Suppression of Black-body Radiation Induced Zeeman Shifts in the Optical Clocks
due to the Fine-structure Intramanifold Resonances

The excitation energies (EE) and M1 RMEs of the resonant transitions associated with the $3s^{2}~{}^{1}S_{0}$ and $3s3p~{}^{3}P_{0}$ clock states of Al⁺, obtained from the CI, CI+MBPT, and MCDHF methods, are summarized in Table 1. The experimental values for EEs taken from the National Institute of Science and Technology (NIST) database NIST are also listed in the same table for comparison. The convergence of the MCDHF calculations with the increase of the configuration space is examined in detail. We plot the convergence behaviors of the EE calculations and M1 RMEs of the two strongest M1 lines corresponding to the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ and $3s3p~{}^{3}P_{0}-3s3p~{}^{1}P_{1}$ transitions in Figs. 1 and 2, respectively.

In comparison with the CI calculations, the evaluated energies from the CI$+$MBPT method show significant improvement in the accuracy. The deviations of the calculations from the experimental results are reduced by approximately one or two orders of magnitude when we include all VC and CC correlation effects through the MBPT method. The accuracies of the EEs from the MCDHF method are generally seen to be lying in between the results from the CI and CI$+$MBPT methods, as the MCDHF calculations have included the VC correlations involving the $2p$ core shell but ignored other VC and CC correlations. In our test calculations, we found that the VC correlations involving the excitations from the $1s$ and $2s$ orbitals and the CC correlations from the SD excitations of the $2p$ electrons do not improve the calculations significantly. Therefore, we do not consider more VC and CC correlations in the MCDHF calculations, but we turn to include the triple (T) excitations into the VC correlation model. Note that the triple excitations are not included in the CI+MBPT calculations. It was found that considering the SDT excitations can give reasonably accurate results for many levels. As shown in Table 1 and Fig. 1(a), the MCDHF method in the SDT approximation gives the most accurate fine-structure splitting for the $3s3p~{}^{3}P_{0,1}$ states, with only $-$0.7 cm^-1 deviation from the experimental value. Although the calculated M1 RME values for the strongest $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ transition is very consistent from all the three considered methods, the improved energy kind of gives us an assurance about the reliability in the determined M1 RME value using the MCDHF method. Another advantage of considering results from the MCDHF method over the CI+MBPT method is that we can estimate the corrections from the Breit interaction and QED effects using the MCDHF method. We present these corrections to the M1 RMEs in Table 1. Considering possible uncertainties from many dominant sources, we present our recommended values of the M1 RMEs associated with the $3s^{2}~{}^{1}S_{0}$ and $3s3p~{}^{3}P_{0}$ clock states of Al⁺ in the last column of Table 1.

We determine the M1 polarizabilities [see Eq. (6)] of the $3s^{2}~{}^{1}S_{0}$ and $3s3p~{}^{3}P_{0}$ clock states of Al⁺ by combining the theoretical M1 RMEs with the experimental energies in order to minimize the uncertainty in the results. The experimental energies of low-lying levels of Al⁺ taken from NIST database NIST generally have uncertainties $\leqslant$ 0.06 cm^-1 Martin-JPCRD-1979 , which correspond to fractional uncertainties of $\lesssim 1\times 10^{-6}$ to the EEs listed in Table 1 except for the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ transition which has a very small EE of 60.88 cm^-1. Since the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ transition has the largest M1 RME, at least three orders of magnitude larger than other M1 transitions associated with both the clock states. Thus, an error of $\thickapprox$ 0.06 cm^-1 in its EE of 60.88 cm^-1 can introduce a fractional error of $\thickapprox$ 0.1% to the dynamic M1 polarizability of the $3s3p~{}^{3}P_{0}$ state in the BBR frequency region. Therefore, we alternatively use its EE inferred from a high-precision laser spectroscopic measurement of the ${}^{3}P_{0,1}$ fine-splitting, whose value is given in Ref. Brewer-PRA-2019 as 1.8241180(2) THz [or 60.846027(7) cm^-1] in the absence of the hyperfine interaction. The use of high-precision energies in the evaluation of M1 polarizabilities ensure that nominal uncertainties to these quantities can arise only from the calculated M1 RMEs.

The frequency-dependent dynamic M1 polarizabilities $\beta(\omega)$ of both the clock states of Al⁺ are plotted in Fig. 3 in the range from $\omega=0$ (static limit) to $\omega=0.19$ a.u.. We also plot the distribution of the BBR spectrum in the same horizontal coordinate in order to facilitate the analysis, and the positions of the laser wavelengths applied in the ⁴⁰Ca⁺-²⁷Al⁺ quantum-logic clock Guggemos-NJP-2019 ; Hannig-RSI-2019 ; Ma-APB-2020 ; Shao-CPL-2019 ; Cui-EPJD-2022 are marked. As can be seen from the figure, the $\beta(\omega)$ curve of the $3s^{2}~{}^{1}S_{0}$ state almost overlap with the $\beta(\omega)=0$ line. The resonant lines of the $3s^{2}~{}^{1}S_{0}$ state are all located in the extreme ultraviolet region due to their high EEs no less than 91274.50 cm^-1 [$>$ 0.41 a.u. (see Table 1)]. Thus, they are not covered in the range considered in this work. There are two resonant lines of the $3s3p~{}^{3}P_{0}$ state located at the 164 $\mu$m and 445 nm, both of which are transitions within the $3s3p$ configuration and are two strongest lines among all possible resonances of the clock states. The dominant 164 $\mu$m $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance is responsible for making large $\beta(\omega)$ values of the $3s3p~{}^{3}P_{0}$ state in the low-frequency range from $\omega=0$ to the middle infrared region but with different signs on both sides of the resonance [see Figs. 3(b)]. In the blue-detuning region to the 164-$\mu$m resonance, the $\beta(\omega)$ value is generally negative except in the close vicinity of the 445 nm corresponding to the $3s3p~{}^{3}P_{0}-3s3p~{}^{1}P_{1}$ resonance [see Figs. 3(a) and 3(c)]. The other resonances in the higher-frequency region (not shown) are negligibly weak. It is worth noting that the BBR spectrum at 300 K are mainly distributed in the blue-detuning region to the 164 $\mu$m resonance, which is in contrast with the dynamic E1 polarizabilities Wei-CPB-2022 ; Middelmann-PRL-2012 , where the BBR spectrum are generally distributed in the red-detuning region to all possible E1 resonances of a typical optical clock state Middelmann-PRL-2012 .

For estimating the shift due to the isotropic magnetic field of a clock transition, we are interested in the determination of the differential polarizabilities, $\Delta\beta(\omega)=\beta_{{}^{3}P_{0}}(\omega)-\beta_{{}^{1}S_{0}}(\omega)$, at different frequencies. Numerical results show that the $\beta_{{}^{1}S_{0}}(\omega)$ values in the whole frequency range of interest are even smaller than the uncertainties of $\beta_{{}^{3}P_{0}}(\omega)$, such that we have $\Delta\beta(\omega)\thickapprox\beta_{{}^{3}P_{0}}(\omega)\gg\beta_{{}^{1}S_{0}}(\omega)$. The contribution of the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance can dominate in the evaluation of $\Delta\beta(\omega)$ [or $\beta_{{}^{3}P_{0}}(\omega)$] by almost 100% in the low-frequency range ($\gtrsim$ 1000 nm) and by over 97% in the visible region except in the vicinity of 445-nm resonance, whereas the residual fractional contributions are almost 100% from the $3s3p~{}^{3}P_{0}-3s3p~{}^{1}P_{1}$ resonance and the contributions of all other resonances are negligibly small. We summarize the leading contributions from individual transitions to $\Delta\beta(\omega)$ at some selected wavelengths in Table 2. The intensity of the BBR at temperature 300 K is larger than 10% of the peak intensity in the range from 114 $\mu$m to 5.78 $\mu$m, and close to zero in the visible region. On the other hand, the $\Delta\beta(\omega)$ values in the visible region are a few orders of magnitude smaller than that in the main BBR region. These values can also be used to estimate the laser-induced Zeeman shifts in Al⁺ clocks. Taking the ⁴⁰Ca⁺-²⁷Al⁺ quantum-logic clock Guggemos-NJP-2019 ; Hannig-RSI-2019 ; Ma-APB-2020 ; Shao-CPL-2019 ; Cui-EPJD-2022 as an example, we used the $\Delta\beta(\omega)$ values at laser wavelengths 866, 397, and 267 nm to estimate first the laser-induced Zeeman shifts in our ⁴⁰Ca⁺-²⁷Al⁺ clocks at APM Shao-CPL-2019 ; Cui-EPJD-2022 and found these shifts to be less than 1 $\mu$Hz.

To verify reliability of our calculations, we use the differential static M1 polarizabiliy $\Delta\beta(0)$ to evaluate the quadratic Zeeman shift coefficient ($C_{2}$), which can be given by $C_{2}=-\frac{\Delta\beta(0)}{2}$. Our computational result is $C_{2}=-71.591(7)$ Hz/mT². This value agrees rather good with two previous measurements as $-$71.988(48) Rosenband-Science-2008 and $-$71.944(24) Hz/mT² Brewer-PRA-2019 , despite the discrepancy ($\approx 0.5$ %) being over the uncertainties. However, it is worth noting that a theoretical value $-71.927$ Hz/mT² calculated in a very simple nonrelativistic limit in Ref. Brewer-PRA-2019 is consistent better with the measurements. In their work, only the nonrelativistic value of M1 RME of the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ transition was used to calculate the $C_{2}$ coefficient. In fact, the nonrelativistic M1 RME value [$\sqrt{2}(1+2a)\mu_{B}$ = 1.41749 a.u.] is found to be larger than all the relativistic values given in Table 1 and Fig. 2 (b) by approximately 0.25%. The underlying reason for the discrepancy of the experimental values from the high-precision relativistic calculations is worth expolring by carrying out further studies using other relativistic many-body methods like relativistic coupled-cluster theory or including higher-order relativistic effects. We note that the role of negative-energy states on the M1 and E2 polarizabilities have been discussed in the literature Wu-arxiv-2023-1 ; Wu-arxiv-2023-2 ; Porsev-arxiv-2023 and these contributions are found to be important only in the visible region where the multipolar polarizabilities are generally very small. Nonetheless, the M1 polarizability values determined in this work for the BBR region are accurate enough to estimate the BBRz shift of the clock transition of Al⁺.

As aforementioned, the BBR spectrum are mainly distributed in the blue-detuning region to the 164 $\mu$m $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance, where the dynamic M1 polarizabilities are smaller than the static value and have an opposite sign. This has suggested the invalidation of the static limit for the differential M1 polarizability to estimate the BBRz shift as discussed in section II. To understand how the BBRz shift fails to achieve this limit in the Al⁺ clock frequency, we discuss the numerical results from both the static and dynamic M1 polarizabilities here. Using the $\Delta\beta(0)$ value listed in Table 2, the BBRz shift is calculated to be $-$551 $\mu$Hz at 300 K. The temperature dependence of the BBRz shift in the static polarizability limit is given by the blue-dashed curve in Fig. 4 (a). Our final estimation of the BBRz shift is obtained by using the all-frequency dynamic polarizability integral given by Eq. (5) and the result is found to be 11.1 $\mu$Hz at 300 K. The temperature dependence of the BBRz shift from the dynamic polarizability integral approach is shown by the red curve in Fig. 4 (b). It is obvious that there are large contradictions between these two results. The curve in Fig. 4 (a) following the static-limit formula Eq. (11) is a simple quartic function curve, whereas the curve in Fig. 4 (b) drawn from the dynamic polarizability integral formula Eq. (5) bends in the opposite direction. The BBRz shift at 300 K estimated from the dynamic integral is only about 2% of that from the static limit with an opposite sign. This is mainly because the absolute value of $\beta_{{}^{3}P_{0}}(\omega)$ descends fast with the increase of the frequency in the blue-detuning region to the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance [see Fig. 3(b)]. Since the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance dominates the contribution of $\Delta\beta(\omega)$ [$\approx\beta_{{}^{3}P_{0}}(\omega)$] in the BBR region, it can be inferred from Eq. (6) that $|\Delta\beta(\omega)|\lesssim\Delta\beta(0)$ in the region $\omega\gtrsim\sqrt{2}\omega_{{}^{3}P_{0}-^{3}P_{1}}=86$ cm^-1. Moreover, the wave number for the peak intensity of the BBR at 300 K is approximately 588 cm^-1, which is about ten times larger than the wave number of the ${}^{3}P_{0}-^{3}P_{1}$ transition ($\omega_{{}^{3}P_{0}-^{3}P_{1}}$ = 61 cm^-1). As a consequence, the BBRz shift is highly suppressed by blue-detuning to the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance for which $|\Delta\beta(\omega)|\approx\Delta\beta(0)(\frac{\omega_{{}^{3}P_{0}-^{3}P_{1}}}{\omega})^{2}\approx\frac{\Delta\beta(0)}{100}$.

Apart from the main components of the BBR spectrum that are distributed in the blue-detuning region to the $3s3p~{}^{3}P_{0}-3s3p~{}^{3}P_{1}$ resonance, there are also small weak components distributed in the red-detuning region [see Fig. 3 (b)]. They lead to a small cancellation of the contributions to the BBRz shift due to opposite values of $\Delta\beta(\omega)$ on both sides of the resonance. When the temperature declines, the intensity of the BBR decreases and meanwhile their distributions are red-shifted, such that there are more cancellated proportions in the contributions to the BBRz shift. In a more detailed view of the curve in Fig. 4 (b), we can find a reversal behavior along with the temperature change in a low-temperature range, which is caused by variations of the cancellations between the contributions from both sides of the resonant line when the resonant line across the main BBR spectrum. The reversal behavior inevitably results in a vanishing BBRz shift at a zero-crossing point, which is predicted at a critical temperature $T_{c}=34$ K for the Al⁺ clock. At this point, the distribution of the BBR spectrum will be divided by the resonant line approximately in halves. From the Wien’s displacement law, we can infer that the peak intensity of the BBR at 34 K is actually at $\omega_{\textrm{peak}}=66$ cm^-1, which is very close to the resonant frequency as $\omega_{\textrm{reson}}=61$ cm^-1. When the temperature drops below $T_{c}$, contributions from the red-detuned components go beyond that from the blue-detuned components. When the temperature further declines, the BBR spectrum can be even shifted to the red-detuning region, and the curve describing the temperature dependence of the BBRz shift in Fig. 4(b) can approach the curve in Fig. 4(a), which means that the static polarizability limit can be applicable in the low-temperature condition.