Towards a Precise Test of King's Linearity in Ca$^+$

We report progress towards a precise measurement of the isotope shifts in the 4$^2$S$_{1/2}\rightarrow$3$^2$D$_{3/2}$ 732 nm electric quadrupole transition in Ca$^+$. We co-trap two isotopes and simultaneously excite both ions using frequency sidebands on a single laser, dramatically reducing systematic uncertainties from many sources such as laser frequency drift and magnetic field instabilities. Such measurements have the potential to reach Hz-level precision or better, as with our recent ppb measurement of the parallel 4$^2$S$_{1/2}\rightarrow$3$^2$D$_{5/2}$ transition. When combined into a King Plot, these two measurements will test King's linearity with heretofore unprecedented precision, offering a path toward probing new physics beyond the Standard Model and also providing benchmarks for ever-improving theory of atomic and nuclear structure.