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

With the development of atomic clock technology, atomic clocks have played an important role in both basic physics Dzuba-NP-2016 ; Safronova-RMP-2018 ; Wcislo-NA-2016 and practical applications Hinkley-science-2013 . Because of their long-term stability and relatively simple structure, microwave clocks play an irreplaceable role in areas such as satellite navigation Bandi-EL-2011 , deep space exploration Prestage-IEEE-2007 ; Tjoelker-PTTI-2012 , and time keeping Diddams-science-2004 . To date, cesium fountain clocks have reigned supreme in state-of-the-art performance regarding frequency uncertainty. Uncertainties have been reduced to the $10^{-16}$ level, the main limitation being collision shifts, black-body radiation (BBR) shifts, and cavity phase shifts Weyers-Metrologia-2018 . Among microwave atomic clocks, those based on ions have attracted extensive research because of unique advantages. First, ion clocks have small collision shifts because of the strong Coulomb repulsion between ions. Second, ions confined in the trap occupy a small space volume, making it easier to measure the temperature with high precision by reducing the uncertainty associated with BBR shifts. Third, ion clocks need no microwave cavity, and therefore uncertainties from cavity phase shifts do not arise. In fountain clocks, such shifts arise from the asymmetry of the electric fields in the microwave cavity when atoms move up and down.

To date, ¹¹³Cd⁺ Jelenkovic-PRA-2006 ; KMiao-OL-2015 , ¹⁷¹Yb⁺ Phoonthong-APB-2014 ; Mulholland-APB-2019 , and ¹⁹⁹Hg⁺ Berkeland-PRL-1998 ; Burt-ITUFFC-2016 ions have been used in the development of microwave ion clocks. A great deal of progress has been achieved. In the past ten years, our group has been committed to developing a microwave atomic clock based on ¹¹³Cd⁺ KMiao-OL-2015 ; JWZhang-PRA-2012 ; SGWang-OE-2013 ; JZHan-EPJD-2019 ; YNZuo-APL-2019 ; JZHan-PRA-2019 . At present, the short-term frequency instability has reached $6.1\times 10^{-13}/\sqrt{\tau}$, and the uncertainty associated with ground-state hyperfine splitting has reached $6.6\times 10^{-14}$ KMiao-OL-2015 . Currently, the fundamental limitation halting further improvements in accuracy and stability of microwave ion clocks are the second-order Doppler shifts (SODSs), and the Dick effect resulting from dead times during interrogations JWZhang-CPL-2015 .

In the past few years, sympathetic-cooling technology has greatly advanced the development of the optical ion clock. Optical ion clocks based on Mg⁺ sympathetic cooling of Al⁺ have become one of the most accurate atomic clocks in the world Brewer-PRL-2019 . This cooling technology cools the target ions continuously to a lower temperature through Coulomb interactions with the coolant ion. Sympathetic cooling could be an effective method to overcome the limitations on stability and accuracy of microwave ion clocks. The application of sympathetic cooling to microwave ion clocks was earlier proposed by D. J. Wineland and collaborators from National Institute of Standards and Technology, and a preliminary scheme was studied involving the use of ¹⁹⁸Hg⁺ Larson-PRL-1986 and ²⁶Mg⁺ Bollinger-IEEE-1991 sympathetic cooling ⁹Be⁺ in further developments of the Be⁺ frequency standard in the Penning trap. Microwave ion clocks need to trap a large number of ions stably to improve the signal-to-noise ratio (SNR) during the interrogation of clock signals. Under the sympathetic cooling scheme, Coulomb repulsion of the inner shells of ions makes the stable trapping of large numbers of dual ion systems very challenging experimentally. With the cooling lasers of the coolant ions operating during microwave interrogations, the light shifts caused by these lasers and the fluorescence of the coolant ions must be evaluated carefully.

We had earlier proposed using ²⁴Mg⁺ to sympathetically cool ¹¹³Cd⁺ and have reported results of our experiment YNZuo-APL-2019 . Compared with that scheme, using ⁴⁰Ca⁺ as coolant is more promising because of certain key advantages. First, the mass ratio of ¹¹³Cd⁺ and ⁴⁰Ca⁺ is smaller than ¹¹³Cd⁺ and ²⁴Mg⁺ and therefore increases the cooling efficiency and improves the trapping stability. Second, the reaction rate of ⁴⁰Ca⁺ with residual background H₂ are much smaller than ²⁴Mg⁺. The reaction turns the Mg⁺ into dark ions and is sufficient to limit the cooling efficiency and stability of confinement. Third, the lasers for ⁴⁰Ca⁺ are much easier to obtain than those for ²⁴Mg⁺. All the lasers needed for ⁴⁰Ca⁺ cooling can be generated by diode lasers, apart from the ultraviolet laser for ²⁴Mg⁺ which must be obtained by a fourth-harmonic-generation laser system. The laser system implemented for ⁴⁰Ca⁺ has better reliability and is cost effective compared with that of ²⁴Mg⁺.

In this study, a large number of ¹¹³Cd⁺ ions sympathetically cooled by ⁴⁰Ca⁺ and stably confined in a linear Paul trap was experimentally achieved. The number of ¹¹³Cd⁺ ions and ⁴⁰Ca⁺ ions reached were $4.2\times 10^{5}$ and $1.7\times 10^{5}$, respectively, and the confinement time constant reached was 84 hours. The temperature of the Cd⁺ ions was as low as $90(10)$ mK , and the corresponding SODS was estimated to be $-1.84(0.2)\times 10^{-16}$. The ac Stark shifts caused by the coolant ions were carefully estimated by using an accurate ab initio approach; the magnitude and uncertainty at wavelengths 397 nm and 866 nm were $5.4(0.5)\times 10^{-17}$ and $2.0(0.2)\times 10^{-19}$, respectively. The Dick-effect-limited Allan deviation was also deduced as there was no need for an extra cooling procedure; its value was decreased to $1.8\times 10^{-14}/\sqrt{\tau}$. These results show that using Ca⁺ as coolant ions to sympathetically cool Cd⁺ is very stable and efficient. The main uncertainty of the systematic frequency shifts was significantly reduced to $10^{-16}$ level.

The entire experimental system consists of three parts: the ion trap, the lasers, and the detection system. The ions are trapped in a linear Paul trap using four electrodes. Every electrode is separated into three segments. The middle acts as radio frequency (RF) electrodes for radial confinement, and the ends act as endcap (EC) electrodes for axial confinement. The driving frequency chosen is 2.076 MHz, and the RF voltage varies from 0 V to 500 V, which covers the common stable areas for ¹¹³Cd⁺ and ⁴⁰Ca⁺ ions. The EC voltage varies from 0 V to 100 V. To reduce RF heating effects and increase the number of trapped ions, the ratio of $R/r_{0}$ was optimized and set at 1.1468 Denison-JVST-1971 , where $2R=14.22$ mm is the outer diameter of the electrode and $r_{0}=6.2$ mm the radial distance from the axis of the trap to the closest surface of the electrodes. The vacuum of the vacuum chamber is maintained by an ion pump, the background pressure being around $5\times 10^{-10}$ mBar. Three pairs of Helmholtz coils are used to generate a static magnetic field to split the Zeeman sublevels and compensate for the geomagnetic field.

We chose photoionization instead of electron bombardment ionization to reduce disturbances from stray charges. For the laser setup of the experiment (Fig. 1), the 423-nm (Ca I $4s^{2}\ {}^{1}S_{0}\rightarrow 4s4p\ ^{1}P_{1}$) laser and the 374-nm lasers are used to ionize neutral ⁴⁰Ca into singly charged ions ⁴⁰Ca⁺. The 397-nm (Ca II $4s\ ^{2}S_{1/2}\rightarrow 4p\ ^{2}P_{1/2}$) laser and the 866-nm (Ca II $3d\ ^{2}D_{3/2}\rightarrow 4p\ ^{2}P_{1/2}$) laser are used for Doppler cooling and repumping the ⁴⁰Ca⁺ ions. For ¹¹³Cd⁺, we use the 228-nm (Cd I $5s^{2}\ {}^{1}S_{0}\rightarrow 5s5p\ ^{1}P_{1}$) laser for two-photon ionization of neutral ¹¹³Cd. Because of the hyperfine structure of ¹¹³Cd⁺, excited dark states are generated. To improve the cooling and fluorescence detection efficiency, we use a circular polarization laser beam to connect two Zeeman sublevels $5s~{}^{2}S_{1/2}$ (F=1 m_F=1) $\rightarrow$ $5p~{}^{2}P_{3/2}$ (F=2 m_F=2) to prevent ions from transitioning into dark states $5p~{}^{2}P_{3/2}$ (F=1), and a 15.2-GHz microwave RF signal to repump the ions in dark states $5s~{}^{2}S_{1/2}$ (F=0) back to the cycling transition state. The energy levels of ¹¹³Cd⁺ are shown in Fig. 2.

For the detection system, we use an electron-multiplying charge-coupled device (EMCCD) to image the structure of the dual-ion system, and a photomultiplier tube (PMT) to determine the number of ion-scattered photons quantitatively. The lens between the EMCCD and PMT are well designed to reduce aberration. The wavelengths used for detection are 397 nm and 214.5 nm for ⁴⁰Ca⁺ and ¹¹³Cd⁺, respectively.

To apply a sympathetic cooling scheme to a microwave ion clock, two conditions must be met simultaneously: i) a large number of target ions must be stably trapped in the ion trap, and ii) a high sympathetic cooling efficiency must be maintained to keep the target ions at a low temperature.

The measurement of the ion temperature is achieved by measuring Doppler broadening of the laser cooling transition Larson-PRL-1986 ; Bollinger-PRL-1984 . Fig. 3 is a typical measurement result. To reduce cooling and heating effects of the detection laser, the laser power is maintained below 20$\rm{\mu}$W. The fitting curve is a Voigt profile, expressed as origin

where $F_{0}$ denotes the offset, $\nu$ the laser frequency, $\nu_{c}$ the laser center frequency, $A$ the area, $\omega_{L}$ the Lorentzian width, $\omega_{G}$ the Gaussian width of Doppler broadening, and $*$ means convolution. For ¹¹³Cd⁺, $\omega_{L}=60.13$ MHz, which is the natural linewidth of the D₂ transition of ¹¹³Cd⁺; the fitted $\omega_{G}=28.68(1.4)$ MHz. The ion temperature is then determined using YNZuo-APL-2019 ; Fritz-book-2006

where $M$ denotes the mass of ¹¹³Cd⁺, $c$ the speed of light, and $k_{B}$ the Boltzmann constant. The corresponding ion temperature is $93(1.4)$ mK, which is the upper limit to the temperature of the ions.

Although we have optimized the experimental setup for the dual-ion system, we still need to find out the best experimental parameter settings to achieve stable trapping of a large number of ions in the dual-ion system and efficient cooling of target ions. The main heating source stems from RF heating effects, and hence we measured with different RF voltages the temperature of the ¹¹³Cd⁺ ions sympathetically cooled by ⁴⁰Ca⁺. From the results (Fig. 4), we see there is a positive correlation between the ion temperature and the RF voltage, signifying RF heating strongly influences ion temperatures. When the RF voltage is above 300 V, we see the ion temperature rising significantly with increasing RF voltage. These hot ⁴⁰Ca⁺ ions cannot be maintained in the liquid phase, and the cloud states cannot cool the ¹¹³Cd⁺ ions to very low temperatures. When the RF voltage is below 250 V, the temperature increases with decreasing RF voltage which we attribute to a decrease in the trap depth. The edge of the RF trap potential may contribute additional heating effects through imperfections in the potential wells. When the RF potential is below 200 V, the ¹¹³Cd⁺ ions cannot be stably confined, and the number of ions decreases rapidly.

Another experimental parameter is the endcap voltage but has no direct influence on the temperature of ions. Nevertheless, as the endcap voltage increases, the ions are squeezed towards the trap axis, and thus more ions will deviate from the trap center, causing an increase in RF heating effects. We measure the temperature of the sympathetic cooled ¹¹³Cd⁺ ions while changing the endcap voltage (Fig. 5). We find three distinct stages between endcap voltages 30 V–50 V. This includes a sharp rise in ¹¹³Cd⁺ temperature, which may indicate a phase transition of ¹¹³Cd⁺ ions into a cloud state. When the endcap voltage is above 60 V, the ⁴⁰Ca⁺ ions cannot be maintained in the liquid phase, and they also cause a sharp increase in temperature of the ¹¹³Cd⁺ ions.

According to Ref. YNZuo-APL-2019 , there is a complicated relationship between the ion temperature and the RF or EC voltage because a decrease in ion number can strongly influence the ion temperature. In this experiment, ions are more stably confined in the ion trap, and hence the number of ions barely decreases. Without the influence of ion number, the relationship between ion temperature and the RF or EC voltage is more intuitive. The experimental results are simply understood from RF heating effects.

We were able to obtain a set of optimized experimental parameters for the ⁴⁰Ca⁺ sympathetic cooling ¹¹³Cd⁺ scheme, the RF and EC voltage being 250 V and 10 V, respectively. At these parameter settings, the temperature of the ¹¹³Cd⁺ ions was measured to be 90(10) mK.

To find the total number of both ion species in the ion trap, we need to know the volume and density of each species. According to Ref. Hornekr-PRL-2001 ; Wineland-PCCSHCIW-1987 , the configurations of the inner and outer layer of ions are cylindrical and ellipsoidal, and the gap between the inner surface of the ellipsoid shell and the outer surface of the inner cylinder is proportional to the square root of the mass ratio. We use these two configurations to fit the dual-ion images taken with the EMCCD [Fig. 6(b)]. From the fitted dimension of the Ca⁺ and Cd⁺ ion clouds [Fig. 6(c)], the total ion volume was calculated to be 5.3 mm³ and 34.7 mm³ for ⁴⁰Ca⁺ and ¹¹³Cd⁺, respectively. In contrast to results in Ref. YNZuo-APL-2019 , we see from Fig. 6(a) and (b) that, in this experimental scheme, there are nearly no dark ions and the ion signals are much more stable, indicating that using ⁴⁰Ca⁺ as coolant to sympathetically cool ¹¹³Cd⁺ is more efficient and stable.

For ions trapped at low temperature, their density is usually estimated using the zero-temperature approximation Wineland-PCCSHCIW-1987 ; Hornekr-PRL-2001 ,

where $\varepsilon_{0}$ denotes the permittivity of vacuum, $V_{RF}$ the amplitude of RF voltage, $M$ the ion mass, $\Omega$ the trap driving frequency, and $r_{0}$ the minimum distance from the trap center axis to the surfaces of the electrodes. In this experiment, $V_{RF}=250$V, $\Omega=2\pi\times 2.076$ MHz. The number density for ¹¹³Cd⁺ and for ⁴⁰Ca⁺ were calculated as $n_{Cd}=1.2\times 10^{13}$ m^-3 and $n_{Ca}=3.3\times 10^{13}$ m^-3, respectively. Therefore, the total number of ⁴⁰Ca⁺ and ¹¹³Cd⁺ ions in the ion trap are $N_{Cd}=4.2\times 10^{5}$ and $N_{Ca}=1.7\times 10^{5}$, respectively.

To analysis the stability of trapping this dual-ion system qualitatively, we measured the loss of ¹¹³Cd⁺ ions from the ion trap. With the RF voltage and EC voltage fixed, and the power and frequency of the lasers stabilized, we assume the PMT counts to be proportional to the ion number. To avoid laser cooling effects during detection, the detection laser is blocked for 10 min after each 20 s measurement [Fig. 7(b)]. The ion-loss measurement results are shown in Fig. 7(a); the counts from background fluorescence have been subtracted. From Fig. 7(a), we see that the PMT counts decrease rapidly at the beginning. This is because when the number of ions is large, more ions are located at positions far from the center of the trap well and cause confinement of these ions at the edge of the potential well that are not very stable. After about 25 min, the trapping of ¹¹³Cd⁺ becomes stable, and the rate of ion loss becomes very slow. The fitted curve is an exponential decay curve, and the time constant is about 84 h. After 3.3 h, the PMT count decreased to 40% (from 18306$\rightarrow$7021), indicating that there are still $1.7\times 10^{5}$ ¹¹³Cd⁺ ions in the ion trap. This suffices to provide a good SNR for the microwave ion clock. The measurement of the ion number loss shows that under sympathetic cooling this dual-ion system is very stably within the ion trap. Low-temperature stable trapping of ¹¹³Cd⁺ is paramount for further developments to microwave ion clocks.

We achieved long-term low-temperature trapping of a large number of sympathetically cooled ¹¹³Cd⁺ ions using laser-cooled ⁴⁰Ca⁺ as coolant. RF trapping was very stable for this Ca⁺-Cd⁺ system. After three hours in the trap, up to $10^{5}$ Cd⁺ ions were still confined. The temperature of the ¹¹³Cd⁺ ions was as low as 90 mK, which aided the suppression of the uncertainty associated with the SODS. Using the equipartition theorem, this uncertainty was roughly estimated to be at the $10^{-17}$ level. The ac Stark shifts incurred by the Ca⁺ cooling lasers and fluorescence were carefully estimated. The ac Stark polarizabilities were calculated by high-accuracy ab initio calculations. Because the frequency differences between the two laser frequencies of Ca⁺ and all possible transitions of Cd⁺ are large, the magnitude and uncertainty were down to $10^{-17}$ and $10^{-18}$ levels, and hence can be ignored in these circumstances. The results indicate that using Ca⁺ sympathetic cooling of Cd⁺ is expected to be an effective experimental method to improve the accuracy of the Cadmium-ion microwave frequency standard to order $10^{-16}$. In this sympathetic cooling scheme, one cooling procedure is no longer needed. By decreasing dead time, the Dick-effect-limited Allan deviation is also reduced. All of the experimental results and theoretical analysis show that a microwave clock based on sympathetically cooled Cd⁺ ions holds promise to attain ultrahigh frequency accuracy and stability.

We would like to thank Z. Y. Peng and K. Beloy for their helpful discussions and suggestions. This work is supported by the National Key Research and Development Program of China (No. 2016YFA0302100), and the Beijing Natural Science Foundation (1202011). Y. M. Yu acknowledges the National Natural Science Foundation of China (No. 11874064), and the Strategic Priority and Research Program of the Chinese Academy of Sciences (No. XDB21030300). J. Z. Han thanks S. C. Wang for image processing.