Controlling systematic frequency uncertainties at the $10^{-19}$ level in linear Coulomb crystals

Trapped ions are ideally suited for precision spectroscopy, as is evident from the remarkably low systematic uncertainties of single-ion clocks. The major weakness of these clocks is the long averaging time, necessitated by the low signal of a single atom. An increased number of ions can overcome this limitation and allow for the implementation of novel clock schemes. However, this presents the challenge to maintain the excellent control over systematic shifts of a single particle in spatially extended and strongly coupled many-body systems. We measure and deduce systematic frequency uncertainties related to spectroscopy with ion chains in a newly developed rf trap array designed for precision spectroscopy on simultaneously trapped ion ensembles. For the example of an In${}^+$ clock, sympathetically cooled with Yb${}^+$ ions, we show in our system that the expected systematic frequency uncertainties related to multi-ion operation can be below $1\times10^{-19}$. Our results pave the way to advanced spectroscopy schemes such as entangled clock spectroscopy and cascaded clock operation.