Based upon an over-one-month frequency comparison of two ${}^{40}{\mathrm{Ca}}^{+}$ optical clocks, the frequency difference between the two clocks is measured to be $3.2\times {10}^{-17}$ with a measurement uncertainty of $5.5\times {10}^{-17}$, considering both the statistic ($1.9\times {10}^{-17}$) and the systematic ($5.1\times {10}^{-17}$) uncertainties. This is the first performance of a ${}^{40}{\mathrm{Ca}}^{+}$ clock better than that of Cs fountains. A fractional stability of $7\times {10}^{-17}$ in 20 000 s of averaging time is achieved. The evaluation of the two clocks shows that the shift caused by the micromotion in one of the two clocks limits the uncertainty of the comparison. By carefully compensating the micromotion, the absolute frequency of the clock transition is measured to be 411 042 129 776 401.7(1.1) Hz.

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.

We present a robust, transportable ${\mathrm{Ca}}^{+}$ optical clock, with a systematic uncertainty of $1.3\times {10}^{-17}$ limited by the black-body radiation (BBR) field evaluation and an uptime rate of $>75\%$ over a 20-day period. The clock is then installed in an air-conditioned car trailer, making it more convenient for applications. Referenced to a stationary laboratory clock, geopotential measurements are made with the transportable clock with a total uncertainty of 0.33 m (statistically 0.25 m and systematically 0.22 m) and agree with the spirit level measurement. After being moved $>1200$ km, the absolute frequency of the ${\mathrm{Ca}}^{+}$ optical clock transition is measured as 411 042 129 776 400.41(23) Hz, with a fractional uncertainty of $5.6\times {10}^{-16}$, which is about one order of magnitude smaller than our previous measurement. The transportable built can be used for sub-meter-level elevation measurements, comparing intercontinental optical clocks, verifying basic physical theories, etc.

We demonstrate experimentally the existence of magic wavelengths and determine the ratio of oscillator strengths for a single trapped ion. For the first time, two magic wavelengths near 396 nm for the ${{}^{40}\mathrm{Ca}}^{+}$ clock transition are measured simultaneously with high precision. By tuning the applied laser to an intermediate wavelength between transitions $4s_{1/2}\to 4p_{1/2}$ and $4s_{1/2}\to 4p_{3/2}$, the sensitivity of the clock transition Stark shift to the oscillator strengths is greatly enhanced. Furthermore, with the measured magic wavelengths, we determine the ratio of the oscillator strengths with a deviation of less than 0.5%. Our experimental method may be applied to measure magic wavelengths for other ion clock transitions. Promisingly, the measurement of these magic wavelengths paves the way to building all-optical trapped ion clocks.

We recently reported a ${}^{40}{\mathrm{Ca}}^{+}$ optical clock comparison with an uncertainty at $5.5\times {10}^{-17}$, mainly limited by the excess micromotion shift. Here we report the progress made to reduce this shift and its uncertainty below $1\times {10}^{-18}$ by precise measurement of the “magic” rf drive frequency ${\mathrm{\Omega }}_0$ at which the micromotion-induced scalar Stark shift and second-order Doppler shift cancel each other. ${\mathrm{\Omega }}_0$ is measured as $2\pi \times 24.801\left(2\right)\mathrm{MHz}$, and the differential static scalar polarizability $\mathrm{\Delta }{\alpha }_0$ of the ${}^{40}{\mathrm{Ca}}^{+}$ ion clock transition is measured as $-7.2677\left(21\right)\times {10}^{-40}\mathrm{J}{\mathrm{m}}^2{\mathrm{V}}^{-2}$. The blackbody radiation shift is then calculated to be 0.37913(12) Hz at 300 K considering the dynamic correction. The contribution of the blackbody shift coefficient to the uncertainty of the optical clock at room temperature has been reduced to the $3\times {10}^{-19}$ level. With further improvements made to reduce the servo error, the total clock uncertainty is reduced to $2.2\times {10}^{-17}$, limited by the blackbody radiation field evaluation.

The hyperfine structures of the $2{{}^{3}S}_1$ states of the ${}^6{\mathrm{Li}}^{+}$ and ${}^7{\mathrm{Li}}^{+}$ ions are investigated theoretically to extract the Zemach radii of the ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$ nuclei by comparing with precision measurements. The obtained Zemach radii are larger than the previous values of Puchalski and Pachucki [ Phys. Rev. Lett. 111, 243001 (2013)] and disagree with them by about 1.5 and 2.2 standard deviations for ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$, respectively. Furthermore, our Zemach radius of ${}^6\mathrm{Li}$ differs significantly from the nuclear physics value, derived from the nuclear charge and magnetic radii [ Phys. Rev. A 78, 012513 (2008)] by more than $6\sigma$, indicating an anomalous nuclear structure for ${}^6\mathrm{Li}$. The conclusion that the Zemach radius of ${}^7\mathrm{Li}$ is about 40% larger than that of ${}^6\mathrm{Li}$ is confirmed. The obtained Zemach radii are used to calculate the hyperfine splittings of the $2{{}^{3}P}_J$ states of ${{}^{6,7}\mathrm{Li}}^{+}$, where an order of magnitude improvement over the previous theory has been achieved for ${}^7{\mathrm{Li}}^{+}$.