Evaluation of trap-induced systematic frequency shifts for a multi-ion optical clock at the ${10}^{-19}$ level

In order to improve the short-term stability of trapped-ion optical clocks, we are developing a frequency standard based on ${}^{115}$In${}^{+}$ / ${}^{172}$Yb${}^{+}$ Coulomb crystals. For this purpose, we have developed scalable segmented Paul traps which allow a high level of control for multiple ion ensembles. In this article, we detail on our recent results regarding the reduction of the leading sources of frequency uncertainty introduced by the ion trap: 2nd-order Doppler shifts due to micromotion and the heating of secular motion, as well as the black-body radiation shift due to warming of the trap. We show that the fractional frequency uncertainty due to each of these effects can be reduced to well below ${10}^{-19}$.