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Chinese Physics B

Evaluation of the electric quadrupole shift in the ytterbium single-ion optical clock

Chao-Wei Wang, Shao-Yang Dai, Ya-Ni Zuo, Kun Liu, Wen-Bo Zhao, Yue-xiang Han, Fang Fang

Published Apr 17, 2026• 0 citations
Publisher

Abstract

Abstract Optical clocks with ultralow uncertainty are strong candidates for the next redefinition of the SI second. The 171 Yb + single-ion clock is particularly attractive owing to its two clock transitions (E2 and E3) and its potential for low uncertainty. A major contribution to the uncertainty budget arises from the electric-quadrupole shift, caused by the interaction between the atomic quadrupole moment and the electric-field gradient at the trap center and strongly dependent on the orientation of the quantization axis. To evaluate this shift, a simplified orthogonal magnetic-field method was employed: three mutually orthogonal magnetic-field directions are generated together with an auxiliary non-orthogonal field. With a dual-condition locking scheme, we measure the E2 transition frequency differences between magnetic-field configurations, infer the electric-field gradient, and apply corrections for the associated quadratic Zeeman shifts. The uncertainty due to magnetic-field misalignment is quantified using a calibrated current-to-field model. The resulting electric-quadrupole shift correction for the 2 S 1/2 → 2 F 7/2 (E3) transition is 3.24 mHz, corresponding to a fractional uncertainty of 1.1×10 −18 in the absolute frequency measurement.

DOI
10.1088/1674-1056/ae60eb

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Physical Review A•Apr 22, 2011•21 citations
Abstract Optical clocks with ultralow uncertainty are strong candidates for the next redefinition of the SI second. The 171 Yb + single-ion clock is particularly attractive owing to its two clock transitions (E2 and E3) and its potential for low uncertainty. A major contribution to the uncertainty budget arises from the electric-quadrupole shift, caused by the interaction between the atomic quadrupole moment and the electric-field gradient at the trap center and strongly dependent on the orientation of the quantization axis. To evaluate this shift, a simplified orthogonal magnetic-field method was employed: three mutually orthogonal magnetic-field directions are generated together with an auxiliary non-orthogonal field. With a dual-condition locking scheme, we measure the E2 transition frequency differences between magnetic-field configurations, infer the electric-field gradient, and apply corrections for the associated quadratic Zeeman shifts. The uncertainty due to magnetic-field misalignment is quantified using a calibrated current-to-field model. The resulting electric-quadrupole shift correction for the 2 S 1/2 → 2 F 7/2 (E3) transition is 3.24 mHz, corresponding to a fractional uncertainty of 1.1×10 −18 in the absolute frequency measurement.
An optical clock based upon a single mercury ion in a radio-frequency trap has the potential for both high stability and high accuracy. The systematic uncertainty of this frequency standard is limited by contributions due to two types of external fields. The clock has previously been operated with a magnetic bias field that destabilizes dark states that limit the rates of laser cooling and state detection. This bias field has previously resulted in a substantial second-order Zeeman shift. We now destabilize the dark states with a polarization-scrambling technique that allows us to operate with only a minimal magnetic field. The second uncertainty that we address here is the magnitude of the quadrupole shift due to the interaction of stray electric-field gradients with the quadrupole moment of the ion. We describe the steps that we are taking to evaluate this shift in the near future. 1
Frequency standards based on optical transitions between electronic states of trapped ions have reached uncertainties below 10−18 [1]. The Stark shift caused by room-temperature blackbody radiation (BBR) causes the largest correction for most high-performance optical clock. The systematic uncertainty of 171Yb+ clocks based on the 2S1/2 (F = 0) → 2D3/2 (F = 2) electric quadrupole (E2) and 2S1/2 (F = 0) → 2F7/2 (F = 3) electric octupole (E3) transitions is currently limited to 28×10−18 and 1.5×10−18, respectively, by the fractional accuracy of the differential polarizability Δαdc of approximately 2% [2]. In the scope of this thesis, a dual species clock based on 171Yb+ and 88Sr+ ions was set up, operated and evaluated, to overcome this barrier. Since Δαdc is known with a smaller fractional uncertainty of 0.04% for the 2S1/2 → 2D5/2 E2 transition of 88Sr+ [3], this allows for a transfer of Δαdc from 88Sr+ to 171Yb+ following a method proposed in [4]. The optical intensity of a laser is calibrated with the Stark shift it causes on the 88Sr+ clock transition, allowing for the determination of Δαdc of the 171Yb+ clock transitions. This allows for a reduction of the fractional uncertainty of Δαdc for both 171Yb+ clock transitions below 0.4% and clock operation on the E2 and E3 transition with a fractional BBR shift uncertainty of 0.84×10−18 and 0.24×10−18 is enabled, respectively. However the value of Δαdc determined in this thesis does not agree with [2], a fractional offset of approximately 14% is observed. To investigate this discrepancy further, Δαdc is measured in the same manner as in [2] for the 88Sr+ clock transition. The same offset of approximately 14% is found, calling the accuracy of at least one of these methods into question. Finally, the first optical frequency ratio ℛ between the 88Sr+ clock transition and the 171Yb+ E3 transition is measured. The fractional uncertainty of ℛ is 23×10−18. ℛ in combination with the absolute frequency of the E3 transition allows for the determination of the 88Sr+ absolute frequency νSr+ with an uncertainty limited by the caesium references. The value of νSr+ falls outside of the recommended range, but is corroborated by measurements of the same quantity by other institutes [5, 6]. [1] M. Marshall et al., Phys. Rev. Lett. 135, 033201 (2025) [2] N. Huntemann et al., Phys. Rev. Lett. 116, 063001 (2016) [3] T. Lindvall et al., Phys. Rev. Lett. 135, 043402 (2025) [4] M. D. Barrett et al., Phys. Rev. A 100, 043418 (2019) [5] C. Marceau et al., Metrologia 62.4, 045001 (2025) [6] T. Lindvall et al., Phys. Rev. Appl. 24, 044082 (2025)
We explore the potential of the electric quadrupole transitions 7s 2S1/27s\,^2S_{1/2}7s2S1/2​ - 6d 2D3/26d\,^2D_{3/2}6d2D3/2​, 6d 2D5/26d\,^2D_{5/2}6d2D5/2​ in radium isotopes as single-ion optical frequency standards. The frequency shifts of the clock transitions due to external fields and the corresponding uncertainties are calculated. Several competitive A^AARa+^++ candidates with A=A=A= 223 - 229 are identified. In particular, we show that the transition 7s 2S1/2 (F=2,mF=0)7s\,^2S_{1/2}\,(F=2,m_F=0)7s2S1/2​(F=2,mF​=0) - 6d 2D3/2 (F=0,mF=0)6d\,^2D_{3/2}\,(F=0,m_F=0)6d2D3/2​(F=0,mF​=0) at 828 nm in 223^{223}223Ra+^++, with no linear Zeeman and electric quadrupole shifts, stands out as a relatively simple case, which could be exploited as a compact, robust, and low-cost atomic clock operating at a fractional frequency uncertainty of 10−1710^{-17}10−17. With more experimental effort, the 223,225,226^{223,225,226}223,225,226Ra+^++ clocks could be pushed to a projected performance reaching the 10−1810^{-18}10−18 level.
The electric quadrupole shift is presently the most significant source of uncertainty on the systematic shifts for several single-ion optical frequency standards. We present a simple method for cancelling this shift based on measurements of the Zeeman spectrum of the clock transition. This method is easy to implement and yields very high cancellation levels. A fractional uncertainty of 5 x 10(-18) for the canceled quadrupole shift is estimated for a measurement of the absolute frequency of the 5s (2)S(1/2)-4d (2)D(5/2) clock transition of 88Sr+.
We report experimental investigations of a single-ion optical frequency standard based on 171Yb+{^{171}\text{Y}\text{b}}^{+}171Yb+. The ion is confined in a cylindrically symmetric radiofrequency Paul trap. The reference transition is the ${^{2}S}_{1/2}(F=0)--{^{2}D}_{3/2}({F}^{\ensuremath{'}}=2)$ electric quadrupole transition at 688 THz. Using a differential measurement scheme, we determine the shift of the reference transition frequency that occurs due to the interaction of the electric quadrupole moment of the 2D3/2{^{2}D}_{3/2}2D3/2​ state with the gradient of the electrostatic stray field in the trap. We determine an upper limit for the instability of the quadrupole shift over times between 100 s to 20 h. We also observe the variations in the shift and in the applied stray-field compensation voltages that result from loading a new ion into the trap and during a subsequent storage period of 74 days. This information is utilized to measure the absolute frequency of the reference transition with an uncertainty that is a factor of 3 smaller than that of the previous measurement. Using a fiber laser based optical frequency comb generator and the cesium fountain clock CSF1 of PTB (Physikalisch-Technische Bundesanstalt), the frequency at 300 K temperature is determined as $688\text{ }358\text{ }979\text{ }309\text{ }306.62\ifmmode\pm\else\textpm\fi{}0.73\text{ }\text{Hz}$.
We explore the potential of the electric quadrupole transitions 7s2S1/27s {}^{2}{S}_{1/2}7s2S1/2​-6d2D3/26d {}^{2}{D}_{3/2}6d2D3/2​, 6d2D5/26d {}^{2}{D}_{5/2}6d2D5/2​ in radium isotopes as single-ion optical frequency standards. The frequency shifts of the clock transitions due to external fields and the corresponding uncertainties are calculated. Several competitive A{}^{A}ARa+{}^{+}+ candidates, with A=A=A= 223--229, are identified. In particular, we show that the transition 7s2S1/2(F=2,mF=0)7s {}^{2}{S}_{1/2} (F=2,{m}_{F}=0)7s2S1/2​(F=2,mF​=0)-6d2D3/2(F=0,mF=0)6d {}^{2}{D}_{3/2} (F=0,{m}_{F}=0)6d2D3/2​(F=0,mF​=0) at 828 nm in 223Ra^{223}\mathrm{Ra}223Ra+{}^{+}+, with no linear Zeeman and electric quadrupole shifts, stands out as a relatively simple case, which could be exploited as a compact, robust, and low-cost atomic clock operating at a fractional frequency uncertainty of ${10}^{\ensuremath{-}17}$. With more experimental effort, the 223,225,226Ra^{223,225,226}\mathrm{Ra}223,225,226Ra+{}^{+}+ clocks could be pushed to a projected performance reaching the ${10}^{\ensuremath{-}18}$ level.