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Blackbody radiation shift in the Sr optical atomic clock

We evaluated the static and dynamic polarizabilities of the 5s^2 ^1S_0 and 5s5p ^3P_0^o states of Sr using the high-precision relativistic configuration interaction + all-order method. Our calculation explains the discrepancy between the recent experimental 5s^2 ^1S_0 - 5s5p ^3P_0^o dc Stark shift measurement \Delta \alpha = 247.374(7) a.u. [Middelmann et. al, arXiv:1208.2848 (2012)] and the earlier theoretical result of 261(4) a.u. [Porsev and Derevianko, Phys. Rev. A 74, 020502R (2006)]. Our present value of 247.5 a.u. is in excellent agreement with the experimental result. We also evaluated the dynamic correction to the BBR shift with 1 % uncertainty; -0.1492(16) Hz. The dynamic correction to the BBR shift is unusually large in the case of Sr (7 %) and it enters significantly into the uncertainty budget of the Sr optical lattice clock. We suggest future experiments that could further reduce the present uncertainties.


I. INTRODUCTION
Optical lattice clocks have shown tremendous progress in recent years [1]. An optical frequency standard based on the 5s 2 1 S 0 − 5s5p 3 P o 0 transition of ultracold 87 Sr atoms confined in a one-dimensional optical lattice is pursued by a number of groups [2][3][4][5][6][7]. Its systematic uncertainty has been demonstrated at the 10 −16 fractional frequency level and an order-of-magnitude improvement is expected to be achieved soon [1,4]. A three-dimensional optical lattice clock with bosonic 88 Sr was demonstrated for the first time in [8].
The measured clock transition frequencies must be corrected in practice for the effect of the ambient blackbody radiation (BBR) shift, which is quite difficult to measure directly. The BBR shift can only be suppressed by cooling the clock. At room temperature, the differential BBR shift of the two levels of a clock transition turns out to make one of the largest irreducible contributions to the uncertainty budget of optical atomic clocks. The Sr clock transition has the largest BBR shift of all optical frequency standards that are currently under development (see Ref. [9] for a recent review). The fractional BBR shift ∆ν BBR /ν 0 in Sr is more than a factor of 1000 larger than the fractional BBR shift in the Al + ion clock [10]. The BBR shift of an optical clock can generally be approximated by the dc Stark shift of the clock transition to about 1-2% precision, because optical frequencies are 100 times greater than characteristic BBR frequencies. However, Sr represents an exception, where the so-called dynamic correction [11], that needs to be determined separately from the dc Stark shift, is 7%. Recently, the dc Stark shift in Sr has been measured with 0.003% pre-cision [13], and the dynamic correction was evaluated based on a set of E1 transition rates and the Stark shift measurement. The measured value differed substantially (by almost 4σ) from the previous theoretical determination [11].
In this work, we evaluate the static and dynamic polarizabilities of the 5s 2 1 S 0 and 5s5p 3 P o 0 states of Sr using the high-precision relativistic CI+all-order method. Our calculation explains the above-mentioned discrepancy between the experimental 5s 2 1 S 0 − 5s5p 3 P o 0 dc Stark shift measurement ∆α = 247.374(7) a.u. [13] and the earlier theoretical result of 261(4) a.u. [11]. We found that the E1 matrix elements for the transitions that give dominant contributions to the 3 P o 0 polarizability, in particular the 5s4d 3 D 1 − 5s5p 3 P o 0 , are rather sensitive to the higher-order corrections to the wave functions and other corrections to the matrix elements beyond the random phase approximation. A correction of only 2.4% to the dominant 3 D 1 − 3 P o 0 matrix element leads to 5% difference in the final value of the 3 P o 0 − 1 S 0 Stark shift. In this work, we included the higher-order corrections in an ab initio way using the CI+all-order approach, and also calculated several other corrections omitted in [11]. Our value for the dc Stark shift of the clock transition, 247.5 a.u., is in excellent agreement with the experimental result 247.374(7) a.u. [13].
We have combined our theoretical calculations with the experimental measurements of the Stark shift [13] and magic wavelength [3] of the 5s 2 1 S 0 − 5s5p 3 P o 0 transition to infer recommended values of the several electric-dipole matrix elements that give the dominant contributions to the 3 P o 0 polarizability. We used these values to evaluate the dynamic correction to the BBR shift of the 1 S 0 − 3 P o  [12] and theoretical energy levels of Sr in cm −1 . Two-electron binding energies are given in the first row, energies in other rows are given relative to the ground state. Results of the CI+MBPT and CI+allorder calculations are given in columns labeled "CI+MBPT" and "CI+All (A)". The CI+all-order values with the ground state two-electron binding energy shifted by 200 cm −1 are given in column labeled "CI+All (B)". Corresponding relative differences of these three calculations with experiment are given in the three corresponding columns labeled "Diff." in %. The We determined that the 5s4d 3 D 1 − 5s5p 3 P o 0 transition contributed 98.2% to the dynamic correction for the 3 P o 0 level. Our calculation enables us to propose an approach for further reduction of the uncertainty in the BBR shift. In particular, there is a correlation in the uncertainty of the BBR shift and the lifetime of the 5s4d 3 D 1 state, if branching ratios are known to sufficient accuracy. At present, experimental measurements of the 5s4d 3 D term-averaged lifetime have an uncertainty of about 7% [14,15]. We note that the experiment [15], which was performed at JILA some 20 years ago, has relevance in the determination of the uncertainty budget of one of the world's most accurate clocks now being developed at the same institution -a development probably not envisaged at the time.
A new determination of this (or 3 D 1 ) lifetime with 0.5% uncertainty would provide a value of the Sr clock BBR shift that is accurate to about 0.5%, which would be a factor of 2 improvement in the uncertainty that we state here. This result is determined by the relevant branching ratios needed for the extraction of the 5s4d 3 D 1 − 5s5p 3 P o 0 matrix elements from the lifetime measurement. We have determined these branching ratios with an uncertainty of 0.2%. A further reduction in the uncertainty of the Sr clock BBR shift could be effected by an improved measurement of these branching ratios. The lifetime of the corresponding 6s5d 3 D 1 state in Yb has been recently measured in Ref. [16].

II. METHOD AND ENERGY LEVELS
Calculation of Sr properties requires an accurate allorder treatment of electron correlations. This can be accomplished within the framework of the CI+all-order method that combines configuration interaction and coupled-cluster approaches [10,[18][19][20][21]. To evaluate uncertainties of the final results, we also carry out CI [22] and CI+many-body perturbation theory (MBPT) [23]  The CI+MBPT and CI+all-order results and further corrections to the E1 matrix elements for transitions that give dominant contributions to the polarizabilities of the 5s 2 1 S0 and 5s5p 3 P o 0 states. The CI+MBPT and CI+all-order results including RPA corrections are given in columns labeled "MBPT+RPA" and "All+RPA", respectively. The relative differences between the CI+all-order+RPA and CI+MBPT+RPA results are given in column "Higher orders" in %. The other contributions include the core-Brueckner (σ), two-particle (2P), structural radiation (SR), and normalization (Norm) corrections. Total relative size of corrections beyond CI+all-order+RPA is given in column "Corr." in %. The recommended values for the 5s 2 1 S0 − 5s5p 1 P o 1 matrix element was obtained from the 1 P o 1 lifetime measurement [17], and the recommended values for all other transitions are from the present work (see Section V). calculations. These methods have been described in a number of papers [10,18,22,23] and we provide only a brief outline of these approaches. We start with a solution of the Dirac-Fock (DF) equations where H 0 is the relativistic DF Hamiltonian [18,23] and ψ c and ε c are single-electron wave functions and energies. The calculations are carried out in the V N−2 potential. The wave functions and the low-lying energy levels are determined by solving the multiparticle relativistic equation for two valence electrons [22], The effective Hamiltonian is defined as where H FC is the Hamiltonian in the frozen-core approximation. The energy-dependent operator Σ(E) which takes into account virtual core excitations is constructed using second-order perturbation theory in the CI+MBPT method [23] and using a linearized coupled-cluster singledouble method in the CI+all-order approach [18]. It is zero in a pure CI calculation. We refer the reader to Refs. [18,23] for detailed description of the construction of the effective Hamiltonian.
Unless stated otherwise, we use atomic units (a.u.) for all matrix elements and polarizabilities throughout this paper: the numerical values of the elementary charge, |e|, the reduced Planck constant,h = h/2π, and the electron mass, m e , are set equal to 1. The atomic unit for polarizability can be converted to SI units via , where the conversion coefficient is 4πǫ 0 a 3 0 /h and the Planck constant h is factored out in order to provide direct conversion into frequency units; a 0 is the Bohr radius and ǫ 0 is the electric constant.
As a first test of the accuracy of our calculations, we compare our theoretical energies with experiment for a number of the even and odd parity states. Comparison of the energy levels (in cm −1 ) obtained in the CI+MBPT, and CI+all-order approximations with experimental values [12] is given in Table I. Ground state two-electron binding energies are given in the first row of Table I, energies in other rows are measured from the ground state. The relative differences of the CI+MBPT and CI+allorder calculations with experiment (in %) are given in columns labeled "Diff". Since the CI+all-order values are systematically higher than the experimental values, a large fraction of the difference from experiment can be attributed to the difference in the value of the ground state two-electron binding energy. We find that shifting the CI+all-order value of the ground state two-electron binding energy by only 200 cm −1 (see results in column CI+all-order (B)) brings the results into excellent agreement with experiment for most of the states. We give the 5s4d 3 D 1 − 5s5p 3 P o 0 transition energy in the last row of Table I. This transition is particulary important to the subject of this work, since it contributes 61% to the static polarizability and 98% to the dynamic correction to the BBR shift of the 5s5p 3 P o 0 state. In fact, the accidentally small value of this transition energy is the source of the anomalously large (7%) dynamic correction to the BBR shift of the 1 S 0 → 3 P o 0 transition in Sr. We see considerable improvement of the accuracy in this transition energy from the CI+MBPT to CI+all-order approximation, by a factor of 6. The CI+MBPT and CI+all-order values differs from the experiment by 11% and 1.7%, respectively.

III. AB INITIO CALCULATION OF ELECTRIC-DIPOLE MATRIX ELEMENTS
The reduced electric-dipole matrix elements are obtained with the CI+all-order wave functions and effective electric-dipole operator D eff in the random-phase approximation (RPA). The effective operator accounts for the core-valence correlations in analogy with the effective Hamiltonian [24,25]. We include additional corrections beyond RPA in the calculation of the E1 matrix  [11] 261(4) Expt. [13] 247.374 (7) a CI+all-order+RPA values (no other corrections). b CI+all-order+RPA + other corrections. c Obtained using experimental 5s5p 1 P o 1 lifetime from [17]. d Obtained using recommended value for the 1 S0 polarizability and the experimental value of the Stark shift [13].
elements in comparison with [11,26]. These contributions include the core-Brueckner (σ), two-particle (2P) corrections, structural radiation (SR), and normalization (Norm) corrections [24,25]. While we find some cancelation between the various corrections, these cannot be omitted at the 1% level of accuracy. Partial cancelation of the structural radiation and normalization corrections was discussed in Ref. [27]. Detailed analysis of the structure radiation correction was carried out in the same work [27].
The results for several transitions that give dominant contributions to the 1 S 0 − 3 P o 0 dc Stark shift are summarized in Table II. The percentage differences between the CI+all-order+RPA and CI+MBPT+RPA calculations are given in the column labeled "Higher orders". We note that it is positive for some transitions and negative for other transitions. Our final ab initio values are given in column labeled "Final". We find that total relative size of corrections beyond CI+all+RPA given in column labeled "Corr" is small, 0.04-1.7%, but significant. We estimate the uncertainties in the ab initio values of the matrix elements to be 1% based on the comparison of the CI+MBPT+RPA and CI+all-order+RPA values and combined size of other corrections.
We also provide the recommended values for these transitions. The recommended value for the 5s 2 1 S 0 − 5s5p 1 P o 1 matrix element was obtained in [11,26] from the 1 P o 1 lifetime measurement from photoassociation spectra [17], the recommended values for all other transitions are obtained in the present work in Section V.

IV. POLARIZABILITIES
We evaluated the static and dynamic polarizabilities of the 5s 2 1 S 0 and 5s5p 3 P o 0 states of Sr using the highprecision relativistic CI+all-order method. The scalar polarizability α 0 (ω) is separated into a valence polarizability α v 0 (ω), ionic core polarizability α c , and a small term α vc that modifies ionic core polarizability due to the presence of two valence electrons. The valence part of the polarizability is determined by solving the inhomogeneous equation in valence space, which is approximated static polarizability α0(ω = 0) and dynamic polarizability α0(ω) at the 813.4 nm magic wavelength. The dominant contributions to the valence polarizabilities are obtained with experimental energies and recommended values of the matrix elements. The electric-dipole reduced matrix elements are given in column labeled "D recom ". The experimental [12] transition energies are given in column labeled "∆Eexpt". The remaining contributions to valence polarizability are given in row labeled "Other". The contributions from the core and αvc terms are listed together in row labeled "Core + Vc".
for the state v with total angular momentum J and projection M . The wave function Ψ(v, M ′ ) is composed of parts that have angular momenta of J ′ = J, J ± 1 that allows us to determine the scalar and tensor polarizability of the state |v, J, M [28]. The effective dipole operator D eff includes RPA corrections. The core and α vc terms are evaluated in the random-phase approximation. Their uncertainty is determined by comparing the DF and RPA values. The small α vc term is calculated by adding α vc contributions from the individual electrons, i.e. α vc (5s 2 ) = 2α vc (5s), and α vc (5s5p) = α vc (5s) + α vc (5p). The frequency dependence of the core and α vc terms is negligible, and we use their static values in all calculations. While we do not use the sum-over-states approach in the calculation of the polarizabilities, it is important to establish the dominant contributions to the final values. We combine the electric-dipole matrix elements and energies according to the sum-over-states formula for the valence polarizability [29]: to calculate the contributions of specific transitions.
Here, J is the total angular momentum of the state v, D is the electric-dipole operator, E i is the energy of the state i, and frequency ω is zero in the static polarizability calculations.
We have carried out several calculations of the dominant contributions to the polarizabilities using different sets of the energies and E1 matrix elements in or- der to understand the difference of the theoretical predictions for the Stark shift 5s 2 1 S 0 − 5s5p 3 P o 0 ∆α = 261(4) a.u. and recent experimental measurement ∆α = 247.374(7) a.u. as well as to provide a recommended value for the 5s5p 3 P o 0 − 5s4d 3 D 1 matrix element. The results are summarized in Table III. Other theoretical calculations of Sr polarizabilities were recently compiled in review [29]. The ground-state polarizability of Sr was calculated using relativistic coupled-cluster (RCC) method in [30]. Their value 199.7(7.3) a.u. is in agreement with our calculations.
In Table III the absolute values of the corresponding reduced electric-dipole matrix elements are listed in columns labeled "D" in a.u.. The theoretical and experimental [12] transition energies are given in columns ∆E th and ∆E expt . The remaining valence contributions are given in rows labeled "Other". The contributions from the core and α vc terms are listed together in row labeled "Core +Vc". The dominant contributions to α 0 listed in columns α 0 [A] and α 0 [B] are calculated with CI + all-order +RPA (no other corrections) matrix elements and theoretical [A] and experimental [B] energies [12], respectively.
Our α 0 [A] result agrees with the earlier calculation of [11] which was carried out using CI+MBPT approach with energy fitting that approximated missing higherorder corrections to the wave functions. We note that this may be fortuitous since the calculation of [11] was carried out in V N potential, while we are using V N −2 potential since the present version of the CI+all-order method is formulated for V N −2 potential. The E1 matrix elements in [11,26] included RPA but omitted all other corrections calculated in the present work. We find that replacing the theoretical energies with experimental -0.1477 (23) values reduces the Stark shift by 2.3%. We note that in the case of Sr all of the states contributing to the polarizabilities are included in our computational basis and this procedure is not expected to cause problems with basis set completeness, as in the case of Yb [31]. The dominant contributions to α 0 listed in column α 0 [C] are calculated with experimental energies and final ab initio matrix elements. Inclusion of the small corrections further reduces the value of the Stark shift by 3.1%, and our resulting value obtained with our final ab initio matrix elements is in excellent agreement with experiment [13].

V. DETERMINATION OF RECOMMENDED VALUES OF ELECTRIC-DIPOLE MATRIX ELEMENTS
To further improve our values of the other matrix elements, we use the measurement of the magic wavelength for the 3 P o 0 − 1 S 0 clock transition to determine recommended values of the 5s5p 3 P o 0 − 5s6s 3 S 1 , 5s5p 3 P o 0 − 5s5d 3 D 1 , and 5s5p 3 P o 0 − 5p 2 3 P 1 matrix elements. The 1 S 0 polarizability at the 813.4 nm magic wavelength is 286.0 a.u. which is essentially fixed by the value of the 5s5p 1 P o 1 lifetime [26]. The contributions to the 3 P o 0 polarizability at the magic wavelength are listed in Table IV. Since the contribution of the 5s5p 3 P o 0 − 5s6s 3 S 1 transition is dominant, the magic wavelength limits the value of this matrix element within about 0.5%. Since we appear to systematically overestimate the correction to matrix elements beyond CI+all+RPA approximation, we adjust the values of two other 5s5p 3 P o 0 − 5s5d 3 D 1 and 5s5p 3 P o 0 − 5p 2 3 P 1 matrix elements in a similar way as the 5s5p 3 P o 0 − 5s6s 3 S 1 one. We plot the dynamic polarizabilities of the 1 S 0 and 3 P o 0 states in the vicinity of the magic wavelength on Fig. 1 to illustrate that the crossing point is extremely sensitive to the matrix element values. The 0.5% change in the values of the matrix elements (corresponding to 1% change in the value of the 3 P o 0 polarizability) shifts the crossing point by more than 4 nm.
After we determined the values of these three matrix elements, we used them to obtain the value of the 5s5p 3 P o 0 − 5s4d 3 D 1 matrix element from the experimental value of the Stark shift. We determine its uncertainty from the uncertainty of all the other contributions to the 3 P o 0 polarizability value (listed in the last column of Table IV). Since our theoretical values may experience a systematic shift in one direction, we (somewhat conservatively) simply add all of the uncertainties, totaling to 3.3 a.u, instead of adding them in quadrature. Assigning this value to be the uncertainty in the dominant 3 D 1 contribution of 272.7 a.u., we estimate the uncertainty in the recommended value of the 5s5p 3 P o 0 − 5s4d 3 D 1 matrix element to be 0.5%. Since the contributions to both static and dynamic polarizabilities from 5s5p 3 P o 0 − 5s7s 3 S 1 and 5s5p 3 P o 0 − 5s6d 3 D 1 transitions are small, we use ab initio CI+all+RPA values and assign them 1.5% uncertainty. Combined with experimental energies and other small contributions, the set of recommended matrix elements reproduces recommended values for both the 5s5p 3 P o 0 static and dynamic polarizability at 813.4 nm magic wavelength.

VI. BLACKBODY RADIATION SHIFT
The leading contribution to the multipolar black body radiation (BBR) shift of the energy level g can be expressed in terms of the electric dipole transition matrix elements [32] Here k B is the Boltzmann constant, y n ≡ (E n − E g )/(k B T ), and F 1 (y) is the function introduced by Farley and Wing in [32]. Its asymptotic expansion is given The Eq. (3) can be expressed in terms of the dc polarizability α g (ω = 0) of the state g as [11] Here ∆E dyn g is determined as and η represents a "dynamic" fractional correction to the total shift that reflects the averaging of the frequency dependence of the polarizability over the frequency of the blackbody radiation spectrum. Corresponding shift in the clock transition frequency, ∆ν dyn )/h, is referred to as dynamic correction to the BBR shift. The quantity η can be approximated by [11] η = η 1 + η 2 + η 3 = 80 63 (2J g + 1) Dynamic corrections to the BBR shift of the 5s 2 1 S 0 − 5s5p 3 P o 0 clock transition in Sr at T = 300 K (in Hz) are given in Table V. The dynamic correction to the BBR shift of the 3 P o 0 level is dominated by the contribution from the 5s5p 3 P o 0 − 5s4d 3 D 1 transition, which contributes 98.2% of the total. Our final result −0.1492 (16) Hz is in excellent agreement with recent value −0.1477(23) Hz of Ref. [13].
Our result enables us to propose an approach for further reduction of the uncertainty in the BBR shift: a measurement of the 5s4d 3 D 1 lifetime with 0.5% uncertainty would provide the value of the BBR shift in Sr clock that is accurate to about 0.5%, which would be a factor of 2 improvement in the uncertainty stated here. Such a determination assumes accurate knowledge of the branching ratios. The 5s4d 3 D 1 level decays to all three 5s5p 3 P o 0,1,2 states, but the branching ratio to the 3 P o 2 level is very small. The lifetime of a state a is calculated as The E1 transition rates A ab are calculated using where λ is the wavelength of the transition inÅ and S is the line strength. We find that the branching ratios are essentially independent of the correlation corrections to the matrix elements. We note that line strength ratios are close to the non-relativistic ones (5/9, 5/12, 1/36), with the differences being −0.23%, +0.22%, and 1.4% for the 3 D 1 − 3 P J transitions, respectively. We illustrate this point in Table VI, where we list the relevant energies, line strengths S, transition rates A, and branching ratios in the CI, CI+MBPT, and CI+all-order approximations.
The RPA corrections to the matrix elements are included in all cases. We used experimental energies in all calculations for consistency. We find that the difference in the CI, CI+MBPT, and CI+all-order branching ratio results is less than 0.1%. Since all the other corrections are small, their uncertainties should be even smaller. As a result, the accuracy of our branching ratios should be better than 0.2%. The recommended values for the 5s5p 3 P o 1,2 − 5s4d 3 D 1 matrix elements are obtained using the recommended matrix element for the 5s5p 3 P o 0 − 5s4d 3 D 1 transition and CI+all-order branching ratios. The recommended values for the transition rates and the 5s4d 3 D 1 lifetime, 2172(24) ns, are obtained using the recommended values of the matrix elements and experimental energies. We also list the recommended values 3 D 2 , 3 D 3 , and term-averaged 3 D lifetimes in Table VII. The 3 D term-averaged lifetime is compared with experiment [14,15].

VII. CONCLUSION
We have evaluated the static and dynamic polarizabilities of the 5s 2 1 S 0 and 5s5p 3 P o 0 states of Sr and explained the discrepancy between the recent experimental 5s 2 1 S 0 − 5s5p 3 P o 0 dc Stark shift measurement [13] and the earlier theoretical result [11]. Our theoretical value for the dc Stark shift of the clock transition, 247.5 a.u., is in excellent agreement with the experimental result. We have provided the recommended values of the matrix elements for transitions that give dominant contributions to the clock Stark shift and evaluated their uncertainties. We evaluated the dynamic correction to the BBR shift of the 1 S 0 − 3 P o 0 clock transition at 300 K to be -0.1492 (16) Hz and proposed an approach for further reduction of the uncertainty in the BBR shift.