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We report sympathetic cooling of $^{113}$Cd$^+$ by laser-cooled $^{40}$Ca$^+$in a linear Paul trap for microwave clocks. Long-term low-temperatureconfinement of $^{113}$Cd$^+$ ions was achieved. The temperature of these ionswas measured at $90(10)$ mK, and the corresponding uncertainty arising from thesecond-order Doppler shifts was estimated to a level of $2\times10^{-17}$. Upto $4.2\times10^5$ Cd$^+$ ions were confined in the trap, and the confinementtime constant was measured to be 84 hours. After three hours of confinement,there were still $10^5$ Cd$^+$ ions present, indicating that thisCa$^+$--Cd$^+$ dual ion system is surprisingly stable. The ac Stark shift wasinduced by the Ca$^+$ lasers and fluorescence, which was carefully estimated toan accuracy of $5.4(0.5)\times10^{-17}$ using a high-accuracy \textit{abinitio} approach. The Dick-effect-limited Allan deviation was also deducedbecause deadtimes were shorter. These results indicate that a microwave clockbased on this sympathetic cooling scheme holds promise in providing ultra-highfrequency accuracy and stability.

Sympathetic cooling of $^{113}$Cd$^+$ by laser-cooled $^{40}$Ca$^+$ in a linear Paul trap for Microwave Ion Clocks