---
abstract: '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 the $^{113}$Cd$^+$ ion has been reported as 15199862855.0192(10) Hz \[Opt. Lett. [**40**]{}, 4249 (2015)\]. Fractional systematic due to the black-body radiation (BBR) shift ($\beta$) arising from the Stark effect in the above clock transition was used as $-1.1 \times 10^{-16}$ from our unpublished preliminary estimation. We present here a precise value as $\beta=-1.815(77) \times 10^{-16}$ by carrying out rigorous calculations of third-order polarizabilities of the hyperfine levels associated with the clock transition. This is determined by evaluating matrix elements of the magnetic dipole hyperfine interaction Hamiltonian, electric dipole operator and energies between many low-lying states of $^{113}$Cd$^+$. We employ all-order relativistic many-body theories in the frameworks of Fock-space coupled-cluster and relativistic multi-configuration Dirac-Fock methods.'
author:
- 'Yan-mei Yu'
- 'B. K. Sahoo'
title: 'Estimation of BBR shift due to Stark effect for the Microwave $^{113}$Cd$^{+}$ Ion Clock'
---

The today’s definition of unit of time in the International System of Units (SI) is based on the microwave transition frequency of the ground state hyperfine splitting of $^{133}$Cs [@lombardi]. However, singly charged ions have now paved the way to carry out many precise experiments with the advent of new technologies for cooling and trapping a singly charged ion. In fact single trapped ions $^{171}$Yb$^{+}$ [@huntemann] and Al$^+$ [@chou] provide now clock frequencies with fractional uncertainties below $10^{-17}$. The advantage of considering an ion incur to its nature for being controlled easily by the electromagnetic radiation. Thus, it can be isolated from the external perturbations better than a neutral atom. As a matter of fact, several singly charged ions such as Hg$^+$, Ca$^+$, Sr$^+$, Yb$^+$ among others are under consideration in the laboratory for atomic clocks (e.g. see [@bks-chapter]). However it is aimed at to construct optical frequency clocks using these ions. Both microwave and optical frequency based clocks have advantages in their own perspectives [@riehle]. Since frequencies in both the categories differ by several orders in magnitude, their applications could be totally diverse in nature. It means development of both are indispensable. Also, it is convenient to build up microwave-frequencies using compact electronic devices than the optical frequencies that are required for the deep-space science and commercial purposes [@prestage1; @gill]. In fact, trapping of the Cd$^+$ ion in a semi-conductor chip has already been demonstrated [@stick]. Also, cooling and crystallization of $^{113}$Cd$^+$ has been performed successfully [@guang] and a proof-of-principle to realize a transportable clock using this ion has been discussed [@zhang]. Such technologies are useful for better quantum computers [@Blinov-Nature-2004; @schindler]. Micro-chip clocks are also of immense interest for various reasons. Micro-chip clocks using neutral atoms are based on the coherent-population trapping of atoms that avoids use of microwave cavity to probe the atomic resonance [@gorecki]. As a result, this provides a very compact clock. On the other hand, confinement of more number of neutral atoms can cause large collisional shifts in the clock frequency. This can be minimized by using singly charged ions trapped in a micro-chip.

            Transition             $n=5$     $n=6$     $n=7$     $n=8$      $n=9$      $n=10$     $n=11$     $n=12$
  ------------------------------ --------- --------- --------- ---------- ---------- ---------- ---------- ----------
   $np~^2P_{1/2}-5s~^2S_{1/2}$    $-1.97$   $0.08$    $-0.08$    $0.07$    $-0.05$      0.04     $-0.02$    $-0.01$
   $np~^2P_{3/2}-5s~^2S_{1/2}$    $2.79$    $-0.03$   $0.08$     $0.07$     $0.06$    $-0.05$    $-0.03$    $-0.02$
   $np~^2P_{1/2}-6s~^2S_{1/2}$    $1.65$    $-5.11$   $-0.13$   $-0.006$    $0.02$    $-0.02$      0.02       0.01
   $np~^2P_{3/2}-6s~^2S_{1/2}$     -2.52    $-7.15$    0.35       0.08       0.02     $-0.005$    0.001      0.002
   $np~^2P_{1/2}-7s~^2S_{1/2}$    $-0.48$    3.83     $-9.07$     0.40     $-0.08$      0.02       0.03      0.007
   $np~^2P_{3/2}-7s~^2S_{1/2}$     0.69      5.77      12.66      0.82       0.25     $-0.11$    $-0.05$    $-0.02$
   $np~^2P_{1/2}-8s~^2S_{1/2}$     0.26     $-0.96$   $-6.72$   $-14.13$     0.74     $-0.19$      0.05     $-0.001$
   $np~^2P_{3/2}-8s~^2S_{1/2}$    $-0.38$   $-1.37$    10.08    $-19.67$   $-1.42$      0.46       0.17       0.05
   $np~^2P_{1/2}-9s~^2S_{1/2}$    $-0.18$    0.51      1.59     $-10.27$   $-20.63$     1.17     $-0.31$    $-0.06$
   $np~^2P_{3/2}-9s~^2S_{1/2}$     0.25      0.71     $-2.26$   $-15.37$    28.71     $-2.17$    $-0.66$    $-0.19$
   $np~^2P_{1/2}-10s~^2S_{1/2}$    0.13     $-0.32$   $-0.80$     2.36     $-13.95$   $-29.82$     1.29       0.26
   $np~^2P_{3/2}-10s~^2S_{1/2}$   $-0.18$   $-0.45$    1.12       3.36      20.87      41.61       2.42       0.57
   $np~^2P_{1/2}-11s~^2S_{1/2}$    0.09     $-0.23$   $-0.50$     1.18     $-3.38$     16.01      44.06     $-0.06$
   $np~^2P_{3/2}-11s~^2S_{1/2}$   $-0.13$   $-0.32$    0.70       1.66       4.85     $-23.93$    61.84       0.37
   $np~^2P_{1/2}-12s~^2S_{1/2}$    0.09     $-0.22$   $-0.44$     0.91     $-2.08$      5.07     $-14.94$    63.89
   $np~^2P_{3/2}-12s~^2S_{1/2}$   $-0.13$   $0.30$     0.61       1.28       2.97     $-7.39$    $-22.16$    90.10

A number of ions are under consideration for microwave clocks such as $^{199}$Hg$^+$, $^{137}$Ba$^+$, $^{171}$Yb$^+$, $^{113}$Cd$^+$, $^9$Be$^+$ etc. [@prestage2]. Among these ions, high precision measurement of frequency of the ground state hyperfine splitting in the $^{113}$Cd$^+$ ion is under progress [@zhang2]. The advantage of this ion, except in $^{199}$Hg$^+$ [@berkeland], for microwave clock is its ground state hyperfine splitting is relatively larger; about 15.2 GHz. Moreover, it requires only one laser for cooling, pumping and detection purposes, which is a big advantage from the experimental set-up point of view. Clock frequency of the $|F=0,M_F=0 \rangle \rightarrow |F'=1, M_F'=0\rangle $ transition in the ground state $5s~^2S_{1/2}$ of $^{113}$Cd$^+$ has already been achieved as $v_0=15199862855.0192(10)$ Hz with a fractional precision of 6.6$\times10^{-14}$ [@Miao-OL-2015; @Wang-OE-2013]. The major uncertainties in the systematics of these measurements come from the Zeeman shift, but uncertainty due to the black-body radiation (BBR) shift due to stray electric fields also contribute significantly. It is feasible to minimize uncertainties due to the Zeeman shifts in few years by improving magnetic shielding and other instrumental errors by optimizing controlling sequences and suppressing laser technical noise [@Miao-OL-2015]. In case these uncertainties reach below 10$^{-16}$ level, it would be necessary at that point to reduce uncertainty due to the BBR shift observed at the room temperature. Thus, it is imperative to estimate this quantity more precisely for the Cd$^+$ ion. Measuring BBR shift in an atomic system at the level of 10$^{-16}$ precision level is strenuous. Previously, a number of theoretical calculations have been performed on the BBR shifts for many of the atomic clock candidates with ultra-high precision [@bks-chapter; @Beloy-PRL-2006; @sahoo; @kallay; @arora]. However, the BBR shift due to the Stark effect in the Cd$^+$ clock transition has not been estimated so far. The fractional BBR shift used in the reported clock frequency by Miao et al [@Miao-OL-2015] was based on our unpublished work. We had given them a ball-park figure of this quantity from our preliminary estimation. Therefore, knowledge of a more reliable value of BBR shift in this clock transition is of immense interest.

The clock transition in the ground state of $^{113}$Cd$^+$ has null differential second-order dipole polarizabilities owing to the same scalar polarizabilities of the hyperfine levels. However, the hyperfine interaction induced third-order dipole polarizabilities can offer finite contributions to the BBR shifts in the above clock transition. In this Rapid Communication, we estimate the fractional uncertainty due to the BBR shift of the $^{113}$Cd$^+$ clock transition by evaluating third-order dipole polarizabilities of the associated hyperfine levels. For this purpose, we employ a relativistic coupled-cluster (RCC) method in the Fock-space framework by considering Dirac-Coulomb-Breit Hamiltonian along with lower-order corrections due to quantum electrodynamics effects, as in our previous work [@yu], and relativistic multi-configuration Dirac-Fock methods (MCDF) method using the GRASP2K package [@GRASP] to evaluate the required matrix elements of the electric dipole (E1) and magnetic dipole (M1) hyperfine interaction operators.

  Transition                                Value
  ---------------------------------------- -------
  $4d^95s^2~^2D_{3/2} - 5p~^2P_{1/2}$       0.24
  $4d^95s^2~^2D_{3/2}-5p~^2P_{3/2}$         0.21
  $4d^95s^2~^2D_{3/2}-6p~^2P_{1/2}$         0.25
  $4d^95s^2~^2D_{3/2}-6p~^2P_{3/2}$         0.07
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^4P_{3/2}$    0.28
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^4P_{1/2}$    0.22
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^4F_{3/2}$    0.03
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^4D_{3/2}$    0.23
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^4D_{1/2}$    0.80
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^2P_{3/2}$    -1.05
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^2P_{1/2}$    0.68
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^2D_{3/2}$    0.16
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^2P_{3/2}$    -1.23
  $5s~^2S_{1/2}-4d^95s(^3D)5p~^2P_{1/2}$    0.67
  \[E1MCDF\]                               

  : Values of the E1 matrix elements (in a.u.) for some of the transitions associated with the states having $4d^95s^2~^2D_{3/2}$ and $4d^95s5p$ configurations in Cd$^+$ from the MCDF method of GRASP2K package [@GRASP]. \[tab:MCDFEEE1\]

The light shift due to the third-order Stark effect in the hyperfine level $|F,m_F\rangle$, for $F=I+J$ with the nuclear spin $I$ and total electron momentum $J$ and $m_F$ is the projection of $F$ on the quantization axis, of a $S$-state by a laser field with strength ${\cal E}$ can be given by $$\begin{aligned}
\label{eqs2}
\delta E_F^{(2,1)}(\omega)&=&-\frac{{\cal E}^2}{2} \alpha_F^{s(3)} .\end{aligned}$$ Here, the third-order scalar polarizability $\alpha_F^{s(3)}$ can be conveniently expressed as [@Beloy-PRL-2006] $$\begin{aligned}
\alpha^{s(3)}_F(\omega)&=&\frac{2}{3}\sqrt{(I)(I+1)(2I+1)} \Bigg\{ \begin{array}{c c c} J &I&F\\I& J&1  \end{array} \Bigg\} \\ \nonumber
                 & \times & \frac{\mu_I}{I} (-1)^{F+I+J}\left [ 2T(\omega)+C(\omega)+R(\omega) \right ],\end{aligned}$$ where $\mu_I$ is the nuclear magnetic moment in unit of Bohr magnetron $\mu_N$. We use here $\mu_I/I=-1.2446 \mu_N$ with $I=1/2$ for $^{113}$Cd$^+$ [@stones]. Other factors are given by $$\begin{aligned}
T(\omega)&=& \frac{1}{2(2J+1)}\sum_{J',J''}\delta_{J,J''} (-1)^{J+J'} \nonumber \\
&\times& \frac{ \langle \gamma J ||{\bf D} || \gamma' J' \rangle \langle \gamma' J' ||{\bf D} || \gamma'' J'' \rangle \langle \gamma'' J'' || {\bf {\cal T}}_{hf} || \gamma J \rangle}{(E_{\gamma J}-E_{\gamma'' J''})} \nonumber \\
&\times& \Bigg[\frac{1}{(E_{\gamma J}-E_{\gamma' J'}+\omega)}+\frac{1}{E_{\gamma J}-E_{\gamma' J'}-\omega}\Bigg] ,
\label{eqtt}\\
C(\omega)&=&\frac{1}{2} \sum_{J',J''} (-1)^{J'-J''}\Bigg\{\begin{array}{c c c} 1&J& J\\ 1 & J' & J'' \end{array} \Bigg\}  \nonumber \\
&\times& \langle \gamma J ||{\bf D} || \gamma' J' \rangle \langle \gamma' J' ||{\bf  {\cal T}}_{hf} ||\gamma'' J'' \rangle \langle \gamma'' J'' ||{\bf D} || \gamma J\rangle \nonumber \\
&\times& \Bigg[\frac{1}{(E_{\gamma J}-E_{\gamma' J'} +\omega)(E_{\gamma J}-E_{\gamma'' J''}+\omega)} \nonumber \\
&+&\frac{1}{(E_{\gamma J}-E_{\gamma' J'}-\omega)(E_{\gamma J}-E_{\gamma'' J''}-\omega)}\Bigg]
\label{eqc}\end{aligned}$$ and $$\begin{aligned}
R(\omega)&=&\frac{1}{2(2J+1)} \langle \gamma J|| {\bf {\cal T}}_{hf} || \gamma J \rangle \sum_{J'} |\langle \gamma J || {\bf D} || \gamma' J' \rangle|^2 \nonumber \\
&\times& \left [\frac{1}{(E_{\gamma J}-E_{\gamma' J'}+\omega)^2}+\frac{1}{(E_{\gamma J}-E_{\gamma' J'}-\omega)^2}\right ],  \ \ \
\label{eqr}\end{aligned}$$ where $E$ with subscript $\gamma J$ represent for the energies of the state with angular momentum $J$ and additional quantum numbers $\gamma$, and ${\cal T}_{hf}(r)= -i \sqrt{2} \left (\mbox{\boldmath$\alpha$} \cdot {\bf C}^{(1)}(\hat{r}) \right )/(c r^2)$ is the electronic component of the M1 hyperfine interaction operator with the Dirac operator $\mbox{\boldmath$\alpha$}$, Racah operator $C^{(1)}$ and speed of light $c$, for which the M1 hyperfine structure constant is defined as $$A_{hf}= \frac{\mu_I}{I} \frac{\langle J\| {\bf {\cal T}}_{hf} \|J\rangle}{\sqrt{J(J+1)(2J+1)}}.
\label{eqa}$$

  Matrix element                                                Value
  ----------------------------------------------------------- ----------
  $5s~^2S_{1/2}-5s~^2S_{1/2}$                                   228.27
  $5s~^2S_{1/2}-6s~^2S_{1/2}$                                  $104.52$
  $5s~^2S_{1/2}-7s~^2S_{1/2}$                                  $-65.56$
  $5p~^2P_{1/2}-5p~^2P_{1/2}$                                   36.67
  $5p~^2P_{1/2}-5p~^2P_{3/2}$                                   $4.14$
  $5p~^2P_{1/2}-6p~^2P_{1/2}$                                  $19.09$
  $5p~^2P_{1/2}-6p~^2P_{3/2}$                                    2.17
  $5p~^2P_{1/2}-7p~^2P_{1/2}$                                  $-12.56$
  $5p~^2P_{1/2}-7p~^2P_{3/2}$                                   $1.44$
  $5p~^2P_{3/2}-5p~^2P_{3/2}$                                   19.66
  $5p~^2P_{3/2}-6p~^2P_{1/2}$                                   $2.11$
  $5p~^2P_{3/2}-6p~^2P_{3/2}$                                  $10.49$
  $5p~^2P_{3/2}-7p~^2P_{1/2}$                                  $-1.38$
  $5p~^2P_{3/2}-7p~^2P_{3/2}$                                    6.98
  $4d^95s^2 \ ^2D_{5/2} - 4d^95s^2 \ ^2D_{5/2}$                 69.73
  $4d^95s^2 \ ^2D_{3/2} - 5s~^{2}S_{1/2}$                      $-0.03$
  $4d^9 5s( ^3D)5p \ ^4P_{3/2}-4d^9 5s( ^3D)5p \ ^4P_{3/2}$     135.68
  $4d^9 5s( ^3D)5p \ ^4P_{1/2}-4d^9 5s( ^3D)5p \ ^4P_{1/2}$     63.21
  $4d^9 5s( ^3D)5p \ ^4F_{3/2}-4d^9 5s( ^3D)5p \ ^4F_{3/2}$    $-35.59$
  $4d^9 5s( ^3D)5p \ ^4D_{3/2}-4d^9 5s( ^3D)5p \ ^4D_{3/2}$     115.04
  $4d^9 5s( ^3D)5p \ ^4D_{1/2}-4d^9 5s( ^3D)5p \ ^4D_{1/2}$    $-32.53$
  $4d^9 5s( ^3D)5p \ ^2P_{3/2}-4d^9 5s( ^3D)5p \ ^2P_{3/2}$     212.02
  $4d^9 5s( ^3D)5p \ ^2P_{1/2}-4d^9 5s( ^3D)5p \ ^2P_{1/2}$     11.11
  $4d^9 5s( ^3D)5p \ ^2D_{3/2}-4d^9 5s( ^3D)5p \ ^2D_{3/2}$     87.38
  $4d^9 5s( ^1D)5p \ ^2P_{3/2}-4d^9 5s( ^1D)5p \ ^2P_{3/2}$     73.40
  $4d^9 5s( ^1D)5p \ ^2P_{1/2}-4d^9 5s( ^1D)5p \ ^2P_{1/2}$     44.73
  \[m1hyf\]                                                   

  : Diagonal and off-diagonal reduced matrix elements of the ${\bf {\cal T}}_{hf}$ operator (in $10^{-8}$ a.u.) between some of the important low-lying states of Cd$^+$ from the RCC and MCDF methods. Only the states with $4d^95s^2~^2D_{3/2}$ and $4d^95s5p$ configurations are obtained by the MCDF method. \[tab:HFSEOM\]

  ------------------------------- -------------------------- ----------------------- -------------------------- ----------------------- --------------
                                                                                                                                        
  $J''$                            $J'=5-12p~^2P_{1/2,3/2}$  $J''$                    $J'=5-12p~^2P_{1/2,3/2}$  $J'$                    
  $6s~^2S_{1/2}$                          13350.910          $5p~^2P_{1/2}$                   723.234           $5p~^2P_{1/2}$          10300.727
  $7s~^2S_{1/2}$                           2319.697          $5p~^2P_{3/2}$                  $-853.041$         $5p~^2P_{3/2}$          18624.610
  $8s~^2S_{1/2}$                           784.107           $6p~^2P_{1/2}$                    0.027            $6p~^2P_{1/2}$          3.912
  $9s~^2S_{1/2}$                           374.568           $6p~^2P_{3/2}$                    0.006            $6p~^2P_{3/2}$          0.542
  $10s~^2S_{1/2}$                          220.400           $7p~^2P_{1/2}$                    0.021            $7p~^2P_{1/2}$          2.780
  $11s~^2S_{1/2}$                           92.335           $7p~^2P_{3/2}$                   $-0.016$          $7p~^2P_{3/2}$          2.773
  $12s~^2S_{1/2}$                          109.521           $8-12p~^2P_{1/2,3/2}$            $-1.216$          $8-12p~^2P_{1/2,3/2}$   7.647
  $[4d^95s^2~^2D_{3/2}]$ states             48.97            $[4d^95s5p]$ states             $-546.265$         $[4d^95s5p]$ states     1349.462
  Uncertainty                           $\pm$1087.159        Uncertainty                    $\pm$244.49         Uncertainty             $\pm$593.406
                                                                                                                                        
  Tail                                     142.741           Tail                            $-10.640$          Tail                    2.580
  Core-valence                             $24.937$          Core-valence                      0.417            Core-valence            $-1.480$
  Valence-core                            $-226.153$         Valence-core                      0.417                                    
  Core-core                                 21.122           Core-core                        $-0.188$          Core                    429.383
  Final                                  17263(1129)         Final                          $-686(245)$         Final                   30772(637)
  ------------------------------- -------------------------- ----------------------- -------------------------- ----------------------- --------------

The BBR shift due to the Stark shift in the $|F,m_F\rangle$ level can be estimated by [@Itano-PRA-1982] $$\begin{aligned}
\label{eqbbr}
\delta E_F^{BBR} & \simeq & -\frac{1}{2}(831.9~V/m)^2 \left (\frac{T(K)}{300} \right )^4 \alpha_F^{s(3)},\end{aligned}$$ where $T(K)$ is the laboratory temperature in kelvin (K) with respect to the room temperature $300K$ and $\alpha_F^{s(3)}$ is in atomic unit (a.u.). Hence, the fractional differential BBR shift for the clock transition can be given by $$\begin{aligned}
\label{eqkbeta}
\beta_{FF'}^{BBR} &=& \frac{1}{\nu_0}(\delta E_F^{BBR} - \delta E_{F'}^{BBR}) \nonumber \\
 &\simeq& - \frac{(831.9~V/m)^2}{2\nu_0} \left (\frac{T(K)}{300} \right )^4  [\alpha_{F}^{s(3)}-\alpha_{F'}^{s(3)}] . \ \\end{aligned}$$

Evaluation of $T$, $C$ and $R$ require many more matrix elements between different possible intermediate states of both the even and odd parities. They mainly involve E1 matrix elements between the $ns ~ ^2S_{1/2}$ and $mp ~ ^2P_{1/2,3/2}$ states with $n$ and $m$ representing the principal quantum numbers of different atomic states. It can also involve E1 matrix elements between the ground state and possible states with the $4d^9 ~ 5s 5p$ configurations. Moreover, it also requires E1 matrix elements involving the $4d^9 ~ 5s^2$ states with the $mp ~ ^2P_{1/2,3/2}$ states and states with the $4d^9 ~ 5s 5p$ configurations. Similarly, both the diagonal and off-diagonal matrix elements between these states of the ${\bf {\cal T}}_{hf}$ operator will be required. Contributions to the sum of $\langle J || O_1 || J' \rangle \langle J' || O_2 || J'' \rangle \langle J'' || O_3
||J \rangle$, for $O_1$, $O_2$ and $O_3$ being either $D$ or ${\bf {\cal T}}_{hf}$ operators, from all possible excited states and core orbitals are evaluated in several steps. The $T$ and $C$ values are evaluated by dividing into valence, core-core, core-valence, and valence-core contributions when both $J'$ and $J''$ belong to virtual states, when both $J'$ and $J''$ belong to core orbitals, when $J'$ is from virtual and $J''$ is from core, and when $J'$ is from core and $J''$ is from virtual, respectively. Similarly, the sum of square of the matrix element $\langle J || D || J' \rangle$ for the evaluation of $R$ are estimated by dividing contributions as valence, core-valence and core contributions following the procedures described in Refs. [@arora; @jasmeet]. As can be seen from Eqs. (\[eqtt\])-(\[eqr\]), the low-lying excited states will contribute most to $T$, $C$ and $R$ while the core, core-valence and valence-core contributions will be relatively small. Thus, we evaluate these contributions by the Dirac-Hartree-Fock (DHF) method. For evaluating the valence correlations, we have calculated up to $12p ~ ^2P_{1/2,3/2}$ states, that correspond to excitation energies about 131300 cm$^{-1}$ from the ground state. All the above excited states, except those having configurations $4d^9 ~ 5s 5p$ and $4d^9 ~ 5s^2$, are calculated using the RCC method, described in our earlier works [@yu; @bks4], with the singles and doubles excitation approximation (CCSD method). To determine matrix elements associated with the states with $4d^9 ~ 5s 5p$ and $4d^9 ~ 5s^2$ configurations, we apply the MCDF method using the GRASP2K package [@GRASP]. Contributions above than the $12p ~ ^2P_{1/2,3/2}$ states are estimated by the DHF method and refer to as “Tail” contributions.

We have compared calculated energies with the National Institute of Science and Technology database (NIST) [@NIST] for the states determined by the RCC and MCDF methods, which are given in the Supplemental material. The CCSD values are found to be within 1% accuracy and the MCDF results for the states with $4d^9 ~ 5s 5p$ and $4d^9 ~ 5s^2$ configurations are found to be about 4% compared to the NIST data. We also give the E1 matrix elements among various low-lying states in Table \[tab:EDM\] from the CCSD method. Experimental values for the E1 matrix elements of the $5s~^2S_{1/2} - 5p~^2P_{1/2}$ and $5s~^2S_{1/2} - 5p~^2P_{3/2}$ transitions are obtained from the precise lifetime measurements of the $5p~^2P_{1/2}$ and $5p~^2P_{3/2}$ states as 1.913(3) a.u. and 2.717(5) a.u., respectively, [@Moehring-PRA-2006]. Our values are in reasonably agreement with these results. Some of the important E1 matrix elements obtained by the MCDF method are given in Table \[E1MCDF\]. These values for the $4d^95s^2~^2D_{3/2} - 5p~^2P_{1/2}$ and $4d^95s^2~^2D_{3/2} - 5p~^2P_{3/2}$ transitions are inferred from experiments as 0.487(17) a.u. and 0.295(37) a.u., respectively, [@Xu-PRA-2004]. They are almost twice than the calculated values. The remaining E1 matrix elements from the MCDF method can be found from the Supplemental material.

Similarly, the reduced matrix elements of ${\bf {\cal T}}_{hf}$ obtained in the RCC and MCDF methods for some of the important contributing states are listed in Table \[m1hyf\]. The rest of the matrix elements for other excited states that are calculated by us are given in the Supplemental material. Using the relation given by Eq. (\[eqa\]), we have also extracted out the reduced expectation values for the $5s~^2S_{1/2}$, $5p~^2_{3/2}$ and $4d^95s^2 \ ^2D_{5/2}$ states as $227.66(1)\times 10^{-8}$ a.u. [@Miao-OL-2015], $18.94\times 10^{-8}$ a.u. [@Tanaka-PRA-1996] and $75.10(3) \times 10^{-8}$ a.u. [@hermann], respectively, from the precise measurements of $A_{hf}$. Comparison of these values with our calculations show good agreement between them.

Using all the reduced matrix elements discussed above, we determine the static values of $T$, $C$ and $R$ for the ground state of $^{113}$Cd$^+$ and give them in Table \[tab:BBR\] after summing contributions from various intermediate states. To reduce uncertainties in these calculations, we use the experimental energies and replace the calculated reduced matrix elements by the precisely known experimental values wherever possible. It to be noted that knowledge of signs of the matrix elements are necessary for the evaluation of $T(0)$ and $C(0)$. However, signs of the experimental E1 amplitudes cannot be determined. In this case, we use the magnitudes from the measurements and signs from the MCDF results. We also give contributions explicitly from the core-core, core-valence, valence-core and core contributions to these quantities in the same table. As can be seen from the table, the $6s~^2S_{1/2}$ and $7s~^2S_{1/2}$ states contribute extremely large to $T(0)$. Other $S$-states also contribute significantly here. Tail, core-valence and valence-core contributions to $T(0)$ are also non-negligible, but core-core contribution is found to be small. The net contributions from the states with $4d^95s^2~^2D_{3/2}$ configurations are also found to be relatively small. The dominant uncertainties come from the matrix elements involving the calculated $S$-states. On other hand, most of the contributions to $C(0)$ comes from the $5P$-states. The net contributions from the states with $4d^95s5p$ configurations are also found to be quite significant. For the $R(0)$ value, the most dominant contributions come from the $5P$-states followed by the states with $4d^9 ~ 5s 5p$ configuration. The core contribution is also found to be quite large. We estimate uncertainties to the above quantities by assigning an overall $\pm$3% errors to the CCSD results and $\pm$20% errors to the MCDF results. Uncertainties due to the other contributions given from the DHF method are estimated to be about 10%.

Accounting for all these values along with the uncertainties, we find the differential BBR shift coefficient $\beta_{FF'}^{BBR}=-1.815(77)\times10^{-16}$ for the $|5s~^2S_{1/2}, F=0,m_F=0 \rangle \leftrightarrow |5s~^2S_{1/2},F'=1,m_{F'}=0 \rangle$ clock transition in the $^{113}$Cd$^+$ ion in contrast to $-1.1\times10^{-16}$ that was used in Ref. [@Miao-OL-2015]. Uncertainty to this value can be further reduced by improving precision of the matrix elements obtained by the MCDF method.

We thank Dr. J. W. Zhang, Joint Institute for Measurement Science, Tsinghua University, China for many useful discussions and providing us experimental information. Y.Y. is supported by the National Natural Science Foundation of China under Grant No. 91536106, the CAS XDB21030300, and the NKRD Program of China (2016YFA0302104). B.K.S. acknowledges financial support from CAS through the PIFI fellowship under the project number 2017VMB0023 and partly by the TDP project of Physical Research Laboratory (PRL), Ahmedabad and the computations were carried out using the Vikram-100 HPC cluster of PRL.

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