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arXiv:2602.05626

Direct comparison of multi-ion optical clocks based on $^{40}$Ca$^+$ and $^{88}$Sr$^+$

Yosef Sokolik, Roee Ozeri, Nitzan Akerman

Published Feb 5, 2026• 0 citations
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Abstract

We report the first direct frequency comparison between two multi-ion optical clocks based on the S$_{1/2}$ to D$_{5/2}$ transition in \Ca and \Sr ions. Using linear chains of up to nine \Ca ions and six \Sr ions, we demonstrate improved stability as a function of the number of ions that are contributing to the laser frequency stabilization servo. The measured joint fractional frequency stability of the two clocks reaches $1.37(12)\times 10^{-15}$ at one second, placing an upper bound on the same stability of one of the clocks at $9.6(8)\times 10^{-16}$ in one second. We measured the frequency ratio of the two clocks to be $R_{\text{Sr/Ca}}=1.082076536381896986(18)$, where the systematic uncertainty is primarily limited by the room temperature blackbody radiation. Our direct measurement represents an order of magnitude improvement compared to existing indirect frequency ratio measurements. Furthermore, by combining our results with recent absolute frequency measurements of the \Sr transition, referenced to a primary frequency standard, we refined the absolute frequency of the \Ca transition to $ν_{\text{Ca}^+}=411042129776400.21(4)$ Hz, reducing its uncertainty by a factor of three. This study presents the first direct comparison between two multi-ion optical clocks, highlighting their significant potential for future applications in fundamental physics tests, geodesy, and precision metrology.

DOI
10.48550/arxiv.2602.05626

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Direct comparison of multi-ion optical clocks based on 40^{40}40Ca+^++ and 88^{88}88Sr+^++
We report the first direct frequency comparison between two multi-ion optical clocks based on the S1/2_{1/2}1/2​ to D5/2_{5/2}5/2​ transition in \Ca and \Sr ions. Using linear chains of up to nine \Ca ions and six \Sr ions, we demonstrate improved stability as a function of the number of ions that are contributing to the laser frequency stabilization servo. The measured joint fractional frequency stability of the two clocks reaches 1.37(12)×10−151.37(12)\times 10^{-15}1.37(12)×10−15 at one second, placing an upper bound on the same stability of one of the clocks at 9.6(8)×10−169.6(8)\times 10^{-16}9.6(8)×10−16 in one second. We measured the frequency ratio of the two clocks to be RSr/Ca=1.082076536381896986(18)R_{\text{Sr/Ca}}=1.082076536381896986(18)RSr/Ca​=1.082076536381896986(18), where the systematic uncertainty is primarily limited by the room temperature blackbody radiation. Our direct measurement represents an order of magnitude improvement compared to existing indirect frequency ratio measurements. Furthermore, by combining our results with recent absolute frequency measurements of the \Sr transition, referenced to a primary frequency standard, we refined the absolute frequency of the \Ca transition to νCa+=411042129776400.21(4)ν_{\text{Ca}^+}=411042129776400.21(4)νCa+​=411042129776400.21(4) Hz, reducing its uncertainty by a factor of three. This study presents the first direct comparison between two multi-ion optical clocks, highlighting their significant potential for future applications in fundamental physics tests, geodesy, and precision metrology.
We report on an evaluation of an optical clock that uses the ^{2}S_{1/2}→^{2}D_{5/2} transition of a single ^{88}Sr^{+} ion as the reference. In contrast to previous work, we estimate the effective temperature of the blackbody radiation that shifts the reference transition directly during operation from the corresponding frequency shift and the well-characterized sensitivity to thermal radiation. We measure the clock output frequency against an independent ^{171}Yb^{+} ion clock, based on the ^{2}S_{1/2}(F=0)→^{2}F_{7/2}(F=3) electric octupole (E3) transition, and determine the frequency ratio with a total fractional uncertainty of 2.3×10^{-17}. Relying on a previous measurement of the ^{171}Yb^{+} (E3) clock frequency, we find the absolute frequency of the ^{88}Sr^{+} clock transition to be 444 779 044 095 485.277(59) Hz. Our result reduces the uncertainty by a factor of 3 compared with the previously most accurate measurement and may help to resolve so far inconsistent determinations of this value. We also show that for three simultaneously interrogated ^{88}Sr^{+} ions, the increased number causes the expected improvement of the short-term frequency instability of the optical clock without degrading its systematic uncertainty.
We recently reported a 5×10-17level frequency comparison of two40Ca+optical clocks, one of which has fractional frequency uncertainty of 3.4×10-17, limited by the BBR effect. Here we present progress towards a40Ca+optical clock with uncertainty at the 10-18level. The differential static scalar polarizability Δα0 of the40Ca+ion clock transition is obtained by precisely measuring the “magic” rf drive frequency. Δα0 is measured as -7.2653(44)x10-40J m2/V2, the blackbody radiation shift is then calculated to be 0.3790(2) Hz at 300 K. The contribution of the blackbody shift coefficient to the uncertainty of the optical clock at room temperature is reduced to 5.6×10-19, the excess micromotion induced clock uncertainty is also reduced to the 10-19level by choosing the appropriate trap drive frequency. With a suppression of the BBR field temperature uncertainty to the sub-Kelvin level and improved clock stability, a robust, relatively low-cost40Ca+ion clock with uncertainty at 10-18level can be made.
Progress of the 40Ca+ ion optical clock based on the 42S1/2 – 3d 2D5/2 electric quadrupole transition is reported. By setting the drive frequency to the “magic” frequency Ω 0, the frequency uncertainty caused by the scalar Stark shift and second-order Doppler shift induced by micromotion is reduced to the 10−19 level. By precisely measuring the differential static scalar polarizability Δα 0, the uncertainty due to the blackbody radiation (BBR) shift (coefficient) is reduced to the 10−19 level. With the help of a second-order integrating servo algorithm, the uncertainty due to the servo error is reduced to the 10−18 level. The total fractional uncertainty of the 40Ca+ ion optical clock is then improved to 2.2× 10−17, whereas this value is mainly restricted by the uncertainty of the BBR shift due to temperature fluctuations. The state preparation is introduced together with improvements in the pulse sequence, and furthermore, a better signal to noise ratio (SNR) and less dead time are achieved. The clock stability of a single clock is improved to 4.8 × 10 − 15 / τ ( in seconds ) .
We report high-precision frequency ratio measurements between optical atomic clocks based on ^{27}Al^{+}, ^{171}Yb, and ^{87}Sr. With total fractional uncertainties at or below 3.2×10^{-18}, these measurements meet an important milestone criterion for redefinition of the second in the International System of Units. Discrepancies in ^{87}Sr ratios at approximately 1×10^{-16} and the Al^{+}/Yb ratio at 1.6×10^{-17} in fractional units compared to our previous measurements underscore the importance of repeated, high-precision comparisons by different laboratories. A key upgrade from our previous work is the use of a common ultrastable reference delivered to all clocks via a 3.6-km phase-stabilized fiber link between two institutions, enabling better accuracy and stability in frequency transfer. Derived from a cryogenic single-crystal silicon cavity, this reference improves comparison stability by a factor of 2-3 over previous systems, with an optical lattice clock ratio achieving a fractional instability of 1.3×10^{-16} at 1 s. Identifying individual clock stabilities with a three-corner-hat measurement, we demonstrate the most stable optical lattice and trapped-ion clocks used in a multispecies comparison. By enabling faster comparisons, this stability will improve sensitivity to nonwhite noise processes and other underlying limits of state-of-the-art optical frequency standards.
A multi-ion optical clock with 5×10−19\mathbf{5 \times 10^{-19}}5×10−19 uncertainty
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Advances in optical clocks motivate a redefinition of the second, requiring rigorous evaluations of systematic uncertainties and robust consistency among the clocks. Here, we report the full evaluation of the systematic frequency shifts of an 27Al+^{27}\mathrm{Al}^+27Al+ single-ion clock, and the measurement of its absolute frequency and frequency ratio with a 87^{87}87Sr optical lattice clock at PTB. The evaluated total systematic fractional frequency uncertainty is 1.6×10−181.6\times10^{-18}1.6×10−18, mainly limited by the accuracy of the relevant atomic coefficients and by background gas collisions. The absolute frequency of the clock has been measured to be νAl+=1121015393207859.19(24) ν_{\mathrm{Al}^+}=1 121 015 393 207 859.19(24)\,νAl+​=1121015393207859.19(24)Hz, obtained by comparison with two primary caesium fountain clocks at PTB. The frequency ratio between the Al and Sr optical clocks has been determined to be νAl+/ν87Sr=2.611701431781462668(36)ν_{\mathrm{Al}^+}/ν_{^{87}\mathrm{Sr}}=2.611 701 431 781 462 668(36)νAl+​/ν87Sr​=2.611701431781462668(36), limited by the accuracy of the Sr clock. This ratio differs by 8.6σ8.6σ8.6σ and 1.2σ1.2σ1.2σ from the 2021 and 2025 frequency ratio published by the BACON collaboration, respectively. These results represent an important contribution toward a future redefinition of the second using optical clocks, and underscore the importance of independent measurements of clock-candidate frequency ratios across different institutions.