Ting-Xian Zhang(張婷賢),Yong-Hui Zhang(張永慧), Cheng-Bin Li(李承斌),?, and Ting-Yun Shi(史庭云)

1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences,Wuhan 430071,China

2University of Chinese Academy of Sciences,Beijing 100049,China

Keywords: hyperfine interaction, Land′e g-factors, electric quadrupole moment, multiconfiguration Dirac–Hartree–Fock(MCDHF)method

1. Introduction

The extraordinary precision of optical clocks allows not only for creating improved time standards, but also for looking for changes in fundamental constants over time, measuring gravity red shift, and probing the existence of forces beyond the Standard Model.[1–3]The fractional uncertainties of a few parts in 10-18,even in 10-19,have been achieved in current leading optical clocks, such as the27Al+clock.[4–7]The dominated uncertainty and frequency shift of the clock transition are caused by the perturbations of the external fields,such as blackbody radiation (BBR) and quadratic Zeeman effect. In comparison to neutral atoms and singly charged ions,the highly charged ions(HCIs)possess optical transitions which can be extremely narrow and less sensitive to the external perturbations, since the electronic cloud is shrunk with the increasing of the ion charge.[8–10]Moreover, the sensitivity of optical transitions in HCIs to effects beyond the Standard Model and Einstein’s Theory of Relativity is enhanced by the nuclear charge and ionization degree. Therefore,HCIs are candidates for the novel clock which can improve the accuracy of the optical frequency standards and subsequent researches,such as exploring changes of the fine structure constant α.[11]In recent years, many HCIs have been suggested as the candidates for making ultra-precise optical clocks and exploring new physics beyond the standard model of particle physics.[10]And experimentally, sympathetic cooling and quantum logic spectroscopy of HCIs have been applied for the Ar13+ion,[12,13]which shed light on building the optical clock based on HCIs.

Several highly charged nickel ions have been proposed for making ultra-precise optical clock with the fractional uncertainties below 10-19level.[9,22,23]The magnetic dipole(M1) transitions between the states belonging to the configuration of the ground state in the61Ni15+and61Ni14+ions were recommended by Yudin et al. The advantages of these ions are the negligible quadrupolar shift,and the simple clock level structures of themself which are benefit to both the theoretical and experimental studies.[9]Soon afterwards,Yu and Sahoo have demonstrated that the projected fractional shift for clock transition 3s23p2Po3/2–2Po1/2 in the61Ni15+ion is below the 10-19level.[22]They also pointed out that the electric quadrupole (E2) transition 3s23p23P0–3P2in58Ni12+ion is suitable for making optical clocks, since the quality factor of this transition is larger than 1015,and the M1 transition 3s23p23P1–3P2can be used for laser cooling.[23]Recently, the six optical clock transitions (in Fig. 1) in four adjacent charge states of highly charged nickel ions, Ni11+, Ni12+, Ni14+,and Ni15+, have been studied both in theory and experiment.Theoretically,the transition energies,transition rates,and enhancement factors for the α-variation for these clock transitions have been calculated by using the multi-configuration Dirac–Hartree–Fock method(MCDHF).Experimentally,with the achievement of producing and extracting the above ions in the low-energy compact Shanghai–Wuhan Electron Beam Ion Trap,[24]the transition wavelengths of four magnetic-dipole(M1) clock transitions have been measured with an accuracy of tens of ppm,which are in agreement with their calculations.

In this work,using the MCDHF method,we calculate the hyperfine interaction constants, Land′e g-factors, and electric quadrupole moment for the states that are involved in the six optical transitions of61Ni HCIs,as shown in Fig.1.The active space approach is adopted to investigate the effects of electron correlations on the atomic parameters in detail. The above investigations allow us to obtain high-precision atomic parameters,and to evaluate reliable uncertainties for the present results.We expect our calculations could support the experimental investigations about the highly charged nickel ion optical clock.

Fig.1. Partial level structures and clock transitions for 61Ni11+, 61Ni12+, 61Ni14+,and 61Ni15+ions. Magnetic-dipole(M1)transitions are shown in red and electric-quadrupole (E2) transitions in blue. The wavelengths labeled in this figure are recommended values by NIST database.[25]

2. Theoretical method

2.1. MCDHF method

Atomic state wave functions(ASFs)in this work are generated by using the MCDHF method.[26,29]For an N-electron atomic system,the Dirac–Coulomb Hamiltonian HDCis given by

where c is the speed of light in vaccum, αiand βiare the 4×4 Dirac matrices, and Vnuc(ri) is the monopole part of the electron–nucleus interaction. In the MCDHF method,an ASF is constructed by configuration state functions(CSFs)|γJMJ〉with the same parity P, total angular momentum J, and its component along the z direction MJ,i.e.,

Here, ciis the mixing coefficient, and γistands for other appropriate quantum number of the CSF. Every CSF is a linear combination of products of one-electron Dirac orbitals. The mixing coefficients and the orbitals are optimized simultaneously in the self-consistent field(SCF)procedure to minimize energies of levels concerned.Once a set of orbitals is obtained,the relativistic configuration interaction(RCI)calculations can be carried out to capture more electron correlations by optimizing the mixing coefficients.[27,28]The Breit interaction in the low-frequency approximation

is included in the RCI Hamiltonian in order to give the correct energy level sequence. In addition, QED corrections are divided into vacuum polarization and self-energy correction and both parts are included in the RCI procedure. The vacuum polarization part is described by the Uehling model potential,and the self-energy correction is given as a sum of the oneelectron corrections,which are evaluated by using a screened hydrogenic model.[29,30]

2.2. Hyperfine interaction

The hyperfine interaction is caused by the interaction between the electrons and the electromagnetic multipole moments of the nucleus. The corresponding Hamiltonian is in the form of

The hyperfine interaction couples the electronic angular momentum J and nuclear spin I to a conserved angular momentum F, F =I+J. In the first-order approximation, the hyperfine energy correction is given by

2.3. Electric quadrupole moment

Because the atomic charge distribution is not always spherical, there is a deviation from the spherical shape, in which case the atomic state has an electric quadrupole moment.The Hamiltonian for the interaction between the electric quadrupole moment and the electric-field gradient is

2.4. Zeeman effect of hyperfine level

The Zeeman interaction of the atom with the magnetic field B can be written as[34,35]For an N-electron atom,the electronic tensor operator is

Here,gJand gFare Land′e g-factors for the fine and hyperfine states.

3. Computational model

For a many-electron atomic system, the description of electron correlations in the computational model plays a key role on the precision of the calculated atomic parameters. The active space approach is utilized to capture the effect of electron correlations systematically.[30,36]In this approach, the CSFs generated through single(S)and double(D)excitations from the occupied orbitals in the single reference(SR)configuration to virtual orbitals capture the first-order electron correlation effects. In order to achieve high accuracy,the higherorder electron correlations beyond the first-order electron correlation also need to be considered. The CSFs generated through SD excitations from the multireference(MR)configurations set can capture the higher-order electron correlation for the complex atomic system,i.e.,MR-SD approach.[37]The MR configurations set is formed by selecting those important CSFs in the first-order ASF. In Table 1, the reference configuration sets, the virtual orbitals, and the number of CSFs corresponding to computational processes of the nickel ions concerned are shown.

For each ion concerned, the computation starts from the Dirac–Hartree–Fock SCF calculation. In this step, the occupied orbitals in the reference configuration are optimized as spectroscopic orbitals(labeled as DHF).As shown in Table 1,the reference configurations for the Ni11+, Ni12+, Ni14+, and Ni15+ions are 3s23p5,3s23p4,3s23p2,and 3s23p,respectively.The 3s and 3p orbitals are the valence orbitals and others are the core orbitals. In the SCF calculations, the correlation between the valence electrons,VV correlation,and that between the valence electrons and core electrons, CV correlation, are included. The configuration expansions are generated by restricting the SD excitations from the single reference configuration. The restriction to the double excitations is that at most one electron in the core subshell can be substituted at one step. The virtual orbitals are augmented layer by layer up to nmax=12 and lmax=6,nmaxand lmaxare the maximum principal quantum number and the maximum angular quantum number of the virtual orbitals, and only the last added virtual orbitals are variable in one step. The orbital set obtained in this CV computational model is used for the subsequent RCI calculations. The correlations in the core electrons, referred to as CC correlations,are included in the RCI computation by allowing the SD excitations from the core orbitals to all the virtual orbitals. In order to analyze the effects from different core electrons on the physical quantities under investigation,we open up the core subshell successively. Corresponding models are labeled with CC2p, CC2s, and CC1s, respectively,and the subscripts 2p,2s,and 1s mean that the CC correlations between these orbitals are involved. So far,the SD-excitation CSFs from the single reference configuration are all included.It means that all the first-order electron correlations are included in our calculations.

As mentioned earlier, the higher-order correlations between the valence orbitals and the core orbitals with n=2 are captured by MR-SD approach. We select the dominant CSFs with the weights of|ci|＞0.01 in the configuration spaces obtained in the CC1smodel to form the multireference configuration set and show them in Table 1. To control the number of CSFs, the SD excitations are allowed from orbitals with n ≥2 in the MR configurations to the first three layers of virtual orbitals. These computational models are marked as MR.Finally, the Breit interaction and QED corrections are evaluated based on the MR model. In practice, we employ the GRASP2018[38]package to perform present calculations.

Table 1. The reference configurations sets,the active orbitals(AO)set,and the number of CSFs(NCSF)for the concerned states of the Ni11+,Ni12+,Ni14+,and Ni15+ ions in various correlation models. The active orbitals are labelled by the maximum principal quantum number nmax and orbital angular momentum l.

4. Results and discussion

4.1. Hyperfine interaction constants

Computational uncertainty is caused by the electron correlations neglected in the computational model. In this work all the first-order electron correlations and the dominant higher-order correlations are included. The main uncertainty arises from the neglected higher-order correlations related to the 1s electrons.But these effects should be smaller than those from the outer shells because of the stronger nuclear Coulomb potential in the inner region. Conservatively,we treat the contribution of the higher-order correlations captured in the MR model as the uncertainty due to the neglected higher-order correlations. For example, the contribution of the higher-order correlation captured in the MR model to the Ahfs(2Po1/2) of the61Ni11+ions is 0.48%, and it is treated as the uncertainty caused by the neglected higher-order correlations. Additionally, the truncation uncertainty is estimated according to the convergence trends of the various correlations. For example,the truncation uncertainty of Ahfs(2Po1/2) for the61Ni11+ions is 0.03%. The total uncertainty of Ahfs(2Po1/2)for the61Ni11+ions is 0.51%. Yu and Sahoo have calculated the hyperfine interaction constants of the61Ni15+ions by using the relativistic coupled-cluster (RCC) method.[22]Our calculated results are in excellent agreement with theirs. Using the hyperfine interaction constants of the MR-BQ models, we evaluate the transition frequency corrections of the concerned clock transitions in Fig. 1 and show them in Table 3. Accordingly, the changes of the wavelength caused by the hyperfine interaction are about 0.001 nm for the c and d lines,and 0.01 nm for the other lines in Fig.1.

Table 2. The hyperfine interaction constants Ahfs and Bhfs (all in MHz)for the concerned states in the 61Ni11+, 61Ni12+, 61Ni14+,and 61Ni15+ions. The numbers in the parentheses are the uncertainties for our results.

Table 3. The hyperfine interaction corrections of the transition frequency Δfhfs (in MHz)for the clock transitions in Fig.1. Numbers in parentheses stand for the uncertainties.

4.2. Land′e g-factor

Table 4. The Land′e g-factors of the concerned states in the 61Ni11+,61Ni12+, 61Ni14+, and 61Ni15+ ions. The numbers in the parentheses are the uncertainties for our results.

4.3. Electric quadrupole moments

In Table 5,the electric quadrupole moments of the clock states in the61Ni11+,12+,14+,15+ions are presented. Similar to the cases of the hyperfine interaction constants and Land′e g-factors, the electric quadrupole moments of the states concerned are sensitive to the VV and CV electron correlations as well. The effects of the CC correlations and higher-order correlations on the electric quadrupole moments under investigation are about 1%,and opposite to each other. Thus,to obtain Θ with high precision, both the CC and the higher-order correlation are indispensable. The Θ(3P2)of the Ni12+and Ni14+ions are more sensitive to the Breit interaction and QED effect than other Θ in Table 5. The corrections of the Breit and QED to the Θ(3P2)of the Ni12+ion are 0.95%and 0.06%, respectively,and 3.3%and 0.25%for the Ni14+ion. For other Θ in Table 5, the influences of the Breit and QED effect are less than 0.1% and 0.01%, respectively. Due to the contributions from the neglected higher-order corrections and truncations of the active orbital sets, 1% uncertainties are given for present electric quadrupole moments. Yu and Sahoo[23]have calculated the Θ(3P2) of the Ni12+ion using the DIRAC program that based on Fock space coupled cluster(FSCC)method.Our calculated results agree with their results.

Table 5. The electric quadrupole moments(in a.u.) of the concerned states in the 61Ni11+, 61Ni12+, 61Ni14+,and 61Ni15+ ions. The numbers in the parentheses are the uncertainties for our results.

4.4. Electric quadrupole shift

The electric quadrupole shift caused by the interaction between the electric quadrupole moments of the clock states with the gradient of the electric field is one of the main systematic shifts in the atomic clock. For the isotopes with nonzero nuclear spin, the quadrupole shift can be either eliminated or strongly suppressed by choosing clock transitions between the hyperfine levels which make the 6j symbol in Eq.(14)equal to zero. For the61Ni11+and61Ni15+ions,it is proper to choose the transitions between2Po3/2F =0 and2Po1/2F =1,2. In the61Ni12+and61Ni14+ions,the suitable states are3P0F =3/2,3P1F =1/2, and3P2F =1/2. Since Θ(2)-qis an even-parity operator of rank 2, the electric quadrupole moments of the states with F ＜1 are zero according to the selection rules.Thus, for the M1 clock transition in61Ni12+,3P1F =1/2–3P2F = 1/2, the quadrupolar shift is zero identically. For the other clock transitions concerned, the electric quadrupolar shifts are suppressed,which are benefited from that the ΘJof the2Po1/2and3P0states are zero. Moreover, the electric quadrupole shift can be canceled by two methods experimentally. Both of them are based on measuring the Zeeman spectrum of the clock transition.[33,39]The averaged value of the transition frequencies that correspond to three magnetic fields with the same magnitude but with mutually perpendicular directions does not contain the electric quadrupole shift. Alternatively,the electric quadrupole shift can be eliminated when an average over all MFis done.

4.5. Zeeman shift

The Zeeman shift of the clock transition frequency is caused by the residual magnetic field B of the environment.Generally,it is sufficient to consider only the first and secondorder Zeeman effects on the clock transitions frequency. The first-order Zeeman shift can be canceled by measuring two transitions with opposite magnetic quantum number MFand averaging the frequencies.[40]Indeed,for the clock transitions between the sublevels2Po1/2F = 0 MF= 0 and2Po3/2F =1,2 MF=0 of the61Ni11+,15+ions, the first-order Zeeman shift equals zero. The quadratic Zeeman shift can be evaluated by ΔE(2)m=CM2B2,where CM2～(gJμB-gIμN(yùn))/h2Ahfs.Using the calculated Land′e g-factors and magnetic dipole hyperfine interaction constants,we estimate theCM2for the clock transitions investigated in this work, and the values are listed in the third column of Table 6. Table 6 also gives the transition frequencies of the clock transitions concerned which are～1014Hz. By assuming a small magnetic field B=1 μT,the relative quadratic Zeeman shifts are evaluated. From Table 6,it can be seen that the quadratic Zeeman shifts are nonnegligible for the six clock transitions concerned,and we must identify the shifts precisely. The g-factor plays a key role in the accurate diagnosis of the magnetic field strength,which is necessary for determining the quadratic shift precisely.

Table 6. The quadratic Zeeman shift coefficients CM2 (in Hz/T2) and the transition frequencies f (in Hz) of the concerned clock transitions in the 61Ni11+, 61Ni12+, 61Ni14+, and 61Ni15+ ions. The relative quadratic Zeeman shifts are evaluated by assuming a small magnetic field B=1 μT.Numbers in square brackets stand for the power of 10.

5. Conclusion

In summary,we calculated the hyperfine interaction constants, Land′e g-factors, and electric quadrupole moments of the sates involved in the clock transitions in61Ni11+,61Ni12+,61Ni14+,and61Ni15+ions using the MCDHF method. Based on the active space approach, the effects of the electron correlations, the Breit interaction, and QED effect on the above atomic parameters were discussed. We found that the VV and CV electron correlations make significant contributions to these physical quantities concerned. Moreover, both the CC and higher-order electron correlations should be included in order to acquire high precision atomic parameters. Reliable uncertainties were evaluated by taking into account the contributions from the neglected high-order corrections and truncations effects of the active orbital sets. In addition, we discussed the electric quadrupole shifts and Zeeman shifts of the clock transitions concerned. For the clock transitions under investigation,the electric quadrupole shifts and linear Zeeman shifts can be eliminated or suppressed. While the quadratic Zeeman shifts should be measured precisely for achieving the uncertainty below the 10-19level.

Acknowledgment

We would like to thank Li Ji-Guang of Institute of Applied Physics and Computational Mathematics for many useful discussions.