We demonstrate precision measurement and control of inhomogeneous broadening in a multi-ion clock consisting of three ${}^{176}{\mathrm{Lu}}^{+}$ ions. Microwave spectroscopy between hyperfine states in the ${}^3{D}_1$ level is used to characterize differential systematic shifts between ions, most notably those associated with the electric quadrupole moment. By appropriate alignment of the magnetic field, we demonstrate suppression of these effects to the $\sim {10}^{-17}$ level relative to the ${{}^{1}S}_0\leftrightarrow {{}^{3}D}_1$ optical transition frequency. Correlation spectroscopy on the optical transition demonstrates the feasibility of a 10-s Ramsey interrogation in the three ion configuration with a corresponding projection noise limited stability of $\sigma (\tau )=8.2\times {10}^{-17}/\sqrt{\tau }$.

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• Atomic, optical & lattice clocks