Roles of electron correlation effects for accurate determination of $g_j$ factors of low-lying states of $^{113}$Cd$^+$ and their applications to atomic clock

TL;DR

This paper accurately calculates the $g_j$ factors of low-lying states in $^{113}$Cd$^+$ using relativistic coupled-cluster theory, considering electron correlation, Breit, and QED effects, with applications to atomic clock frequency shifts.

Contribution

It introduces a comprehensive RCC-based approach including triples, Breit, and QED effects for precise $g_j$ factor determination in Cd$^+$ ions, improving accuracy over previous methods.

Findings

Significant triples contributions improve accuracy of $g_j$ factors.

Estimated Zeeman shift in $^{113}$Cd$^+$ microwave clock transition.

Validated energy levels against experimental data.

Abstract

We investigate roles of electron correlation effects in the determination of $g_j$ factors of the $ns~^2S_{1/2}$ ($n$=5,6,7), $np~^2P_{1/2,3/2}$ ($n$=5,6), $5d~^2D_{3/2,5/2}$, and $4f~^2F_{5/2,7/2}$ states of the singly ionized cadmium (Cd$^+$) ion. Single and double excited configurations along with important valence triple excited configurations through relativistic coupled-cluster (RCC) theory are taken into account for incorporating electron correlation effects in our calculations. We find significant contributions from the triples to the lower $S$ and $P$ states for attaining high accuracy results. The contributions of Breit interaction and lower-order quantum electrodynamics effects, such as vacuum polarization and self-energy corrections, are also estimated using the RCC theory and are quoted explicitly. In addition, we present energies of the aforementioned states from our calculations and compare them with the experimental results to validate $g_j$ values. Using the $g_j$ factor of the ground state, systematical shift due to the Zeeman effect in the microwave clock frequency of the $|5s~^2S_{1/2}, F=0,m_F=0 \rangle \leftrightarrow |5s~^2S_{1/2}, F=1,m_F=0 \rangle$ transition in $^{113}$Cd$^+$ ion has been estimated.

Tables

Table 1. Table 1: Electron affinities (EAs) of the 5 − 7 ​ s ​ S 1 / 2 2 5 7 𝑠 superscript subscript 𝑆 1 2 2 5-7s~{}^{2}S_{1/2} , 5 − 6 ​ p ​ P 1 / 2 , 3 / 2 2 5 6 𝑝 superscript subscript 𝑃 1 2 3 2 2 5-6p~{}^{2}P_{1/2,3/2} , 5 ​ d ​ D 3 / 2 , 5 / 2 2 5 𝑑 superscript subscript 𝐷 3 2 5 2 2 5d~{}^{2}D_{3/2,5/2} , and 4 ​ f ​ F 5 / 2 , 7 / 2 2 4 𝑓 superscript subscript 𝐹 5 2 7 2 2 4f~{}^{2}F_{5/2,7/2} states (in cm -1 ) of 113 Cd + from DHF, second-order relativistic many-body perturbation theory (RMBPT(2)) and RCCSD methods are given. Corrections from the Breit ( Δ Δ \Delta Breit) and low-order QED ( Δ Δ \Delta QED) interactions are quoted separately. The final results are obtained by the RCCSD values adding Δ Δ \Delta Breit and Δ Δ \Delta QED are compared with the experimental values listed in the NIST database NIST . Uncertainties to our final results are estimated using triple excitations in the perturbative approach. Differences between our results from the NIST data are mentioned as Δ Δ \Delta in percentage.

State	DHF	RMBPT(2)	RCCSD	$\Delta$Breit	$\Delta$QED	Final	NIST	$\Delta$
$5s{}_{}{}^{2}S_{1/2}^{}$	-124567.93	-138618.14	-136012.37	84.10	60.14	-135868(587)	-136374.34	0.4
$5p{}_{}{}^{2}P_{1/2}^{}$	-84902.34	-93102.78	-91847.93	83.31	7.58	-91757(306)	-92238.40	0.5
$5p{}_{}{}^{2}P_{3/2}^{}$	-82870.41	-90363.14	-89347.42	50.24	-0.64	-89298(295)	-89755.93	0.5
$6s{}_{}{}^{2}S_{1/2}^{}$	-51061.62	-53896.08	-53315.36	18.02	12.42	-53285(147)	-53383.93	0.2
$5d{}_{}{}^{2}D_{3/2}^{}$	-45146.73	-46880.67	-46602.79	3.85	0.38	-46599(94)	-46685.35	0.2
$5d{}_{}{}^{2}D_{5/2}^{}$	-45009.55	-46667.87	-46445.57	-0.60	0.01	-46446(90)	-46530.84	0.2
$6p{}_{}{}^{2}P_{1/2}^{}$	-39865.05	-41935.34	-41536.83	22.69	1.98	-41512(78)	-41664.22	0.4
$6p{}_{}{}^{2}P_{3/2}^{}$	-39241.67	-41181.53	-40844.47	13.81	-0.25	-40831(75)	-40990.98	0.4
$7s{}_{}{}^{2}S_{1/2}^{}$	-28191.88	-29280.12	-29045.33	7.11	4.87	-29033(58)	-29073.79	0.1
$4f{}_{}{}^{2}F_{5/2}^{}$	-27539.36	-27884.18	-27880.85	0.13	0.11	-27881(22)	-27955.19	0.3
$4f{}_{}{}^{2}F_{7/2}^{}$	-27542.58	-27886.78	-27884.13	0.23	0.11	-27884(22)	-27942.22	0.2

Table 2. Table 2: The g j subscript 𝑔 𝑗 g_{j} factors of the 5 − 7 ​ s ​ S 1 / 2 2 5 7 𝑠 superscript subscript 𝑆 1 2 2 5-7s~{}^{2}S_{1/2} , 5 − 6 ​ p ​ P 1 / 2 , 3 / 2 2 5 6 𝑝 superscript subscript 𝑃 1 2 3 2 2 5-6p~{}^{2}P_{1/2,3/2} , 5 ​ d ​ D 3 / 2 , 5 / 2 2 5 𝑑 superscript subscript 𝐷 3 2 5 2 2 5d~{}^{2}D_{3/2,5/2} , and 4 ​ f ​ F 5 / 2 , 7 / 2 2 4 𝑓 superscript subscript 𝐹 5 2 7 2 2 4f~{}^{2}F_{5/2,7/2} states. The Dirac contributions to the g j subscript 𝑔 𝑗 g_{j} factor denoted by g j D superscript subscript 𝑔 𝑗 𝐷 g_{j}^{D} from the DHF and RCCSD methods along with corrections due to the perturbatively triple excitation terms and the Breit and QED electronic interactions are given. Contributions to Δ ​ g j Q Δ superscript subscript 𝑔 𝑗 𝑄 \Delta g_{j}^{Q} from the DHF and RCCSD method are quoted correspondingly. The final values, g j D + Δ ​ g j Q superscript subscript 𝑔 𝑗 𝐷 Δ superscript subscript 𝑔 𝑗 𝑄 g_{j}^{D}+\Delta g_{j}^{Q} by adding corrections to g j D superscript subscript 𝑔 𝑗 𝐷 g_{j}^{D} , are taken from the RCCSD method. The uncertainties are mentioned in the parentheses by estimating as half of the contributions due to the perturbatively triple excitation terms.

State	Contributions to $g_j^D$		Corrections to $g_j^D$			Contributions to $\Delta g_j^Q$		Final
	DHF	RCCSD	Triples	Breit	QED	DHF	RCCSD
$5s{}_{}{}^{2}S_{1/2}^{}$	1.999876	2.001624	-0.001064	-0.000020	-0.0000050	0.002320	0.002324	2.00286(53)
$5p{}_{}{}^{2}P_{1/2}^{}$	0.666604	0.666568	0.001662	0.000013	0.0000007	-0.000773	-0.000773	0.66747(83)
$5p{}_{}{}^{2}P_{3/2}^{}$	1.333283	1.333522	0.000860	-0.000003	-0.0000006	0.000773	0.000773	1.33515(43)
$6s{}_{}{}^{2}S_{1/2}^{}$	1.999967	2.000202	-0.000253	-0.000003	-0.0000007	0.002320	0.002321	2.00227(13)
$6p{}_{}{}^{2}P_{1/2}^{}$	0.666647	0.666657	0.000407	0.000004	0.0000003	-0.000773	-0.000773	0.66630(20)
$6p{}_{}{}^{2}P_{3/2}^{}$	1.333315	1.333396	0.000189	-0.000001	-0.0000003	0.000773	0.000773	1.33436(1)
$7s{}_{}{}^{2}S_{1/2}^{}$	1.999984	2.000070	-0.000099	-0.000001	-0.0000003	0.002320	0.002320	2.00229(5)
$5d{}_{}{}^{2}D_{3/2}^{}$	0.799982	0.800013	0.000645	0.000002	-0.0000001	-0.000464	-0.000464	0.80020(32)
$5d{}_{}{}^{2}D_{5/2}^{}$	1.199982	1.200094	0.000375	-0.000002	-0.0000014	0.000464	0.000464	1.20093(19)
$4f{}_{}{}^{2}F_{5/2}^{}$	0.857136	0.857140	-0.000027	0.000000	0.0000001	-0.000331	-0.000331	0.85678(1)
$4f{}_{}{}^{2}F_{7/2}^{}$	1.142850	1.142855	-0.0000427	0.000000	-0.0000001	0.000331	0.000331	1.14314(2)

Table 3. Table 3: The contributions to g j D superscript subscript 𝑔 𝑗 𝐷 g_{j}^{D} obtained in the perturbative triples using the RCCSDpT method.

States	$T_2^{†}O{S}_{3v}^{pert}$	$S_{1v}^{†}{T}_2^{†}O{S}_{3v}^{pert}$	$T_3^{pert†}O{T}_3^{pert}$	$S_{3v}^{pert†}O{S}_{3v}^{pert}$
$5s{}_{}{}^{2}S_{1/2}^{}$	-0.001070	-0.000037	0.000154	-0.000111
$5p{}_{}{}^{2}P_{1/2}^{}$	0.000830	0.000050	0.000033	0.000749
$5p{}_{}{}^{2}P_{3/2}^{}$	0.000137	0.000011	0.000073	0.000639
$6s{}_{}{}^{2}S_{1/2}^{}$	-0.000286	-0.000030	0.000026	0.000037
$6p{}_{}{}^{2}P_{1/2}^{}$	0.000173	-0.000018	0.000008	0.000244
$6p{}_{}{}^{2}P_{3/2}^{}$	-0.000001	-0.000014	0.000018	0.000186
$7s{}_{}{}^{2}S_{1/2}^{}$	-0.000107	-0.000011	0.000010	0.000009
$5d{}_{}{}^{2}D_{3/2}^{}$	0.000025	$\sim 0.0$	0.000027	0.000586
$5d{}_{}{}^{2}D_{3/2}^{}$	-0.000056	$\sim 0.0$	0.000041	0.000389
$4f{}_{}{}^{2}F_{5/2}^{}$	-0.000027	$\sim 0.0$	$\sim 0.0$	$\sim 0.0$
$4f{}_{}{}^{2}F_{7/2}^{}$	-0.000043	$\sim 0.0$	$\sim 0.0$	$\sim 0.0$

Equations

$$H_{mag}$$

$$\displaystyle{\vec{\cal M}}=\sum_{i,q=-1,0,1}\mbox{\boldmath${\vec{\mu}}$}_{q}^{(1)}(r_{i})=\sum_{i}r_{i}\left\{\mbox{\boldmath${\vec{\alpha}}$}_{i}\otimes{\vec{\textbf{C}}}^{(1)}\right\}^{(1)}.$$

$$g_j^D=-\frac{1}{{\mu }_B}\frac{\vec{\mathcal{M}}}{\vec{\text{J}}}$$

$$g_j^D=-\frac{1}{2{\mu }_B}\frac{⟨J||\vec{\mathcal{M}}||J⟩}{\sqrt{J(J+1)(2J+1)}}.$$

$$\displaystyle\langle\kappa_{f}||\mbox{\boldmath${\mu}$}^{(1)}||\kappa_{i}\rangle$$

$$⟨{\kappa }_f||{\text{C}}^{(k)}||{\kappa }_i⟩$$

$$\Pi (l_{{\kappa }_f},k,l_{{\kappa }_i})$$

$$g_f$$

$$\Delta H_{mag}$$

$$\Delta g_j^Q=0.001160\frac{⟨J||\Delta \vec{\mathcal{M}}||J⟩}{\sqrt{J(J+1)(2J+1)}},$$

$$\displaystyle\langle\kappa_{f}||\Delta\mbox{\boldmath${\mu}$}^{(1)}||\kappa_{i}\rangle$$

$$\Delta g_j^{\mathrm{NR}}=\frac{(\alpha Z{)}^2}{n^2}\frac{m}{M},$$

$$H^{DC}=\sum_{i}\left[c\mbox{\boldmath$\alpha$}_{i}\cdot\textbf{p}_{i}+(\beta_{i}-1)c^{2}+V_{n}(r_{i})\right]+\sum_{i,j>i}\frac{1}{r_{ij}},\\$$

$$V^{B}(r_{ij})=-\frac{[\mbox{\boldmath$\alpha$}_{i}\cdot\mbox{\boldmath$\alpha$}_{j}+(\mbox{\boldmath$\alpha$}_{i}\cdot\mathbf{\hat{r}_{ij}})(\mbox{\boldmath$\alpha$}_{j}\cdot\mathbf{\hat{r}_{ij}})]}{2r_{ij}},$$

$$|{\Psi }_v⟩=e^T\{1+S_v\}|{\Phi }_v⟩,$$

$$⟨{\Phi }_0^{\ast }|{\overline{H}}_N|{\Phi }_0⟩=0$$

$$\langle\Phi_{v}^{*}|\big{(}\overline{H}_{N}-\Delta E_{v}\big{)}S_{v}|\Phi_{v}\rangle=-\langle\Phi_{v}^{*}|\overline{H}_{N}|\Phi_{v}\rangle,$$

$$\Delta E_v=⟨{\Phi }_v|{\overline{H}}_N\left\{1+S_v\right\}|{\Phi }_v⟩-⟨{\Phi }_0|{\overline{H}}_N|{\Phi }_0⟩.$$

$$T=T_1+T_2\text{and}{S}_v=S_{1v}+S_{2v}.$$

$$\frac{⟨{\Psi }_v|O|{\Psi }_v⟩}{⟨{\Psi }_v|{\Psi }_v⟩}$$

$$T_{3}^{\rm{pert}}=\frac{1}{6}\sum_{abc,pqr}\frac{\big{(}H_{N}T_{2}\big{)}_{abc}^{pqr}}{\epsilon_{a}+\epsilon_{b}+\epsilon_{c}-\epsilon_{p}-\epsilon_{q}-\epsilon_{r}}$$

$$S_{3v}^{\rm{pert}}=\frac{1}{4}\sum_{ab,pqr}\frac{\big{(}H_{N}T_{2}+H_{N}S_{2v}\big{)}_{abv}^{pqr}}{\Delta E_{v}+\epsilon_{a}+\epsilon_{b}-\epsilon_{p}-\epsilon_{q}-\epsilon_{r}},$$

$$W(0,0,B)$$

$$W(1,0,B)$$

$$W(1,\pm 1,B)$$

$$\displaystyle W(0,0,B)-W(1,0,B)\simeq-hA_{hf}-\frac{[g_{j}-g_{I}]^{2}\mu_{B}^{2}B^{2}}{2hA_{hf}}.\ \ \$$

$$\Delta {\nu }_{Zeem}^{(2)}(B)=-\frac{[g_j-g_I{]}^2{\mu }_B^2{B}^2}{2h^2{A}_{hf}}.$$

$$\delta [\Delta {\nu }_{Zeem}^{(2)}(B)]$$

$$W(0,0,B)-W(1,1,B)$$

$$\displaystyle W(0,0,B)-W(1,-1,B)\simeq-hA_{hf}+\frac{[g_{j}+g_{I}]\mu_{B}B}{2}.\ \ \$$

Full text

I Introduction

Urge of high precision atomic clocks for both scientific and commercial applications are well acknowledged in many areas of physics. Today optical atomic clocks offer the most precise frequencies to recalibrate unit of time (i.e. second). Howbeit, use of microwave atomic clocks are extremely robust in numerous scientific and industrial fields including satellite navigation systems, network synchronization, timekeeping applications and defense systems etc. Weyers-MT-2018 owing to their several order lower frequencies than the optical clock frequencies. Mostly microwave clocks are based on the neutral atoms, but making these clocks using singly charged ions have many advantages. They can be more compact in size and consume low power, that are the imperative criteria for making a portable atomic clock. Compared with the other developed trapped ion microwave clocks, e.g. 199Hg+ Phoonthong-APB-2014 ; Burt-trans-2016 , microwave clock using 113Cd*+* has a unique feature as its cooling and pumping lines have the frequency difference of only 800 MHz. Therefore, its cooling, pumping and detecting processes can be carried out by the same laser. This is advantageous for making miniaturized atomic clock for aerospace applications Zhangjw-PRA-2012 . Since 2012, we have progressed a lot in making high-performance miniaturized cadmium ion atomic clocks Zhangjw-PRA-2012 ; Wangsg-OE-2013 ; Zhangjw-APB-2014 ; Miaok-OL-2015 . We achieved the frequency uncertainty and stability for the ground state hyperfine splitting to $6.6\times 10^{-14}$ and $6.1\times 10^{-13}/\sqrt{\tau}$, where $\tau$ is the average time, respectively, in 2015 Miaok-OL-2015 . We have accomplished sympathetic cooling of Cd+ ions in the mean time, which lowers the second-order Doppler shift and dead-time drastically Zuoyn-arxiv-2019 . In order to improve uncertainty of the clock frequency to below $10^{-16}$ or making comparable with the currently available 133Cs fountain clocks, one of the important tasks for us is to estimate the second-order Zeeman shift accurately.

In the microwave atomic clocks a small external magnetic field is applied to break the degeneracy of the ground state hyperfine levels. The fluctuation arising from the background magnetic field can be suppressed down to $10^{-9}$ Tesla or even lower by using multi-layer magnetic shields. When the stability of the background magnetic field is well controlled, it becomes important to calibrate the applied magnetic field very strictly. This can affect precise estimate of the Zeeman shift, hence accurate determination of the clock transition. For this purpose precise knowledge of the $g_{j}$ factor of the ground state of Cd+ ion is strongly desired. There has also been immense interest to understand roles of various physical effects for the accurate determination of the $g_{j}$ factors of atomic states. Most of the $g_{j}$ factor studies are concentrated on the highly charged ions (HCIs) with few electrons in which relativistic effects play the crucial roles for their accurate determination. It demands for rigorous treatments of quantum electrodynamics (QED) to higher-orders and nuclear recoil (NR) effects. For example, agreement between the theoretical and experimental values of the $g_{j}$ factors in the H-like C5+, O7+, and Si13+ ions Verdu-PRL-2004 ; Sturm-PRA-2013 ; Sturm-nature-2014 ; Kohler-JPB-2015 , and in the Li-like Si11+ and Ca17+ Wagner-PRL-2013 ; Volotka-PRL-2014 ; Kohler-ncom-2016 HCIs at the 8th or even at the lower decimal places serve as the most stringent test of the bound-state QED theory. Such studies yield unprecedented values of the ratio between the mass of an electron and the mass of a proton, and the fine structure constant Sturm-atoms-2017 . Contrasting to the great success in achieving high precision values of the $g_{j}$ factors in few-body systems, accurate determination of these factors in many-electron systems is challenging owing to strong electron correlation effects associated with this property. In the neutral atoms or singly charged ions, these interactions contribute predominantly to the $g_{j}$ factors over the QED interactions Veseth-PRA-1980 ; Veseth-JPB-1983 ; Dzuba-PS-1985 ; Gossel-PRA-2013 . In this context, the relativistic coupled-cluster (RCC) theory, which is currently known as one of the leading quantum many-body methods and has been referred to as the gold standard for treating electron correlations, is apt to determine atomic properties including $g_{j}$ factors accurately. This theory was employed to study the $g_{j}$ factors of the ground states of Li, Be+, and Ba+ by Lindroth and Ynnerman Lindroth-PRA-1993 , but they had determined only the corrections to the $g_{j}$ factors ($\Delta g_{j}$) due to electron correlation effects with respect to the bare Dirac values. Recently, the roles of electron correlation effects to the net $g_{j}$ factors of ground and few excited states of Ca+ were demonstrated by employing RCC theory SahooKumar-PRA-2017 .

In this work, we have applied the RCC theory to calculate the $g_{j}$ factors of the ground state $5s~{}^{2}S_{1/2}$ and some of the low-lying $ns~{}^{2}S_{1/2}$ ($n$=6,7), $np~{}^{2}P_{1/2,3/2}$ ($n$=5,6), $5d~{}^{2}D_{3/2,5/2}$, and $4f~{}^{2}F_{5/2,7/2}$ excited states of Cd+. We have also determined electron affinities (EAs) of the valence electrons of the above states with the $[4d^{10}]$ closed-shell configuration by considering singles and doubles excitations approximation in the RCC theory (RCCSD method) and compared them with the available experimental results to validate our calculations. We have incorporated contributions from the important valence triples excitations in a perturbative approach in the RCCSD method (RCCSDpT method) only in the $g_{j}$ factor evaluation expression as described in SahooKumar-PRA-2017 . Further, systematic shift due to the Zeeman effect in the microwave clock frequency of 113Cd+ ion has been estimated by using the calculated $g_{j}$ factor of its ground state.

II Theory

In the presence of an external homogeneous magnetic field ${\vec{\bf B}}$, the interaction Hamiltonian of electrons in an atomic system is given by Sakurai1967

[TABLE]

where $e$ is the electric charge of the electron, $c$ is the speed of light, ${\vec{\alpha}}$ is the Dirac operator, and ${\vec{\textbf{A}}}$ is the vector field seen by the electron located at ${\vec{\bf r}}$ due to the applied magnetic field. We can rewrite the above expression as

[TABLE]

where ${\vec{\textbf{C}}}^{(1)}$ is the Racah coefficient of rank one and magnetic moment operator given by

[TABLE]

Using this operator, the Dirac contribution to the Lande $g_{j}$ factor of a bound-state electron in an atomic system is given by

[TABLE]

with total angular momentum of the state J and the Bohr magneton $\mu_{B}=e\hbar/2m_{e}$ for mass of the electron $m_{e}$ and Planck constant $\hbar$. Using the reduced matrix element, it can be expressed as

[TABLE]

For the calculation of this factor, the single particle reduced matrix element of \mbox{\boldmath{\mu}}_{q}^{(1)} is given by

[TABLE]

where $P(r)$ and $Q(r)$ are the large and small components of the radial parts of the single particle Dirac orbitals, respectively, and $\kappa$ is the relativistic angular momentum quantum number. The reduced matrix element of the Racah ${\vec{\textbf{C}}}^{(1)}$ operator is calculated as

[TABLE]

with

[TABLE]

for the corresponding orbital momentum $l_{\kappa}$ of the orbital with relativistic quantum number $\kappa$.

For evaluating $g_{j}^{D}$ using Eq. (5), it is necessary to calculate wave functions of the states in an atomic system considering relativistic effects. It is also known that the Dirac value of Lande $g$ factor of a free electron ($g_{f}^{D}$) has significant corrections from the QED theory. The net value with the QED effects is approximately given by Czarnecki2005

[TABLE]

where $\alpha_{e}$ is the fine structure constant. To account for this correction along with $g_{j}^{D}$ (denoted by $\Delta g_{j}^{Q}$) for the net result as $g_{j}=g_{j}^{D}+\Delta g_{j}^{Q}$, we consider the additional interaction Hamiltonian with the magnetic field as Akhiezer1965

[TABLE]

where $\beta$ is the Dirac matrix, ${\vec{\Sigma}}$ is the four-component spinor and {\Delta\vec{\cal M}}=\sum_{i,q=-1,0,1}\Delta\mbox{\boldmath{\vec{\mu}}}_{q}^{(1)}(r_{i})=\sum_{i}\beta_{i}\mbox{\boldmath{\vec{\Sigma}}}_{i}. Using this Hamiltonian, we determine $\Delta g_{j}^{Q}$ as Cheng1985

[TABLE]

for which the single particle matrix element is given by

[TABLE]

Contribution due to the NR effect to the bound state $g_{j}$ factors in Cd+ can be estimated using the formula Shabaev-JPCRD-2015

[TABLE]

where $m$ and $M$ are masses of an electron and atomic nucleus, $Z$ is the atomic number, and $n$ is the principal quantum number of the interested states. This is found to be of the order of $\sim 10^{-7}$; which is neglected because such uncertainty is much below than the intended precision that can be achieved in the present work.

III Method of calculations

We consider the Dirac-Coulomb (DC) Hamiltonian to calculate the wave functions of the atomic states, which is given in atomic unit (a.u.) by

[TABLE]

where $\textbf{p}_{i}$ is the momentum operator, $V_{n}(r)$ denotes the nuclear potential, and $\frac{1}{r_{ij}}$ represents for the Coulomb potential between the electrons located at the $i$ and $j$ positions. The Breit interaction contribution is estimated by incorporating the interaction potential

[TABLE]

where $\mathbf{\hat{r}_{ij}}$ is the unit vector along $\mathbf{r_{ij}}$. Similarly, we also include effective potentials for vacuum polarization (VP) and self-energy (SE) interactions as discussed in our previous work Yuym-PRA-2019 to account for QED interactions in the determination of the atomic wave functions.

The investigated states of Cd*+* ion can be expressed in the RCC theory as Yuym2017 ; Licb2018

[TABLE]

where $|\Phi_{v}\rangle=a^{\dagger}_{v}|\Phi_{0}\rangle$ is the reference state with valence orbital $v$ for the Dirac-Hartree-Fock (DHF) wave function of the $[4s^{2}4p^{6}4d^{10}]$ closed-shell configuration. The RCC excitation operators $T$ and $S_{v}$ are responsible for exciting electrons from the $|\Phi_{0}\rangle$ and $|\Phi_{v}\rangle$ reference states, respectively. The amplitudes of these RCC operators are evaluated by solving the following equations

[TABLE]

and

[TABLE]

where $|\Phi_{0}^{*}\rangle$ and $|\Phi_{v}^{*}\rangle$ are the singly and doubly excited state configuration with respect to $|\Phi_{0}\rangle$ and $|\Phi_{v}\rangle$, respectively. The notation $\overline{H}_{N}$ is defined as $\overline{H}_{N}=(He^{T})_{l}$ with subscript $N$ means normal order form of the operator and $l$ means that all the terms are linked. The quantity $\Delta E_{v}$ corresponds to EA of the state with the valence electron $v$. We evaluate $\Delta E_{v}$ by

[TABLE]

In the RCCSD method, the singles and doubles excitations are denoted by

[TABLE]

After obtaining amplitudes of the RCC operators, the expectation value of an operator $O$ is evaluated as

[TABLE]

We adopt an iterative procedure to include contributions from the non-terminative terms from the above expressions. Here, $O$ stands for both the ${\bf{\cal M}}$ and $\Delta{\bf{\cal M}}$ operators for the evaluations of the $g_{j}^{D}$ and $\Delta g_{j}^{Q}$ contributions, respectively. In our previous work, we had observed that triples excitations play important roles in the determination of the $g_{j}$ factors SahooKumar-PRA-2017 . Inclusion of these excitations require huge computational resources, which we are lacking at present. Therefore, we take into account these contributions in the RCCSDpT method only by defining excitation operators in the perturbative approach as

[TABLE]

and

[TABLE]

where $\{a,b,c\}$ and $\{p,q,r\}$ represent for the occupied and virtual orbitals, respectively, and $\epsilon$s are their single particle orbital energies. Contributions from these operators to the $g_{j}$ factors are then given in terms of $T_{2}^{\dagger}OT_{3}^{pert}$, $S_{2v}^{\dagger}OT_{3}^{pert}$, $S_{2v}^{\dagger}OS_{3v}^{pert}$, $T_{2}^{\dagger}OS_{3v}^{pert}$, $S_{1v}^{\dagger}T_{2}^{\dagger}OS_{3v}^{pert}$, $T_{3}^{pert\dagger}OT_{3}^{pert}$, and $S_{3v}^{pert\dagger}OS_{3v}^{pert}$ along with their possible complex conjugate (c.c.) terms as part of Eq. (23) in the RCCSD expression.

IV Results and Discussion

To verify accuracies of the wave functions of the atomic states, whose $g_{j}$ factors are investigated here, we have evaluated EAs and compared them with their available experimental values. In Table 1, we give EAs of 11 low-lying states of Cd+ from the DHF, second-order relativistic many-body perturbation theory (RMBPT(2)) and RCCSD methods. As can be seen, the DHF method gives lower values while the RMBPT(2) method gives larger values compared to the results from the RCCSD method. The RCCSD results are in close agreement with the experimental results. Corrections from the Breit and QED interactions are quoted explicitly from the RCCSD method, and those corrections are found to be comparatively small. Uncertainties to our final results are given by estimating contributions from the valence triple excitations in the perturbative approach, which are reasonably large. This shows that the triple excitations are important to improve accuracies of our calculations. Our results are compared with the experimental values listed in the National Institute of Science and Technology (NIST) database NIST . Differences between our final results from these experimental values are given as $\Delta$ in percentage in the same table. This shows that our calculations agree with the experimental values at the sub-one percentage level in all the states. These uncertainties can be reduced further by incorporating full triple excitations in our RCC method. Nonetheless, this analysis shows that we shall be able to obtain $g_{j}$ factors with similar accuracies for the considered states as these quantities are sensitive to accuracies of the wave functions in the far nuclear region as for the case of energies.

After the investigation of roles of the electron correlation effects in the EAs, we present the calculations of $g_{j}$ factors of the above aforementioned states of 113Cd+. We give the $g_{j}^{D}$ values in Table 2 from the DHF and RCCSD methods along with the corrections from the terms involving perturbative triple operators of the RCCSDpT method. The estimated $\Delta g_{j}^{Q}$ corrections to the $g_{j}^{D}$ values are also quoted from the DHF and RCCSD methods in the same table. They seem to be the decisive contributions to obtain the final results. Correlation contributions to the $g_{j}^{D}$ values due to the perturbative triple excitations, and contributions from the Breit and QED interactions are shown explicitly. The final $g_{j}$ values, $g_{j}=g_{j}^{D}+\Delta g_{j}^{Q}$, of the respective states are obtained by adding all these corrections. In our calculations we find the perturbatively triple excitation terms have the sizable contributions to the $g_{j}$ factors. This suggests full account of triple and other higher level excitations would improve the results further. Nonetheless, we anticipate additional contributions from these higher level excitations will be within the half of the estimated contributions due to the perturbative triple excitations. On this basis we assign uncertainties as 50% of the perturbative excitation contributions to the final values of the $g_{j}$ factors. There are no experimental values of the $g_{j}$ factors of the considered states in Cd+ available to compare with our calculations.

The contributions to $g_{j}^{D}$ of the perturbatively triple excitations terms originate mainly from the $T_{2}^{\dagger}OS_{3v}^{pert}$, $S_{1v}^{\dagger}T_{2}^{\dagger}OS_{3v}^{pert}$, $T_{3}^{pert\dagger}OT_{3}^{pert}$, and $S_{3}^{pert\dagger}OS_{3v}^{pert}$ terms , which is given explicitly in Table 3. Computing these terms is very time consuming. As can be seen that these triple contributions are as large as compared to the correlation contributions due to the RCCSD method. Contributions from $T_{2}^{\dagger}OS_{3v}^{pert}$ are found to be relatively larger compared to the other terms followed by the $T_{3}^{pert\dagger}OT_{3}^{pert}$, $S_{1v}^{\dagger}T_{2}^{\dagger}OS_{3v}^{pert}$, and $S_{3}^{pert\dagger}OS_{3v}^{pert}$ and there are strong cancelations among the contributions from the above terms. The computation of $S_{3}^{pert\dagger}OT_{3}^{pert}$ is extremely costly, and hence it is neglected in the $g_{j}^{D}$ calculation of $4f$ states. As shown in 3 the contributions of the triple excitation terms for the $4f$ states are far less than the other states. In Fig. 1, we plot magnitudes of these contributions from the individual terms explicitly in order to highlight their roles. It can be seen from these figures that the triples excitations have substantially big contributions for the ground state and $n=5$ excited states and smaller contributions for higher-lying excited states.

Another motivation of the present work is to estimate the typical order of systematic effect due to the applied magnetic field to the microwave clock frequency of the ground state hyperfine structure splitting in 113Cd+. The present level of accuracy of this clock is $\sim 10^{-14}$ Miaok-OL-2015 and our desired precision level is $10^{-16}$. In 113Cd+, the ${}^{2}S_{1/2}~{}(F=0,m_{F}=0)\to(F=1,m_{F}=0)$ transition is considered for making clock, where $F=I\pm J$ for the nuclear spin $I$ and electron angular momentum $J$ and with its projection $m_{F}$. Under the low magnetic field, the energies $W(F,m_{F},B)$ of different hyperfine-Zeeman sublevels $|(J,m_{J})F,m_{F}\rangle$ for a given magnetic field strength $B$ can be approximately estimated as Breit1931 ; Vanier1989 ; Itano2001

[TABLE]

and

[TABLE]

where $g_{j}$ and $g_{I}$ are the electronic and nuclear $g$ factors, $\mu_{B}$ is the Bohr magneton, and $A_{hf}$ is the magnetic dipole hyperfine structure constant. Using Eqs. (26) and (27), the energy difference between the clock states of the ground state in 113Cd+ is given by

[TABLE]

As can be seen, the energy shift depends on the stray magnetic field, where the second term in Eq. (29) is referred to as the second-order Zeeman shift. Thus, the change in frequency due to the second-order Zeeman shift is expressed as

[TABLE]

As can be seen, uncertainty to this quantity depends on the uncertainties in both the $g_{j}$ factor and the applied magnetic field $B$. In our experiment, the magnetic field is generated by a pair of Helmholtz coils and the current controls the strength of the magnetic field. Thus, it is not straightforward for us to measure the applied magnetic field strength very accurately. Under such circumstances, the value of $B$ needs to be calibrated.

Assuming the $B$ value is known precisely, the fractional frequency uncertainty in $\Delta\nu^{(2)}_{Zeem}(B)$ can be obtained by

[TABLE]

where $\delta g_{j}$ is the estimated uncertainty of $g_{j}$. The above value is estimated by considering $B=10^{-7}$ Tesla for our typical condition of the experiment, $A_{hf}\simeq-15.2$ GHz Miaok-OL-2015 , and $g_{I}$=$0.6223009(9)\times 10^{-3}$ Spence1972 . By substituting our determined values $g_{j}=2.00286$ and $\delta g_{j}=\pm 0.00053$, we find that uncertainty in the second-order Zeeman shift will not affect to the clock frequency at the $10^{-16}$ precision level if the applied magnetic field is lower than the above assumed value. It is also imperative to estimate the maximum level of fluctuation in the applied in order to sustain the aforementioned precision level of the clock frequency. For this purpose, we use two magnetic-field sensitive transitions, i.e., $\nu_{+}$ and $\nu_{-}$ correspond to the transitions from $(F,m_{F})=(0,0)$ to $(F,m_{F})=(1,1)$ and from $(F,m_{F})=(0,0)$ to $(F,m_{F})=(1,-1)$, respectively. Using Eqs.(26) and (28), their energy differences at the first-order level of $B$ can be given by

[TABLE]

and

[TABLE]

This yields frequency difference as

[TABLE]

Using this expression, the magnetic field strength $B$ can be determined as

[TABLE]

This leads to uncertainty in the calibration of magnetic field with respect to the $g_{j}$ factor as

[TABLE]

Here, we have used as $\delta\nu_{\mp}\simeq 2.8$ kHz for $B=10^{-7}$ Tesla Miaok-OL-2015 along with values for other variables as defined above. According to this, the calibration of the $B$ value will be affected by the uncertainty in the value of $g_{j}$. Using our estimated $g_{j}$ value, we anticipate the uncertainty in $B$ would be less than $10^{-10}$ Tesla. This is sufficiently low to maintain the uncertainty in the fractional second-order Zeeman shift with respect to the clock frequency lower than $10^{-17}$ for the assumed magnetic field. This is good enough for ensuring the $10^{-16}$ precision level measurement of clock frequency.

V Conclusion

The $g_{j}$ factors of the ground $5s~{}^{2}S_{1/2}$ state, and the excited $6-7s~{}^{2}S_{1/2}$, $5-6p~{}^{2}P_{1/2,3/2}$, $5d~{}^{2}D_{3/2,5/2}$ and $4f~{}^{2}F_{5/2,7/2}$ states of 113Cd+ are calculated by using the relativistic coupled-cluster theory. We have also evaluated energies of these states and compared them with their experimental results to verify reliability of our calculations. We observed triples effects are significant for high precision calculation of the $g_{j}$ factors of the aforementioned states, especially for the low-lying states, in Cd+. Using the precisely determined $g_{j}$ factor of the ground state, we have analyzed typical order of systematic shift due to the second-order Zeeman effect in the clock frequency of the $|5s~{}^{2}S_{1/2},F=0,m_{F}=0\rangle\leftrightarrow|5s~{}^{2}S_{1/2},F=1,m_{F}=0\rangle$ transition in 113Cd+ ion and found it will be within the desired fractional uncertainty $10^{-16}$ level.

Acknowledgement

We would like to thank V. M. Shabaev and V. A. Dzuba for their helpful suggestions. This work is supported by the National Key Research and Development Program (NKRDP) of China, Grant No. 2016YFA0302100, the National Natural Science Foundation of China under Grant No. 11874064, and the Strategic Priority, and Research Program of the Chinese Academy of Sciences (CAS), Grant No. XDB21030300, and B.K.S. would like to acknowledge use of Vikram-100 HPC of Physical Research Laboratory (PRL), Ahmedabad, and J.Z.H., J.W.Z. and L.J.W. acknowledge the Initiative Program of State Key Laboratory of Precision Measurement Technology and Instruments.