High-accuracy multi-ion spectroscopy with mixed-species Coulomb crystals

Multi-ion optical clocks offer the possibility of overcoming the low signal-to-noise ratio of single-ion clocks, while still providing low systematic uncertainties. We present simultaneous spectroscopy of up to four ${}^{115}$In${}^{+}$ clock ions in a linear Coulomb crystal, sympathetically cooled with ${}^{172}$Yb${}^{+}$ ions. In first clock comparisons, we see agreement below $1\times {10}^{-17}$ with results obtained using a single In${}^{+}$ ion, for which we have evaluated the systematic uncertainty to be $2.5\times {10}^{-18}$. Operation with four clock ions reduces the instability from $1.6\times {10}^{-15}\mathrm{/}\sqrt{t\mathrm{/}(1\mathrm{s})}$ to $9.2\times {10}^{-16}\mathrm{/}\sqrt{t\mathrm{/}(1\mathrm{s})}$. We derive a model for decay-related dead time during state preparation, which matches the observed scaling of instability with clock ion number $N$, and indicates that $1\mathrm{/}\sqrt{N}$ scaling can be achieved with the addition of a repump laser.