We present a liquid-nitrogen-cooled $\mathrm{Ca}$${}^{+}$ optical clock with an overall systematic uncertainty of $3.0\times {10}^{-18}$. In contrast to the room-temperature $\mathrm{Ca}$${}^{+}$ optical clock that we have reported previously, the cryogenic black-body radiation (BBR) shield in vacuum is cooled to $82\pm 5$ K using liquid nitrogen. We also implement an ion trap with a reduced heating rate and improved laser cooling. This allows the ion temperature to fall to the Doppler-cooling limit during the clock operation and the systematic uncertainty associated with the secular (thermal) motion of the ion is reduced to $<1\times {10}^{-18}$. The uncertainty arising from the probe laser light shift and the servo error is also reduced to $<1\times {10}^{-19}$ and $4\times {10}^{-19}$ with the hyper-Ramsey method and the higher-order servo algorithm, respectively. By comparing the output frequency of the cryogenic clock to that of a room-temperature clock, the differential BBR shift between the two is determined with a fractional statistical uncertainty of $7\times {10}^{-18}$. The differential BBR shift is used to calculate the static differential polarizability and the result is found to be in excellent agreement with our previous measurement using a different method. This work suggests that the BBR shift of optical clocks can be suppressed well in a liquid-nitrogen environment. Systems similar to what is presented here can also be used to suppress the BBR shift significantly in other types of optical clocks, such as $\mathrm{Yb}$${}^{+}$, $\mathrm{Sr}$${}^{+}$, $\mathrm{Yb}$, Sr, etc.

• Received 7 April 2021
• Revised 23 December 2021
• Accepted 3 January 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.034041