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

We present a direct optical frequency comparison between two multi-ion trapped-ion clocks based on the S_1/2 - D_5/2 electric quadrupole transitions in Ca⁺ and Sr⁺. The clocks operate with linear ion chains containing up to 10 ions, enabling enhanced stability through the simultaneous interrogation of multiple ions. The measured joint stability reaches 1.37x10^-15 at one second, placing an upper bound of 9.6×10⁻¹⁶ for one of the clocks. In addition, we determine the frequency ratio between the two clocks to less than 3x10^-17, with the uncertainty dominated by the evaluation of room-temperature blackbody radiation shifts. This direct comparison improves the accuracy of existing indirect measurements of the Sr+/Ca+ frequency ratio by approximately one order of magnitude. By combining our result with recent absolute frequency measurements of the Sr+ transition referenced to a primary frequency standard, we can refine the absolute frequency of the Ca+ clock transition, reducing the uncertainty by a factor of three. Our results demonstrate the viability of multi-ion optical clocks for high-precision frequency ratio measurements and highlight their potential for applications in precision metrology, relativistic geodesy, and tests of fundamental physics.