Blackbody radiation shift for the $^1$S$_0$--$^3$P$_0$ optical clock transition in zinc and cadmium atoms

TL;DR

This paper calculates black-body radiation shifts for the $^1$S$_0$--$^3$P$_0$ transition in zinc and cadmium atoms using advanced atomic structure methods, providing insights for optical clock accuracy.

Contribution

It offers the first detailed relativistic calculations of BBR shifts for these specific atomic transitions in zinc and cadmium.

Findings

Computed static polarizabilities of clock levels.

Evaluated electric-dipole matrix elements.

Compared BBR shifts across different neutral divalent atom clocks.

Abstract

Black-body radiation (BBR) shifts of $^3\!P_0-^1\!S_0$ clock transition in divalent atoms Cd and Zn are evaluated using accurate relativistic many-body techniques of atomic structure. Static polarizabilities of the clock levels and relevant electric-dipole matrix elements are computed. We also present a comparative overview of the BBR shifts in optical clocks based on neutral divalent atoms trapped in optical lattices.

Tables

Table 1. Table 1: Black-body radiation shift for the clock transitions between the 1 S 0 ground state and the lowest-energy 3 P 0 o subscript superscript absent 𝑜 0 {}^{o}_{0} state for divalent atoms. Result for Zn and Cd are from this work, for Mg, Ca, Sr, Yb from Ref.  Porsev and Derevianko ( 2006 ) , and for Hg from Ref.  Hachisu et al. ( 2008 ) . δ ​ ν 𝛿 𝜈 \delta\nu is the BBR shift at T = 300 ​ K 𝑇 300 K T=300\,\mathrm{K} with our estimated uncertainties. ν 0 subscript 𝜈 0 \nu_{0} is the clock transition frequency, and δ ​ ν / ν 0 𝛿 𝜈 subscript 𝜈 0 \delta\nu/\nu_{0} is the fractional contribution of the BBR shift. The last column lists fractional errors in the absolute transition frequencies induced by the uncertainties in the BBR shift.

Atom	$\delta \nu$, (Hz)	${\nu }_0$, (Hz)	$\delta \nu /{\nu }_0$	Uncertainty
Zn	-0.244(10)	$9.69\times {10}^{14}$	$-2.5\times {10}^{-16}$	$1\times {10}^{-17}$
Cd	-0.248(15)	$9.03\times {10}^{14}$	$-2.8\times {10}^{-16}$	$2\times {10}^{-17}$
Mg Porsev and Derevianko (2006)	-0.258(7)	$6.55\times {10}^{14}$	$-3.9\times {10}^{-16}$	$1\times {10}^{-17}$
Ca Porsev and Derevianko (2006)	-1.171(17)	$4.54\times {10}^{14}$	$-2.6\times {10}^{-15}$	$4\times {10}^{-17}$
Sr Porsev and Derevianko (2006)	-2.354(32)	$4.29\times {10}^{14}$	$-5.5\times {10}^{-15}$	$7\times {10}^{-17}$
Yb Porsev and Derevianko (2006)	-1.25(13)	$5.18\times {10}^{14}$	$-2.4\times {10}^{-15}$	$3\times {10}^{-16}$
Hg Hachisu et al. (2008)	-0.181	$1.13\times {10}^{15}$	$-1.6\times {10}^{-16}$

Table 2. Table 2: Rescaling parameters λ l subscript 𝜆 𝑙 \lambda_{l} and f k subscript 𝑓 𝑘 f_{k} for the core-valence correlation operators Σ ^ 1 subscript ^ Σ 1 \hat{\Sigma}_{1} and Σ ^ 2 subscript ^ Σ 2 \hat{\Sigma}_{2}

	${\lambda }_s$	${\lambda }_p$	$f_1$
Zn	1.113	1.106	1
Cd	0.8714	0.887	0.8

Table 3. Table 3: Energy levels of Zn and Cd (cm -1 ); comparison of “scaled” theory and experiment. n = 4 𝑛 4 n=4 for Zn and n = 5 𝑛 5 n=5 for Cd.

			Zn		Cd
Config.	State	$J$	Expt.	Theory	Expt.	Theory
$nsnp$	3Po	0	32311	32348	30114	30108
		1	32501	32546	30656	30664
		2	32890	32950	31827	31866
$nsnp$	1Po	1	46745	46908	43692	43721
$ns\left(n+1\right)s$	3S	1	53672	53412	51484	51317
$ns\left(n+1\right)s$	1S	0	55789	55513	53310	53088
$nsnd$	1D	2	62459	62333	59220	59282
$nsnd$	3D	1	62769	62606	59486	59512
		2	62772	62609	59498	59521
		3	62777	62613	59516	59534

Table 4. Table 4: Electric dipole transition amplitudes (reduced matrix elements, a.u.) for Zn and Cd.

Transition	Zn	Cd
1S${}_0{}^{}\to _{}^3$P${}_1{}^{}{}_{}{}^o$	0.045	0.158
1S${}_0{}^{}\to _{}^1$P${}_1{}^{}{}_{}{}^o$	3.320	3.435
3P${}_0{}^o\to _{}^3$S1	1.466	1.486
3P${}_0{}^o\to _{}^3$D1	2.127	2.222

Table 5. Table 5: Static polarizabilities of the 1 S 0 and 3 P 0 o subscript superscript absent 𝑜 0 {}^{o}_{0} states of Zn and Cd (a.u.); comparison with experimental results and other calculations.

Atom	State	Expt. Goebel et al. (1996); Goebel and Hohm (1995)	This work	Ref. Ye and Wang (2008)	Ref. Ellingsen et al. (2001)
Zn	1S0	38.8(8)	38.58	38.12	39.13
Zn	3P${}_0{}^{}{}_{}{}^o$		66.53	67.69	66.50
Cd	1S0	49.65(1.62)	46.52	44.63
Cd	3P${}_0{}^{}{}_{}{}^o$		75.31	75.29

Equations

$$\frac{\delta \nu }{{\nu }_0}=\beta {\left(\frac{T}{300K}\right)}^4,$$

$$\delta E_v\approx -\frac{2}{15}(\alpha \pi {)}^3{T}^4{\alpha }_v(0)\left[1+\eta \right],$$

$$\eta =\frac{(80/63){\pi }^2}{{\alpha }_v(0)T}\sum _p\frac{|⟨p||D||v⟩{|}^2}{(2J_v+1)y_p^3}\left(1+\frac{21{\pi }^2}{5y_p^2}+\frac{336{\pi }^4}{11y_p^4}\right).$$

$${\alpha }_v(0)=\frac{2}{3(2J_v+1)}\sum _n\frac{|⟨v||\mathbf{D}||n⟩{|}^2}{E_n-E_v},$$

$$⟨\delta {\Psi }_v|=\sum _n\frac{⟨v||\mathbf{D}||n⟩}{E_n-E_v}⟨n|.$$

$$({\hat{H}}_0-E_v)\delta {\Psi }_v=-\mathbf{D}{\Psi }_v,$$

$${\alpha }_v(0)=\frac{2}{3(2J_v+1)}⟨\delta {\Psi }_v||\mathbf{D}||v⟩.$$

$${\hat{H}}_0={\hat{h}}_1(1)+{\hat{h}}_1(2)+{\hat{h}}_{12},$$

$${\hat{h}}_1=c(\alpha \cdot \mathbf{p})+m{c}^2(\beta -1)-\frac{Z{e}^2}{r}+{\hat{V}}_{\mathrm{core}}+{\hat{\Sigma }}_1.$$

$${\hat{h}}_{12}=\frac{e^2}{r_{12}}+{\hat{\Sigma }}_2,$$

Full text

One of the factors limiting the accuracy of the modern atomic clocks is the perturbation of the clock frequency by the bath of thermal photons, i.e., by black body radiation (BBR). $10^{-15}$ is the typical value of the fractional BBR correction to optical lattice clocks Porsev and Derevianko (2006) at room temperatures, while the current generation of optical atomic clocks have demonstrated the fractional inaccuracies at the level of $10^{-18}$ or better Brewer et al. (2019); Bothwell et al. (2019); McGrew et al. (2018). Therefore, all the recent advances in atomic clocks address the BBR shift problem either through cryogenic techniques, active temperature stabilization, or specially-designed BBR chambers. All of these techniques can be advanced further by using atoms that have a reduced sensitivity to BBR.

To the leading order, the fractional BBR correction to the unperturbed clock frequency $\nu_{0}$ can be parameterized as

[TABLE]

where $T$ is the bath temperature. There are two issues associated with the BBR shift: (i) one needs to know the coefficient $\beta$ with sufficiently high accuracy so that the uncertainty in $\beta$ does not degrade the clock output and (ii) even if $\beta$ is known precisely, there are uncertainties arising from the ambient temperature fluctuations and imperfect knowledge of the temperature field. Apparently, the smaller the $\beta$, the better.

There are two main classes of optical atomic clocks that are presently well-positioned to eventually replace the primary frequency standard. The first, more mature, class of clocks is based on trapped ions and the second class employs neutral divalent atoms trapped in optical lattices. A comparative overview of the BBR shift for various ion clocks is given in Ref.Rosenband et al. (2006) and for lattice clocks in Ref. Porsev and Derevianko (2006). The NIST group Rosenband et al. (2007) has pointed out that the BBR shift is exceptionally small in Al*+* ion, $\delta{\nu}/{\nu_{0}}\sim 10^{-17}$. For divalent atoms considered in the literature so far (Mg, Ca, Sr, Yb, Hg) the least susceptible are mercury lattice clocks Hachisu et al. (2008), $\delta{\nu}/{\nu_{0}}\sim 10^{-16}$ at room temperatures.

Divalent cadmium and zinc atoms were found recently Ovsiannikov et al. (2007) to have properties suitable for realizing the neutral atom optical lattice clocks. With the BBR shift being one of the most important contributors to the uncertainty budget of the clocks, here we extend the survey of Ref. Porsev and Derevianko (2006) and compute the BBR shifts for the Cd and Zn lattice clocks. The results of our analysis are summarized in Table 1. We find that for Cd and Zn the fractional BBR shifts are comparable to the so-far most favorable Hg. At least from this perspective, these atoms may serve as a competitive alternative to already operational Sr, Yb, and Hg clocks.

Details of calculations — To compute the energy shift due to black-body radiation we use the formalism developed in Ref. Porsev and Derevianko (2006). The electric-dipole contribution to the BBR energy shift of state $v$ is given by

[TABLE]

Here $y_{p}=(E_{p}-E_{v})/T$, $\alpha_{v}(0)$ is the static scalar dipole polarizability, and $\eta$ represents a “dynamic” fractional correction to the total shift. $D$ is the electric-dipole operator. The calculations requires evaluating the static polarizability for both clock levels. The clock transition is between the 1S0 ground state and the lowest-energy 3P${}^{o}_{0}$ state.

The static scalar polarizability $\alpha_{v}(0)$ of an atom in state $v$ is given by

[TABLE]

where summation goes over the complete set of excited many-body states (including continuum and core-excited states). We use the Dalgarno-Lewis method and reduce the summation to solving the inhomogeneous Schrödinger (Dirac) equation (setup is similar to Ref. Derevianko and Johnson (1997)). In this approach, a correction to the atomic wave function due to the external electric field is introduced

[TABLE]

This correction satisfies an inhomogeneous equation

[TABLE]

where $\hat{H}_{0}$ is an effective Hamiltonian of the atom. Once the $\delta\Psi_{v}$ is found, static polarizability is calculated as

[TABLE]

We employ a computational scheme based on combining the configuration interaction method with the many-body perturbation theory (CI+MBPT) Dzuba et al. (1996). The effective Hamiltonian is constructed for the two valence electrons, while excitations from the core are taken into account by means of the MBPT. The Hamiltonian has the form

[TABLE]

where $\hat{h}_{1}$ is a single-electron part of the relativistic Hamiltonian

[TABLE]

Here $c$ is speed of light, and $\mathbf{\alpha}$ and $\beta$ are Dirac matrices, $Ze$ is the nuclear charge, $\hat{V}_{\rm core}$ is the Hartree-Fock potential of the atomic core (including the non-local exchange term) and $\hat{\Sigma}_{1}$ is the correlation potential which describes the correlation interaction between a valence electron and the core (see Refs.Dzuba et al. (1996); Dzuba and Johnson (1998) for details).

The $\hat{h}_{12}$ operator in (6) is the two-electron part of the Hamiltonian:

[TABLE]

where first term is standard Coulomb interaction between valence electrons and second term is the correction to it due to correlations with core electrons.

We use the second-order MBPT to calculate the self-energy operators $\hat{\Sigma}_{1}$ and $\hat{\Sigma}_{2}$ via direct summation over a complete set of single-electron states. This set of basis states is constructed using the B-spline technique Johnson and Sapirstein (1986). We use 40 B-splines of order 9 in a cavity of 40 Bohr radius. The same basis of the single-electron states is also used in constructing the two-electron basis states for the CI calculations. We employ partial waves $\ell=0-4$ for the valence CI subspace and $\ell=0-5$ for internal summations inside the self-energy operator.

Additionally, to mimic the omitted higher-order MBPT effects, we rescale the $\hat{\Sigma}$ operator to fit the experimental energies. The $\hat{\Sigma}_{1}$ operator is replaced in (6) by $\lambda_{l}\hat{\Sigma}_{1}$, where $l=0,1,2$ is a the angular momentum of a single-electron state. The $\hat{\Sigma}_{2}$ operator is replaced by $f_{k}\hat{\Sigma}_{2}$, where $k$ is multipolarity of the Coulomb interaction. The values of the rescaling parameters are presented in Table 2. The resulting energies after the scaling procedure are listed in Table 3. A typical deviation from the experimental values is in the order of 100 $1/\mathrm{cm}$. Even after the scaling, the disagreement remains, as the number of fitting parameters is limited.

*Results — * With the computed wavefunctions, we may evaluate various matrix elements. While computing matrix elements (and polarizabilities) we use single-particle matrix elements dressed in the random-phase approximation. Qualitatively this correspond to the shielding of the applied electromagnetic field by the core electrons. Notice that the static polarizability depends sensitively on the values of the dipole matrix elements for the lowest-energy excitations. Our computed dipole matrix elements for the two lowest-energy excitations originating from the two clock states are presented in Table 4. The inter-combination transition 1S${}_{0}\rightarrow^{3}$P${}^{o}_{1}$ is non-relativistically forbidden. A three-fold increase in the matrix element values when progressing from Zn $Z=30$ to heavier Cd $Z=48$ is consistent with the relevant suppression factor of $(\alpha Z)^{2}$. Similarly to the case of Sr and Yb atoms Yasuda et al. (2006); Takasu et al. (2004), we anticipate that the high-accuracy values for the 1S${}_{0}\rightarrow^{1}$P${}^{o}_{1}$ matrix elements may be derived from photoassociation spectroscopy with ultracold atoms. If such data become available, the accuracy of our values for polarizability may be improved by correcting the matrix elements of Table 3 with the experimental values and correcting $\alpha_{v}(0)$ with Eq.(2).

The computed values of the static polarizabilities of the clock levels are presented in Table 5. The values combine both valence and core polarizabilities. Core polarizabilities are 2.296 a.u. for Zn and 4.971 a.u. for Cd Johnson et al. (1983). For the ground states we compare our values with the experimental resultsGoebel et al. (1996); Goebel and Hohm (1995). For Zn our computed value is within the experimental uncertainty while for Cd the results disagree by about $2\,\sigma$ of the experiment. Our results are consistent with the previous theoretical work Ye and Wang (2008); Ellingsen et al. (2001). These authors employed methods sufficiently different from our approach to warrant additional confidence in the theoretical predictions. Ye and Wang (2008) used a semi-empirical model potential method and Ellingsen et al. (2001) employed a multi-reference configuration-interaction method using a two-electron relativistic pseudo-potential.

Finally, we combine the static polarizabilities using Eq. (1) and arrive at the BBR shifts summarized in Table 1. The results also include the dynamic correction $\eta$; it turns out to be less than $7\times 10^{-4}$ for both Zn and Cd. This small correction can be safely neglected at the present level of accuracy. We also estimate the theoretical error bar for the BBR correction: 4% for Zn and 6% for heavier Cd. The error was evaluated by carrying out two calculations: with and without scaling of self-energy operator to experimental energies. We find that the resulting uncertainty would affect the accuracy of the clock output in the 17th significant figure. Overall fractional BBR shift for both Cd and Zn is slightly larger than in Hg, but 5 times smaller than in Sr and 10 times smaller than in Yb.