J. Z. Han

We reported the realization of a $^{174}\mathrm{Yb}^+$-$^{113}\mathrm{Cd}^+$ bi-species Coulomb crystal comprising $^{174}\mathrm{Yb}^+$ ions as coolant and verified its potential for application as a $^{113}\mathrm{Cd}^+$ microwave frequency standard employing sympathetic cooling.The two species of massive ions stably trapped in a Paul trap make up this large two-component crystal. The $^{113}\mathrm{Cd}^+$ ions are trapped in the center, which reduces considerably RF heating and excess micromotion to which the $^{113}\mathrm{Cd}^+$ ions are subjected. Under this scheme, the uncertainty due to the second-order Doppler effect is reduced to $5\times10^{-16}$, which represents an order of magnitude improvement over sympathetic cooled $^{40}\mathrm{Ca}^+$-$^{113}\mathrm{Cd}^+$ crystal. The uncertainty from the second-order Zeeman effect, which contributes the largest uncertainty to the microwave-ion frequency standard, is reduced to $4\times10^{-16}$. The relevant AC Stark shift uncertainty is estimated to be $4\times10^{-19}$. These results indicate using $^{174}\mathrm{Yb}^+$ as coolant ions for $^{113}\mathrm{Cd}^+$ is far superior and confirm the feasibility of a sympathetic-cooled cadmium-ion microwave clock system employing a $^{174}\mathrm{Yb}^+$-$^{113}\mathrm{Cd}^+$ two-component crystal.

The accurate evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in a linear Paul trap has been studied with experiments and molecular dynamics (MD) simulations. The motion of trapped ions in the trap has three contributions, and we focus on the ion excess micromotion, which is rarely discussed when evaluating the SODFS. Based on the hypothesis that the ion density is uniformly distributed in the radial direction, we propose a new model to accurately evaluate the total SODFS for ion microwave clocks. The effectiveness of the model has been verified both in simulation and experiment, especially for ion ensemble with temperature less than 100 mK. We believe that our new model offers advantages in accurately evaluating the SODFS for the ion trap, especially those of laser-cooled ion microwave clocks based on large ion clouds.

Regarding trapped-ion microwave-frequency standards, we report on the determination of hyperfine splittings and Landé $g_J$ factors of $^{111,113}$Cd$^+$. The hyperfine splittings of the $5p~^2P_{3/2}$ state of $^{111,113}$Cd$^+$ ions were measured using laser-induced fluorescence spectroscopy. The Cd$^+$ ions were confined in a linear Paul trap and sympathetically cooled by Ca$^+$ ions. Furthermore, the hyperfine splittings and Landé $g_J$ factors of the $5s~^2S_{1/2}$ and $5p~^2P_{1/2,3/2}$ levels of $^{111,113}$Cd$^+$ were calculated with greater accuracy using the multiconfiguration Dirac--Hartree--Fock scheme. The measured hyperfine splittings and the Dirac--Hartree--Fock calculation values were cross-checked, thereby further guaranteeing the reliability of our results. The results provided in this work can improve the signal-to-noise ratio of the clock transition and the accuracy of the second-order Zeeman shift correction, and subsequently the stability and accuracy of the microwave frequency standard based on trapped Cd$^+$ ions.

We report on progress towards a microwave frequency standard based on a laser-cooled 171Yb+ ion trap system. The electronics, lasers, and magnetic shields are integrated into a single physical package. With over 1E5 ions are stably trapped, the system offers a high signal-to-noise ratio Ramsey line-shape. In comparison with previous work, the frequency instability of a 171Yb+ microwave clock was further improved to $8.5 \times {10^{ - 13}}/\sqrt τ$ for averaging times between 10 and 1000 s.

We measured the ground-state hyperfine splitting of trapped ${}^{113}$Cd${}^{+}$ ions to be 15199862855.02799(27) Hz with a fractional uncertainty of $1.8\times10^{-14}$. The ions were trapped and laser-cooled in a linear quadrupole Paul trap. The fractional frequency stability was measured to be ${4.2} \times 10^{-13}/\sqrtτ $, obtained from Ramsey fringes of high signal-to-noise ratio and taken over a measurement time of nearly 5 hours, which is close to the short-term stability limit estimated from the Dick effect. Our result is consistent with previous reported values, but the measurement precision is four times better than the best result obtained to date.

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

This paper reports the direct temperature determination of sympathetically cooled 113Cd+ ions with laser-cooled 24Mg+ in a linear Paul trap. The sympathetically cooled ion species distribute in the outer shell of the large ensembles, which contain up to 3.3E5 ions. With optimized parameters, the minimum temperature of the sympathetically cooled 113Cd+ ions was measured to be tens of mK. These results indicate promising performance for microwave atomic clocks. The second order Doppler frequency shift is two orders of magnitudes lower and the Dick effect is suppressed.