$ channels resolves a previously reported deviation in the $6snp$ ($F=3/2$) energies. For the $g$ series the authors find that spin-orbit coupling dominates over exchange, so the states are best described in a $jj$-coupled basis rather than $LS$ coupling. The models reproduce measured energies to the order of a megahertz or better and predict Landé $g$-factors and static dipole polarizabilities of $6snd$ states in agreement with experiment, which is what makes them useful for designing Rydberg-mediated entangling gates.

What carries the argument

The central object is the multichannel quantum defect theory (MQDT) model, a compact parametrization of a Rydberg series as interacting channels, each with a quantum defect and an ionization threshold, whose bound states are found by matching boundary conditions. The load-bearing extension here is the combined-channel MQDT model in which a single rotation angle $\theta_{\rm pf}$ couples the $6s_{1/2}np_{3/2}$ and $6s_{1/2}nf_{5/2}$ channels; for $^{171}$Yb the hyperfine structure is introduced through a frame transformation, and for the $6sng$ states a frame transformation to a $jj$-coupled basis (channels such as $(6s_{1/2})(ng_{9/2})$ and $(6s_{1/2})(ng_{7/2})$) replaces $LS$ coupling. Five core-excited $5d$ perturbing channels pin down the energy dependence of the quantum defects in the $6snf$ model.

What would settle it

Measure the predicted $6sng$ ($F=11/2$) series of $^{171}$Yb, or record low-$n$ ($n<30$) $6snf$ $^{1,3}F_3$ energies with microwave accuracy; if the fitted p-f mixing angle or the unassigned fifth $5d$ channel is wrong, the residuals will exceed the roughly 1 MHz RMS of the current fit and the $g$-factor admixtures near the avoided crossings will disagree with the predictions.

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Extended reading notes

Core claim

The paper's discovery is that the odd-parity Rydberg series of $^{171}$Yb with $F=3/2$ and $F=5/2$ cannot be described by independent $p$ and $f$ channels: configuration interaction mixes $6snp$ and $6snf$ states, most strongly near accidental degeneracies. Including a single p-f mixing angle $\theta_{\rm pf}=1.842(8)\times10^{-2}$ between the $6s_{1/2}np_{3/2}$ and $6s_{1/2}nf_{5/2}$ channels in a combined MQDT model removes the dispersion-like residuals previously seen in the $6snp$ ($F=3/2$) series. For $\ell=4$, the measured $g$-factors show that the two $6sng$ ($J=4$) series have an $LS$-coupling singlet-triplet mixing angle $\theta_{LS}=0.66(4)$, close to the full $LS$-$jj$ rotation value of $0.73$, while in a $jj$-coupled description the mixing is only $\theta_{jj}=-0.06(3)$; this demonstrates that spin-orbit dominates exchange and that $jj$ coupling is the natural basis. The same models predict the static dipole polarizabilities of $6snd$ states, and the predictions match measured Stark shifts, including non-quadratic behaviour near near-degenerate opposite-parity states.

Load-bearing premise

The model assumes that a single constant mixing strength between the $p$ and $f$ series, together with five core-excited extra channels, is enough to capture all the relevant level mixing in the odd-parity $F=3/2$ and $F=5/2$ manifolds.

Editorial extensions

If this is right

  • Energies and wavefunctions of $6snd$ ($F=3/2,5/2$) states in $^{171}$Yb can now be predicted reliably, including non-quadratic Stark shifts near degeneracies with $6snp$ states.
  • The combined $6snp$/$6snf$ models remove the previous $6snp$ ($F=3/2$) discrepancy, so gate-design calculations using those states no longer carry that systematic error.
  • The $jj$-coupling result for $6sng$ indicates that for $\ell\geq4$ the hydrogenic limit is approached: quantum defects are small and simple single-channel models suffice in the measured range.
  • Measured and predicted $g$-factors confirm the MQDT wavefunctions, not just the energies, validating the Rydberg-Rydberg interaction-potential calculations presented for the gate-relevant $s$ and $d$ states.
  • Predicted static dipole polarizabilities of additional $6snd$ series in both isotopes are now available for the design of electric-field-insensitive Rydberg states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One step beyond the paper's data: if the extracted p-f mixing angle reflects a genuine core property, the same near-degeneracy-enhanced mixing should appear in other ytterbium isotopes and in isoelectronic alkaline-earth-like atoms wherever an $np$ and an $nf$ series cross, making $\theta_{\rm pf}$ a benchmark for ab initio treatments of core quadrupole interactions.
  • The paper notes that the energy dependence of the $^1F_3$ quantum defect is unusually large; a natural reading is that an additional core-excited perturber lies just above the modelled range, and low-$n$ microwave data would force the model to include it, shifting the low-$n$ $g$-factor predictions accordingly.
  • The $jj$-coupling crossover at $\ell=4$ suggests that for even higher angular momentum ($h$, $i$) the single-active-electron hydrogenic description becomes quantitatively accurate, which would make microwave spectroscopy of those series a clean probe of core polarization and non-adiabatic effects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reports high-precision microwave spectroscopy of 6snf and 6sng Rydberg states in 174Yb and 171Yb, together with multichannel quantum defect theory (MQDT) models that include channel interactions and hyperfine structure. New 174Yb 6snf 1,3F3 data bridge earlier laser and microwave measurements; a combined 6snp/6snf MQDT model for the 171Yb F=3/2 and F=5/2 odd-parity series introduces a single p-f mixing angle and resolves a previously reported dispersion-like residual in the 6snp (F=3/2) energies. For the ℓ=4 states, the authors show that the two J=4 series are better described in a jj-coupled basis than in LS coupling, and they report the first 171Yb 6sng data. The models are validated by out-of-fit measurements of Landé g-factors and static dipole polarizabilities, with generally good agreement. The paper closes with predicted 6snd Rydberg-Rydberg interaction potentials relevant to entangling gates. The MQDT code is made available through the open-source package rydcalc.

Significance. If the models are accepted, the paper provides a substantially more complete MQDT characterization of ytterbium Rydberg series (ℓ≤4) in both isotopes than previously available, with direct relevance to Rydberg-mediated quantum gates. The strengths of the paper are the extensive new data set with ~100 kHz uncertainties, the independent g-factor and polarizability checks that do not enter the energy fits, the resolution of a known residual in the 171Yb 6snp series, and the public release of the fitting software. The main risk is model uniqueness: the 6snf 1,3F3 model relies on an unidentified fifth 5d perturbing channel and an unusually large energy-dependent quantum defect, and the fit applies two post-hoc data corrections. These choices are acknowledged in the text but their impact on extrapolated wavefunctions—including those used for gate-relevant interaction potentials—is not quantified. The central derivation is sound, but these load-bearing points need additional robustness analysis before the quantitative predictive claims can be fully accepted.

major comments (3)
  1. [Section III A, Table S7] The seven-channel model for the 6snf 1,3F3 series relies on a fifth 5d perturbing channel placed above the lowest ionization threshold whose physical identity is unknown, and the fit requires μ(2)_{1F3} = −12.73, which the text itself calls 'unusually large' and says 'could indicate additional unaccounted perturbing states.' Since these channels and the strong energy dependence of the quantum defect directly determine the wavefunctions used in the polarizability and interaction-potential predictions of Sections VII and VIII, the paper should demonstrate robustness to alternative parameterizations—for example, by removing the fifth channel or by varying its threshold and quantum defect within plausible ranges, and by quantifying how the predicted g-factors, polarizabilities, and the spectra shown in Figure 11 change. Without such a test, the in-sample agreement does not establish that the extrapolated high-n wavefunctions are unbiased.
  2. [Section III A, Fig. 2] The fit excludes the Ref. [31] data with ν>65 as exhibiting 'large scattering' and applies a global −2.4 GHz offset to the Ref. [26] laser data. Both choices are post-hoc and affect the fitted MQDT parameters. Please provide a quantitative justification: show the residuals of the excluded points, test the sensitivity of the fitted parameters and predictions to including those points with larger uncertainties or without the offset, and state explicitly how the offset was determined from the overlap region and whether its uncertainty is consistent with the quoted 3 GHz accuracy of Ref. [26]. As written, the claimed sub-500 kHz agreement applies only after these corrections have been applied.
  3. [Section IV, Fig. 6(b)] The single direct transition-strength check of the p-f mixed wavefunction gives an experimental ratio 11.9(9) versus the MQDT prediction 8.0, a discrepancy of roughly 4σ that is attributed to frequency-dependent microwave intensity. This is the only out-of-fit probe of the mixing-angle wavefunction beyond g-factors, and at the stated precision it does not support the model. Please either measure the ratio with calibrated microwave intensity, or add a quantitative calibration uncertainty to the prediction and discuss what constraint on θ_pf remains. As it stands, the wavefunction validation for the p-f mixing is considerably weaker than the energy validation.
minor comments (5)
  1. [Table S20] The caption states that the transitions are between 6sn′f 3F3 states, but several rows list the lower state as 6snd 3F3 (e.g., '6s35f 3F3 6s35d 3F3'); this appears to be a typo and should be corrected.
  2. [Section III A / Fig. 2] The text says the Ref. [26] data are 'shifted' by 2.4 GHz while the Fig. 2 caption describes a 'global offset of −2.4 GHz to the term values'; please make the sign convention consistent in both places.
  3. [Section IV] The sentence 'The states with predominantly p character therefore have a O(θ2pf) ≈ 10−4 admixture of f states population' is awkward; it should say 'admixture of f-state population' or 'f-state amplitude' to avoid confusion between amplitude and probability.
  4. [Tables S17 and S23] There are typos in the table captions: 'spectrscopic' in Table S17 and 'desribed' in Table S23 should be corrected.
  5. [Section V, Table S1] The 174Yb 6sng 3G5 single-channel parameters are inferred from the 171Yb model, and the 171Yb F=11/2 series is predicted without direct measurement. This is stated, but the main text should reiterate the caveat wherever the 3G5 parameters are used for predictions of polarizabilities or interactions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the MQDT models are fitted to energy data, while the headline validations (g-factors and dc Stark polarizabilities) are independent measurements not used in the fits.

full rationale

The paper's derivation chain is a conventional empirical MQDT analysis: model parameters are obtained by global weighted fits to spectroscopic energies (Sec. II B), and the resulting models are then compared with separately measured observables. Section IV explicitly states that g-factors are used to 'probe the accuracy of the MQDT wavefunctions' after the energy fit, and Section VII compares predicted static dipole polarizabilities of 6snd states with independently measured dc Stark shifts; neither of these observables enters the parameter optimization. The p-f mixing angle in 171Yb is extracted from fitted energies, but the g-factor agreement near the avoided crossings is an independent wavefunction check, and the single transition-strength comparison (11.9(9) measured vs 8.0 predicted) is reported as a discrepancy attributed to microwave calibration, not as a fitted agreement. The seven-channel f model includes a phenomenological fifth 5d perturber above threshold and an 'unusually large' 1F3 quantum-defect curvature; the paper explicitly flags these as possible missing physics, which is a limitation rather than a circular step. The jj-coupling sign for the 171Yb g series is transferred from 174Yb g-factor measurements, a cross-isotope calibration, and the 3G5 parameters for 174Yb are inferred from 171Yb data; these are explicit parameter transfers, not predictions redefined as inputs. The same-group citation Ref. [13] supplies the MQDT framework, prior data compilation, and initial guesses, but the new models are globally refit to both prior and new data, and the central validations do not depend on Ref. [13] as an unverified premise. No equation or parameter in the paper is defined in terms of the quantity it is claimed to predict, so no circular step can be exhibited.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The central claims rest on dozens of fitted MQDT parameters: quantum defects, channel rotation angles, perturbing-channel thresholds, and mixing angles. The energy-level predictions are therefore not first-principles; the independent support comes from g-factor and polarizability measurements that were not included in the fits. The new fifth 5d perturber has no direct evidence, and low-n accuracy is acknowledged to be limited.

free parameters (7)
  • theta_pf (171Yb p-f mixing angle) = 1.842(8) x 10^-2
    Fit to odd-parity F=3/2 and 5/2 energy spectra in Sec. IV; controls the coupling between 6snp and 6snf channels.
  • theta_jj (171Yb G4 mixing angle) = -0.082(6)
    Fit to 171Yb g-state energies in Sec. VI; sign inferred from 174Yb g-factor measurements.
  • theta_LS (174Yb singlet-triplet mixing angle in LS basis) = 0.66(4)
    Fit to 174Yb g-factor data in Sec. V; compared with the analytic value 0.73 from the LS-jj frame transformation.
  • Quantum defect coefficients for 174Yb 6snf 3F2 = mu0 = 0.0718252326, mu2 = -1.00091963
    Single-channel energy-dependent quantum defects fit to microwave data in Tab. S22 over nu=27 to 47.
  • Quantum defect coefficients for 174Yb 6sng +/-G4 = mu0 = 0.0262659964 and -0.148808463; mu2 = 0.0254568575 and -0.134219071
    Fit to g-state energies in Tabs. S27 and S8 using a two-channel MQDT model.
  • Five 5d perturbing channel parameters in the 174Yb 6snf model = mu0 values from 0.175 to 0.239 and rotation angles theta13 through theta27
    Fit to 1,3F3 energies in Tab. S7; includes a fifth channel with no direct spectroscopic identification.
  • Singlet-triplet mixing angle function theta(nu) = -0.02084(4) + 0.239(2)/nu^2
    Fit to 171Yb hyperfine structure and applied to both isotopes in Sec. III A and Sec. IV.
assumptions (5)
  • domain assumption Frame transformation correctly describes the 171Yb hyperfine coupling between the Rydberg electron and the I=1/2 nucleus.
    Sec. II B and Sec. IV; 171Yb models start from 174Yb parameters and add hyperfine interaction through a frame transformation from Refs. [13, 35, 36].
  • domain assumption The p-f mixing arises from effective quadrupole interactions in the ionic core coupling channels with Delta-l = 2.
    Sec. IV cites H2 and noble-gas studies; this mechanism is assumed and only its strength, theta_pf, is fit.
  • ad hoc to paper The five 5d perturbing channels in the 174Yb 6snf model, including a fifth channel above the lowest threshold, are sufficient to capture the energy dependence of the quantum defects.
    Sec. III A; the paper notes the eigenchannel quantum defect remains unusually large and additional perturbing states may be missing.
  • domain assumption For l=4, the spin-orbit interaction dominates over the exchange interaction, so a jj-coupled basis is the correct zero-order description.
    Sec. V and VI; motivated by small quantum defects and prior alkaline-earth work, and supported by the measured g-factors.
  • ad hoc to paper The 2.4 GHz offset applied to Ref. [26] laser data and the exclusion of nu>65 data from Ref. [31] are valid corrections.
    Sec. III A and Fig. 2(b); the offset is within the quoted 3 GHz uncertainty, but the exclusion is a post-hoc choice that affects the fit.
invented entities (1)
  • Fifth 5d perturbing channel in the 174Yb 6snf model
    purpose: Captures the energy dependence of the 1F3 eigenchannel quantum defect that the four observed perturbers do not explain.
    Sec. III A states 'we found it necessary to include a fifth perturbing state with an energy above the lowest ionization threshold'; no direct experimental identification is given, and the paper says additional perturbing states may exist.

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Cite this review

Pith. "Pith review of Microwave spectroscopy and multi-channel quantum defect analysis of ytterbium Rydberg states." pith.science (2026). https://pith.science/paper/B4DLUDU3

@misc{pith2026250711487,
  author       = {Pith},
  title        = {Pith review of: Microwave spectroscopy and multi-channel quantum defect analysis of ytterbium Rydberg states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B4DLUDU3}},
  note         = {Machine review of arXiv:2507.11487}
}
abstract

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 $6sn\ell$ Rydberg states in $^{174}$Yb and $^{171}$Yb to include the $\ell = 3$ ($f$) and $\ell = 4$ ($g$) series. Our measurements reveal $p$-$f$ mixing in odd-parity Rydberg states of $^{171}$Yb, which we incorporate by combined MQDT models for $6snp$ and $6snf$ series. Additionally, we observe that for $\ell = 4$ the spin-orbit interaction dominates over the exchange interaction, such that the $6sng$ states are more accurately described in a $jj$-coupled basis. We validate our models by comparing the predicted Land\'e $g$-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.

Figures

Figures reproduced from arXiv: 2507.11487 by the authors.

Figure 1
Figure 1. (a) Schematic diagram of the spectroscopy setup [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) Lu-Fano-type plot of 174Yb odd-parity 6snf Ry￾dberg states. The colored circles correspond to MQDT model predictions. Black crosses indicate microwave spectroscopic data obtained in Ref. [31]. The black points are experimen￾tally measured energies by microwave spectroscopy reported in this work. The gray crosses are laser spectroscopy data from Refs. [26, 30]. The effect of a global offset of −2.4 GHz to the ter… view at source ↗
Figure 3
Figure 3. Schematic energy level diagram of the odd-parity 6 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: MQDT models for 171Yb odd-parity Rydberg series. (a) Lu-Fano-type plots of the F = 3/2 and (b) F = 5/2 series, respectively. The black points are microwave spectroscopy data measured in this work and in Ref. [13]. The gray crosses are laser spectroscopic data reported …
Figure 5
Figure 5. Figure 5: MQDT models for the 171Yb 6snp (F = 1/2) and 6snf (F = 7/2) Rydberg series. (a),(b) Lu-Fano-type plots. The black points correspond to microwave spectroscopic data, and colored points are MQDT bound state energy predictions. The gray crosses are three-photon laser spec…
Figure 6
Figure 6. Figure 6: (a) Deviation between experimental energies [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: (a) Lu-Fano-type plot of 174Yb ℓ = 4 Rydberg states: +G4 (dark blue), −G4 (light blue), 3G3 (green), and 3G5 (orange). The J = 4 series are fit to a single MQDT channel described in jj coupling. The 3G5 MQDT parame￾ters are inferred from the 171Yb model, as described i…
Figure 8
Figure 8. Figure 8: (a),(b) Lu-Fano-type plots of the G F = 5/2 (yel￾low), F = 7/2 (blue), F = 9/2 (pink), and F = 11/2 (light gray) series in 171Yb, converging to the Fc = 1 and Fc = 0 hy￾perfine thresholds, respectively. The black points correspond to experimentally observed states by m…
Figure 9
Figure 9. Figure 9: Measured (filled circles) and predicted (open circles) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: (a),(b) Measured (filled circles) and pre [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: Predicted 171Yb pair-interaction potentials for target Rydberg states |S⟩ = |ν, L = 0, F = 1/2, mF = −1/2⟩ at a magnetic field strength of 5 G perpendicular to the interatomic axis, shown for (a),(b) ν = 54.28 and (c),(d) ν = 53.30. Each pair of panels is referenced t…
Figure 12
Figure 12. Figure 12: Predicted absolute static dipole polarizabilities of [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Predicted absolute static dipole polarizabilities of (a),(b),(c) [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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    0 0 0 0 0 − √ 7/4 − p 3/7 p 2/7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1/2 √ 3/4 0 0 0 0 0 3 /4 25 Table S12. Six-channel MQDT model parameters for the |ν, L= 2, F= 3/2⟩ series of 171Yb, obtained by fitting to spectroscopic ...

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    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Table S14 (continued) i, ¯α,α 7 (Fc = 0) 8 9 10 11 ( Fc = 0) |i⟩ (6s1/2)(np3/2) 5 d e 5d f 5d g (6s1/2)(nf5/2) Ii (cm−1) 50 442 .795744 83 967 .7 83 967 .7 83 967 .7 50 443 .217463 | ¯α⟩ 6snp 3P2 5d e 5d f 5d g 6snf 3F2 µ...

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    II A and from Ref

    174Yb 1,3F3 Table S19: Spectroscopic data of 6 snf 1,3F3 ↔ 6sn′d 1,3D2 Rydberg-Rydberg transitions obtained by microwave spectroscopy in an atomic beam setup described in Sec. II A and from Ref. [31]. f i ν f ←i (MHz) ˜νexp. (cm−1 ) ˜ νth. (cm−1 ) Eexp. − Eth. Ref.(h · MHz) 6s...

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    174Yb 3F2 Table S22: Spectroscopic data of 6 snf 3F2 ↔ 6sn′d 1,3D2 Rydberg-Rydberg transitions obtained by microwave spectroscopy in an atomic beam setup described in Sec. II A. f i ν f ←i (MHz) ˜νexp. (cm−1 ) ˜ νth. (cm−1 ) Eexp. − Eth. (h · MHz) 6s29f 3F2 6s31d 1D2 −106 543....

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    The transitions are obtained by two-step microwave spectroscopy via an intermediate 6 sn′′f 3F3 state in an atomic beam setup described in Sec

    174Yb 3G3 Table S25: Spectroscopic data of 6 sng 3G3 ↔ 6sn′d 1,3D2 Rydberg-Rydberg transitions. The transitions are obtained by two-step microwave spectroscopy via an intermediate 6 sn′′f 3F3 state in an atomic beam setup described in Sec. II A. f i ν f ←i (MHz) ˜νexp. (cm−1 )...

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    These transitions are obtained by two-step microwave spectroscopy via an intermediate 6 sn′′f 3F3 state in an atomic beam setup described in Sec

    174Yb ±G4 Table S27: Spectroscopic data of 6 sng ±G4 ↔ 6sn′d 1,3D2 Rydberg-Rydberg transitions. These transitions are obtained by two-step microwave spectroscopy via an intermediate 6 sn′′f 3F3 state in an atomic beam setup described in Sec. II A. f i ν f ←i (MHz) ˜νexp. (cm−1...

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    174Yb 3D3 Table S29: Spectroscopic data of 6 snd 3D3 ↔ 6sn′d 3D2 Rydberg-Rydberg transitions obtained by two-photon microwave spectroscopy in an atomic beam setup described in Sec. II A. f i ν f ←i (MHz) ˜νexp. (cm−1 ) ˜ νth. (cm−1 ) Eexp. − Eth. (h · MHz) 6s46d 3D3 6s45d 3D2 ...

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    The experimental values for α0,exp

    174Yb 1,3D2 Table S30: Experimental ( α0,exp.) and predicted ( α0,theo.) static dipole polarizabilities of 6 snd 1D2 Rydberg states of 174Yb for mJ = 0 , 1, scaled by ν7 6s. The experimental values for α0,exp. are obtained by measurement of quadratic Stark shifts. Prediction f...

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    II A and from Ref

    171Yb P and F , F = 3/2 Table S32: Spectroscopic data of the odd-parity F = 3/2 Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A and from Ref. [13]. The transitions are from i = |ν, S, F= 1/2⟩ (a) and i = |ν, D, F= 5/2⟩ (b). νf...

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    171Yb P and F , F = 5/2 Table S34: Spectroscopic data of the odd-parity F = 5/2 Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A. The transitions are from i = |ν, D, F= 3/2⟩ (a) and i = |ν, D, F= 5/2⟩ (b). νf Fc=1 νi Fc=1 νf ←i...

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    171Yb F , F = 7/2 Table S36: Spectroscopic data of the |ν, ℓ= 3, F= 7/2⟩ Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A. The transitions are from i = |ν, D, F= 5/2⟩ states. νf Fc=1 νi Fc=1 νf ←i (MHz) ˜νexp. (cm−1 ) ˜ νth. (c...

  62. [70]

    171Yb G, F = 5/2 Table S38: Spectroscopic data of the |ν, ℓ= 4, F= 5/2⟩ Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A. The transitions are two-step: from a i = |ν, D, F= 5/2⟩ state and via an intermediate |ν, F, F= 7/2⟩ stat...

  63. [71]

    171Yb G, F = 7/2 Table S39: Spectroscopic data of the |ν, ℓ= 4, F= 7/2⟩ Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A. The transitions are two-step: from a i = |ν, D, F= 5/2⟩ state and via an intermediate |ν, F, F= 7/2⟩ stat...

  64. [72]

    171Yb G, F = 9/2 Table S40: Spectroscopic data of the |ν, ℓ= 4, F= 9/2⟩ Rydberg series obtained from microwave spectroscopy in an atomic beam setup described in Sec. II A. The transitions are two-step: from a i = |ν, D, F= 5/2⟩ state and via an intermediate |ν, F, F= 7/2⟩ stat...

  65. [73]

    The experimental values for α0,exp

    171Yb D, F = 3/2 Table S41: Experimental ( α0,exp.) and predicted ( α0,theo.) static dipole polarizabilities of |ν, ℓ= 2, F= 3/2⟩ Rydberg states of 171Yb, scaled by ν7 Fc=1. The experimental values for α0,exp. are obtained by measurement of quadratic Stark shifts. Prediction f...

  66. [74]

    The experimental values for α0,exp

    171Yb D, F = 5/2 Table S42: Experimental ( α0,exp.) and predicted ( α0,theo.) static dipole polarizabilities of |ν, ℓ= 2, F= 5/2⟩ Rydberg states of 171Yb, scaled by ν7 Fc=1. The experimental values for α0,exp. are obtained by measurement of quadratic Stark shifts. Prediction f...

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