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

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.

💡 Research Summary

This work presents the first direct frequency comparison between two multi‑ion optical clocks based on the S₁/₂ → D₅/₂ electric‑quadrupole transitions of 40Ca⁺ and 88Sr⁺. The authors trap linear crystals of up to nine Ca⁺ ions and six Sr⁺ ions in separate segmented linear Paul traps, and they stabilize a common laser system to both species via an optical frequency comb (OFC). A quasi‑continuous dynamical decoupling (QCDD) interrogation scheme is employed to average out the linear Zeeman shift and the quadrupole shift, which are the dominant inhomogeneous shifts in multi‑ion ensembles.

The 729 nm Ca⁺ clock laser is pre‑stabilized to a high‑finesse ultra‑low‑expansion (ULE) cavity and its frequency is offset‑locked to the Ca⁺ transition using an electro‑optic modulator (EOM). This laser also locks the OFC, which in turn transfers its coherence to the 674 nm Sr⁺ clock laser. By delivering both laser beams through a common fiber, common‑mode fiber noise is largely suppressed without active fiber‑noise cancellation.

Stability measurements were performed by servo‑locking the 729 nm laser to different numbers of Ca⁺ ions (1, 3, 9) while the Sr⁺ clock operated with six ions. The overlapping Allan deviation of the frequency ratio shows a clear improvement with ion number: the joint fractional frequency stability reaches 1.37(12) × 10⁻¹⁵ at 1 s for the nine‑ion Ca⁺ case. This corresponds to an upper bound on the individual clock stability of 9.6(8) × 10⁻¹⁶ at 1 s, comparable to the best single‑ion clocks despite the modest pre‑stabilized laser noise (~1 × 10⁻¹⁵). The observed scaling follows approximately √N, but deviates at larger N because laser white‑frequency noise (σ_L ≈ 1.1 × 10⁻¹⁵) begins to dominate over quantum projection noise (σ_N ≈ 2.2 × 10⁻¹⁵).

For the frequency ratio, the authors accumulated 170 h of data over two weeks, alternating the number of Sr⁺ ions (3–6) and varying the Sr⁺ trap RF power. Each run yields a statistical uncertainty of 1–3 × 10⁻¹⁷. The combined result is

R_Sr/Ca = 1.082076536381896986(18),

where the 1.8 × 10⁻¹⁷ relative uncertainty is dominated by systematic effects. The systematic uncertainty budget is dominated by black‑body radiation (BBR) shifts. The BBR sensitivities are Δν_Sr/ν ≈ 3.0616(46) × 10⁻¹¹ T⁴ and Δν_Ca/ν ≈ 4.7066(3) × 10⁻¹¹ T⁴. Although the chamber temperature is measured to 0.1 K, RF heating of the trap raises the effective BBR temperature seen by the ions. By measuring the frequency shift of the Sr⁺ clock versus trap RF power, the authors infer an effective BBR‑induced shift of 6(9) × 10⁻³ Hz W⁻¹, which they use to correct each data point. The residual BBR uncertainty contributes 1.8 × 10⁻¹⁷ to the ratio.

Other systematic contributions include second‑order Zeeman shifts (bias fields of ≈3 G for Sr⁺ and ≈2.6 G for Ca⁺), with uncertainties limited by the knowledge of the quadratic Zeeman coefficients (10⁻³ relative). Magnetic field fluctuations are kept below 10 µG by interleaved Ramsey spectroscopy and active coil feedback. Excess micromotion (EMM) is minimized by sideband spectroscopy in three directions; the residual axial modulation index is β < 0.2, and operation near the “magic” RF frequencies (24.825 MHz for Ca⁺, 16.15 MHz for Sr⁺) largely cancels the second‑order Doppler and scalar Stark shifts associated with micromotion. Height‑difference (gravitational red‑shift) and secular motion second‑order Doppler shifts are also evaluated and corrected.

Combining the measured ratio with recent absolute frequency measurements of the Sr⁺ transition referenced to a primary standard, the authors refine the absolute frequency of the Ca⁺ ¹S₀ → ³D₅/₂ transition to

ν_Ca⁺ = 411 042 129 776 400.21(4) Hz,

reducing its uncertainty by a factor of three.

The study demonstrates that multi‑ion optical clocks can achieve near‑quantum‑projection‑noise‑limited stability with modest laser pre‑stabilization, and that direct frequency ratio measurements between different species can reach 10⁻¹⁸‑level precision. These results open the way for using multi‑ion clocks in applications requiring fast averaging, such as relativistic geodesy, searches for temporal variations of fundamental constants, and tests of physics beyond the Standard Model.