Suppressing inhomogeneous broadening in a lutetium multi-ion optical clock

We demonstrate precision measurement and control of inhomogeneous broadening in a multi-ion clock consisting of three ${}^{176}$Lu${}^{+}$ ions. Microwave spectroscopy between hyperfine states in the ${}^3{D}_1$ level is used to characterise 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 $\sim 10^{-17}$levelrelativetothe$^1S_0\leftrightarrow{}^3D_1$ optical transition frequency. Correlation spectroscopy on the optical transition demonstrates the feasibility of a 10s Ramsey interrogation in the three ion configuration with a corresponding projection noise limited stability of $\sigma (\tau )=8.2\times {10}^{-17}\mathrm{/}\sqrt{\tau }$