Determination of hyperfine splittings and Landé $g_{J}$ factors of $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ states of ^111,113Cd⁺ for a microwave frequency standard

With their improvements in accuracy over time, atomic clocks have played an important role in practical applications Hinkley et al. (2013); Burt et al. (2021) and testing the fundamental physics Dzuba et al. (2016); Wcisło et al. (2016); Safronova et al. (2018). Indeed, the microwave-frequency atomic clock plays a vital role in satellite navigation Bandi et al. (2011), deep space exploration Prestage and Weaver (2007); Burt et al. (2016), and timekeeping Diddams et al. (2004). Among the many clock proposals, trapped-ion microwave-frequency clocks have attracted wide attention from researchers because the ions are well isolated from the external environment in an ion trap. The setup is conducive to improvements in the transportability of atomic clocksSchwindt et al. (2016); Mulholland et al. (2019a, b); Hoang et al. (2021). Such clocks are also considered the next generation of practical microwave clocks Schmittberger and Scherer (2020).

Cadmium ions (Cd⁺) benefit from a simple and distinct electronic structure, which is easily controlled, manipulated, and measurable with high precision. The microwave-frequency standard based on laser-cooled ¹¹³Cd⁺ has achieved an accuracy of $1.8\times 10^{-14}$ and a short-term stability of $4.2\times 10^{-13}/\sqrt{\tau}$ Miao et al. (2021). The high performance and potential for miniaturization make this frequency standard suitable in establishing a ground-based transportable frequency reference for navigation systems and for comparing atomic clocks between remote sites Zhang et al. (2012); Wang S. G (2013); Miao et al. (2015, 2021). Moreover, it has been proposed as a means to achieve an ultra-high level of accuracy down to $10^{-15}$ Han et al. (2021), highlighting the importance of accurately evaluating systematic frequency shifts.

Optical pumping is a fundamental process in operating a trapped-ion microwave frequency standard. The optical pumping efficiency determines directly the signal-to-noise ratio of the “clock signal,” which affects the short-term stability and measurement accuracy of the ground-state hyperfine splitting (HFS) for such frequency standards. Realizing optical pumping for the ¹¹³Cd⁺ microwave-frequency standard requires a blueshift in the laser frequency of the Doppler-cooling transition $5s~{}^{2}S_{1/2}~{}F=1,~{}m_{F}=1$–$5p~{}^{2}P_{3/2}~{}F=2,~{}m_{F}=2$ to reach the $5p~{}^{2}P_{3/2}~{}F=1$ hyperfine level. However, there are no precise measurements available of the HFSs for other excited states Li et al. (2018). A preliminary measurement of the HFS for the $5p~{}^{2}P_{3/2}$ level of the ¹¹³Cd⁺ ion is approximately 800 MHz Tanaka et al. (1996). Therefore, to improve the optical pumping efficiency and hence the performance of the ¹¹³Cd⁺ microwave frequency standard, the HFSs of the $5p~{}^{2}P_{3/2}$ level of the ¹¹³Cd⁺ ion need to be determined with greater accuracy. From the perspective of atomic structure calculations, the high-precision measurements of the HFSs for the $5p~{}^{2}P_{3/2}$ level of ^111,113Cd⁺ can also be used for testing and developing calculation models of the atomic structure.

In a trapped-ion microwave frequency standard, an external magnetic field is applied to provide the quantization axis to break the degeneracy of the ground-state magnetic level. Among all the systematic frequency shifts of a frequency standard, one dominant shift is the second-order Zeeman shift (SOZS) induced by the external magnetic field Berkeland et al. (1998); Phoonthong et al. (2014); Miao et al. (2021). The precise estimation of this SOZS and the calibration of the external magnetic field require accurate knowledge of the ground-state Landé $g_{j}$ factor Han et al. (2019). The external magnetic field in our latest laser-cooled microwave-frequency standard based on trapped ¹¹³Cd⁺ ions is approximately $8000$ nT Miao et al. (2021). However, only two theoretical studies have provided a value of the ground-state Landé $g_{J}$ factor of Cd⁺, one giving $2.00286(53)$, calculated using the relativistic-coupled-cluster (RCC) theory Han et al. (2019), and the other giving $2.002291(4)$, calculated by the $\Lambda$-approach RCC ($\Lambda$-RCC) theory Yu et al. (2020a). The two Landé $g_{J}$ factors have a difference of $0.0006$ that generates a relative frequency shift of $6.6\times 10^{-14}$. This large systematic shift obviously falls short inaccuracy of our latest ¹¹³Cd⁺ microwave-frequency standard ($1.8\times 10^{-14}$) Miao et al. (2021). Therefore, re-determining the ground-state $g_{J}$ factor of ¹¹³Cd⁺ is imperative if further improvements in accuracy for this microwave-frequency standard are to be attained.

In this work, the HFSs of the $5p~{}^{2}P_{3/2}$ level of the ¹¹³Cd⁺ ion is measured using the laser-induced fluorescence (LIF) technique. To maintain a low-temperature environment, the ¹¹³Cd⁺ ions are sympathetic-cooled by laser-cooled ⁴⁰Ca⁺ ions, a technique that improves the accuracy of measurements. Furthermore, the HFSs and Landé $g_{J}$ factors of both the $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ levels were calculated using the multiconfiguration Dirac–Hartree–Fock (MCDHF) method. Electron correlation effects are carefully investigated and taken into account. Off-diagonal terms are also included to improve the calculation accuracy of the HFSs for the $5p~{}^{2}P_{1/2,3/2}$ level in Cd⁺. Cross-checking the measured and calculated HFS results ensures the reliability and accuracy of the results provided in this work. Our results are of great importance for further improving the performance of the Cd⁺ microwave-frequency standard.

To obtain the HFSs of the $5p~{}^{2}P_{3/2}$ level for ^111,113Cd⁺, we first measure the frequency shifts from the $5s~{}^{2}S_{1/2}~{}F=1$–$5p~{}^{2}P_{3/2}~{}F=2$ transition of ^111,113Cd⁺ to the $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transition of ¹¹⁴Cd⁺. Briefly, for the experimental setup (see Ref. Han et al. (2021) for details), we prepared crystals of two ion species consisting of approximately $10^{5}$ Ca⁺ and Cd⁺ ions in a linear Paul trap. The Ca⁺ and Cd⁺ ions are produced by selected-photoionization using laser beams of wavelength 423-nm (Ca $4s^{2}~{}^{1}S_{0}$–$4s4p~{}^{1}P_{1}$) / 374-nm (Ca $4s4p~{}^{1}P_{1}$– Continuum), and 228-nm (Cd $5s^{2}~{}^{1}S_{0}$–$5s5p~{}^{1}P_{1}$). The Ca⁺ are used as coolant ions that are Doppler-cooled using lasers beams of wavelength 397-nm (Ca⁺ $4s~{}^{2}S_{1/2}$–$4p~{}^{2}P_{1/2}$) and 866-nm (Ca⁺ $3d~{}^{2}D_{3/2}$–$4p~{}^{2}P_{1/2}$). The Cd⁺ ions are sympathetically-cooled to less than 0.5 K through Coulomb interactions with the Ca⁺ ions. The frequency shifts of the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transition of ^111,113Cd⁺ and the $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transitions of ¹¹⁴Cd⁺ were measured using scanning frequencies in a weak 214.5-nm probe laser beam. The $5s~{}^{2}S_{1/2}~{}(F=1,m_{F}=1)$–$5p~{}^{2}P_{3/2}~{}(F=2,m_{F}=2)$ transition is a cycling transition that was used to cool and detect the ^111,113Cd⁺ ions. Although the circularly polarized cooling laser beam excites a cycling transition, ions may, as a result of the polarization impurity, still, leak to the $5s~{}^{2}S_{1/2}~{}F=0$ state via $5p~{}^{2}P_{3/2}~{}F=1$ state. To increase detection efficiency, 20-dBm microwave radiation resonant with the ground-state hyperfine transition (15.2-GHz for ¹¹³Cd⁺ and 14.5-GHz for ¹¹¹Cd⁺) is applied during LIF detection. The frequency of each laser beam is measured using a high-precision wavemeter (HighFinesse WS8-2).

In obtaining the measured LIF spectrum (Fig. 2), the beam intensity is maintained below 5 $\mu$W/mm² (saturation parameter 0.0006) to reduce the cooling and heating effects of the probe beam. The fitted curve is the Voigt profile Zuo et al. (2019); Han et al. (2021), expressed as

where $F_{0}$ is the offset, $\nu$ is the laser beam frequency, $\nu_{c}$ is the ion resonance frequency, $A$ is the area, $\nu_{L}$ is the Lorentzian width, $\nu_{G}$ is the Gaussian width of Doppler broadening. The line profile is slightly asymmetrically because of the heating and cooling effects of the probe beam, which lead to a slight increase in the uncertainty of the estimated transition frequency.

Measurements present three sources of uncertainty:

• i)

  Statistical uncertainties. For the Cd⁺ $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transition, $\nu_{L}$ is 60.13 MHz which represent the natural width, the fitted $\nu_{G}$ is approximately 30 MHz, and the ion temperature is estimated to be approximately 100 mK. The statistical uncertainty associated with the transition frequencies of ¹¹⁴Cd⁺ and the ^111,113Cd⁺ is approximately 1 MHz, and thus the statistical uncertainty in their differences is approximately 1.4 MHz;

• ii)

  Instrument uncertainties. The uncertainty arising from the drift in the wavemeter is less than 0.5 MHz in a laboratory environment Liu et al. (2018);

• iii)

  Systematic uncertainties. Most systematic shifts are common to the $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ transitions of both ¹¹⁴Cd⁺ and ^111/113Cd⁺ and thus cancel each other out. Because the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transition of Cd⁺ is sensitive to magnetic fields, the Zeeman shift becomes the dominant contributor to systematic uncertainties.

In a weak field ($\mu_{B}B\sim 0.14$ MHz $\ll$ hyperfine constant $A$), the Zeeman shift for a specific energy level is expressed as

where $g_{F}$ is given by

where $J=L+S$ the total electron angular momentum ($S$ and $L$ the spin and orbital angular momenta), and $F=I+J$ the total angular momentum with $I$ denoting the nuclear spin. In typical conditions of our experiment, $B\sim 8000$ nT Miao et al. (2015, 2021). By introducing values $g_{J}=2.002257$ and $1.334056$ for the levels $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{3/2}$ calculated in this work (see text below), the Zeeman shift for the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transition of Cd⁺ is estimated to be 0.11 MHz. Therefore, the total systematic shifts for the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transitions of Cd⁺ are estimated to be below 0.5 MHz.

The final frequencies for the $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ transitions of ^111,113Cd⁺ and that for $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$ of ¹¹⁴Cd⁺ are determined to be 4649.0(1.6) MHz and 4041.8(1.6) MHz, respectively.

In LS-coupling, the energy shifts after the hyperfine interaction are expressed as Foot et al. (2005)

Therefore, for ^111,113Cd⁺, we have

where $\nu_{s,F=1\rightarrow p,F=2}^{111/113}$ is the transition frequency of $5s~{}^{2}S_{1/2}~{}(F=1)$–$5p~{}^{2}P_{3/2}~{}(F=2)$ of ^111,113Cd⁺; $\nu_{s\rightarrow p}^{111/113}$ is the transition frequency of $5s~{}^{2}S_{1/2}$–$5p~{}^{2}P_{3/2}$; $\nu_{HFS,s}$ is the HFS of $5s~{}^{2}S_{1/2}$; and $\nu_{HFS,p}$ is the HFS of $5p~{}^{2}P_{3/2}$. In reference to ¹¹⁴Cd⁺, through a linear transformation, Eq. (5) may be expressed as

where $\Delta\nu_{s,F=1\rightarrow p,F=2}^{111/113,114}=\nu_{s,F=1\rightarrow p,F=2}^{111/113}-\nu_{s\rightarrow p}^{114}$ and $\Delta\nu_{s\rightarrow p}^{111/113,114}=\nu_{s\rightarrow p}^{111/113}-\nu_{s\rightarrow p}^{114}$. With our measurements, $\Delta\nu_{s,F=1\rightarrow p,F=2}^{111/113,114}$ are respectively 4649.0(1.6) MHz and 4041.8(1.6) MHz, whereas $\Delta\nu_{s\rightarrow p}^{111/113,114}$ are 1314.3(22)[023] MHz and 555.2(23)[008] MHz Hammen et al. (2018). From our previous measurements obtained through double-resonance microwave laser spectroscopy, the $\nu_{HFS,s}^{111/113}$ were accurately measured to be 14530507349.9(1.1) Hz Zhang et al. (2012) and 15199862855.02799(27) Hz Miao et al. (2021), from which we derived $\nu_{HFS,p}^{111/113}$ to be 794.6(3.6) MHz and 835.5(2.9) MHz.

The measured HFSs for the $5p~{}^{2}P_{3/2}$ level and the calculated HFSs and Landé $g_{J}$ factors for the $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/3,3/2}$ levels in this work are listed in Table 3; other experimental and calculated results are also listed for comparison. For the HFSs, our group’s previous high-accuracy measurements for the ^111,113Cd⁺ ground state provided an excellent benchmark for the atomic structure calculation of Cd⁺. The present HFSs for the $5s~{}^{2}S_{1/2}$ state calculated using the MCDHF method show stronger agreement with our previous experimental results than those of previous theoretical results Dixit et al. (2008); Li et al. (2018). The present measured HFSs for the $5p~{}^{2}P_{3/2}$ level is also in agreement with the present theoretical results. The cross-checking between experiment and theory ensures the reliability of the Cd⁺ $5p~{}^{2}P_{3/2}$ HFSs determined in this work. We recommend the adoption of 794.6(3.6) and 835.5(2.9) as the blue-shifted frequencies for optical pumping in the microwave-frequency standard based on ^111/113Cd⁺.

Regarding the Landé $g_{J}$ factors, there are no experimental results for Cd⁺. Accurate calculations of Landé $g_{J}$ factors has proven complicated even for alkali atoms and alkali-like ions because they are sensitive to electron correlations. Those calculated in this work using the MCDHF method show strong deviations from previous RCC results Han et al. (2019). The ground state Landé $g_{J}$ factor calculated in this work ($2.002257(1)$) agrees with the recent result calculated using the $\Lambda$-RCC theory ($2.002291(4)$) Yu et al. (2020b) to the fourth decimal place, although there is no overlap within their margins of uncertainty. To our knowledge, there also exists a significant difference in results between the $\Lambda$-RCC calculations with the configuration interaction and the many-body perturbation (CI+MBPT) calculations in Yb⁺ ground-state Landé $g_{J}$ factor Gossel et al. (2013); Yu et al. (2020b). Comparing the results of the same physical quantity from different calculation methods is also of great significance for developing atomic structure calculation models and understanding the role of electronic correlation effects. Therefore, we encourage more experimental and theoretical research on the Landé $g_{J}$ factors of Cd⁺.

For precaution, we recommend the value 2.00226(4) for the Cd⁺ ground state Landé $g_{J}$ factor in the evaluation of the SOZS of the microwave frequency standard of trapped Cd⁺ ions. The SOZS can be estimated using the Breit–Rabi formula,

for which $B\sim 8000$ nT for the Cd⁺ microwave frequency standard during actual operations. Thus, the fractional frequency shifts incurred when using the value of $g_{J}=2.00226(4)$ is $4.4\times 10^{-15}$. The fractional frequency shifts produced by this $g_{J}$ factor for the Cd⁺ ground-state can meet current accuracy requirements for the best Cd⁺ microwave frequency standard ($1.8\times 10^{-14}$). However, for further developments of this standard, the ground state $g_{J}$ factor of Cd⁺ also needs to be determined more accurately.

We reported on the determination of HFSs and Landé $g_{J}$ factors for the $5s~{}^{2}S_{1/2}$ and $5p~{}^{2}P_{1/2,3/2}$ levels of ^111,113Cd⁺. The HFSs of the $5p~{}^{2}P_{3/2}$ level was measured using the laser-induced-fluorescence technique. The Cd⁺ ions were co-trapped with Ca⁺ ions in the same linear ion trap and sympathetically cooled through the Coulomb interaction with laser-cooled Ca⁺ ions. Furthermore, the HFSs and Landé $g_{J}$ factors for both levels of interest were calculated using the MCDHF calculation. Three computational strategies were followed to account for the electronic correlation effects more comprehensively. The final calculated HFSs were in perfect agreement with the measured HFSs of this work and our previous work, which from cross-checks, demonstrated the reliability of the calculations and the experiments. The HFSs and Landé $g_{J}$ factors determined in this work can further improve the efficiency of the optical pumping procedure and the accuracy of the second-order Zeeman correction, and the stability and accuracy of the microwave frequency standard based on trapped Cd⁺ ions.

We thank Z. M. Tang for the helpful discussions. This work is supported by the National Key R&D Program of China (No. 2021YFA1400243), National Natural Science Foundation of China (Nos. 91436210, 12074081, 12104095).