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

TL;DR

This study delivers the first direct frequency comparison between multi-ion optical clocks based on Ca$^+$ and Sr$^+$, achieving $1.37(12)\times10^{-15}$ stability at $1\,$s for the two-clock ensemble and an upper bound of $9.6(8)\times10^{-16}$ for a single clock. It directly measures the Sr/Ca clock-frequency ratio as $R_{\text{Sr/Ca}}=1.082076536381896986(18)$ with $1.8\times10^{-17}$-level precision, dominated by BBR systematics, and demonstrates a substantial improvement over prior indirect determinations. By combining the ratio with Sr$^+$ absolute-frequency measurements, the Ca$^+$ absolute frequency is refined to $\nu_{\text{Ca}^+}=411042129776400.21(4)$ Hz, reducing uncertainty by a factor of three. The results underscore the viability and advantages of multi-ion clock architectures for precision metrology, geodesy, and fundamental-physics tests, and set the stage for future ultra-stable, potentially transportable optical clocks.

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.

Figures (4)

• Figure 1: Experimental setup for the direct comparison of multi-ion Ca$^+$ and Sr$^+$ optical clocks. The 729 nm Ca$^+$ clock laser is pre-stabilized to a ULE cavity and locked to the Ca$^+$$S_{1/2}\!\rightarrow\!D_{5/2}$ transition via an EOM offset. It stabilizes an OFC, which transfers coherence to the pre-stabilized 674 nm Sr$^+$ clock laser. The light of the two lasers is delivered to the OFC through a common fiber to suppress fiber-induced phase noise. No active stabilization of fiber noise is used in the setup.
• Figure 2: Clock stability vs. number of ions. The main figure presents an overlapping Allan deviation of the frequency ratio of two clocks for three different numbers of Ca$^+$ ions used in the laser servo, and a fixed number of six Sr$^+$ ions. Solid lines represent linear fits on a log-log scale, with data points at longer time scales (indicated by large markers). The inset shows our extracted stability at one second, from the fits, as a function of the number of Ca$^+$ ions. The dashed black line is a fit to $1/\sqrt{N}$, and the solid black line is a fit that includes residual laser noise. The dotted lines indicate the calculated pure QPN of the two clocks.
• Figure 3: Frequency ratio of the $^{88}$Sr$^+$ and $^{40}$Ca$^+$ clock transitions. vhv fh;;;;(a) Individual measurement runs corrected for all the systematic effects collected over two weeks. Symbols indicate the number of Sr$^+$ ions that were used in the measurement. The color corresponds to the two different trap RF powers for which the Sr$^+$ clock was operated, as presented in (b). The mean, standard deviation, and statistical uncertainty of the mean are indicated by the dashed line, red shaded area, and gray shaded area, respectively. (b) Frequency shift of the Sr$^+$ clock, extracted from the measured ratio, as a function of trap RF drive power due to the BBR beyond the measured chamber temperature. The linear fit and uncertainty are indicated by the dashed black line and red shaded area, respectively. We used this analysis to correct for the BBR shift in the ratio results of (a).
• Figure 4: The direct measured frequency ratio from this work compared to indirect combined published values from Zhang(2023)zhang2023absolute for Ca$^+$ and Lindvall(2025)lindvall202588, Marceau(2025)marceau2025absolute and Steinel(2023)steinel2023evaluation for Sr$^+$. The current CIPM2021Margolis2024CIPM recommendation is shown as well. Note the break in the x-axis scale.