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We report on Raman sideband cooling of ${^{25}\text{Mg}^+}$ tosympathetically cool the secular modes of motion in a$^{25}\text{Mg}^+$-$^{27}\!\text{Al}^+$ two-ion pair to near thethree-dimensional (3D) ground state. The evolution of the Fock-statedistribution during the cooling process is studied using a rate-equationsimulation, and various heating sources that limit the efficiency of 3Dsideband cooling in our system are discussed. We characterize the residualenergy and heating rates of all of the secular modes of motion and estimate asecular motion time-dilation shift of ${-(1.9 \pm 0.1)\times 10^{-18}}$ for an${^{27}\text{Al}^+}$ clock at a typical clock probe duration of $150$ ms. Thisis a 50-fold reduction in the secular motion time-dilation shift uncertainty incomparison with previous ${^{27}\text{Al}^+}$ clocks.

Sympathetic ground state cooling and time-dilation shifts in an $^{27}\text{Al}^+$ optical clock