Static electric dipole polarizability (α) of an atomic system provides the information about the distortion of the electron cloud when the system is placed in an electric field. Accurate knowledge of this quantity is essential in many areas of physics. The ytterbium (Yb) atom is a special atom in the sense that its isotopes are rich in the earth and it has long-lived metastable 6s6p and states and a number of allowed transitions with the wavelengths that are easily accessible by the lasers for cooling and trapping processes. The optically trapped Yb atoms provide a promising tool to study the degenerate quantum gases,^[1] optical atomic clock,^[2–4] quantum information processing,^[5] search of the parity nonconservation,^[6] CP-violation,^[7] etc.

There are two transitions in Yb that are the most important. The first is the transition from the ground state 6s²¹S₀ to 6s6p , which is used for the clock transition of a Yb optical lattice clock.^[8] The precision values of the static polarizabilities of 6s²¹S₀ and 6s6p states are needed for the best evaluation of the uncertainty of the clock transition caused by the Stark shift and black-body radiation shift. The intercombination transition 6s²¹S₀–6s6p of Yb has a narrow linewidth around 181 kHz that could cool atoms down to the photon recoil temperature of 4.4 μK.^[9] In order to gain a high-density trapping, a far-off resonant trap (FORT) for Yb atom was then adopted, which overlaps a magneto-optical trap (MOT) by using the ¹S₀ to transition.^[10–12] The high-density trapping and following achievement of Bose-Einstein condensation (BEC) of ¹⁷⁴Yb atoms using FORT experiments are expected to be important steps for the future investigation of new quantum phenomena.^[1] There are no precise experimental α values available for the above states of Yb atom. A preliminary experimental result α of the ground state of Yb has been reported as 142(36) a.u. (The unit a.u. is short for atomic unit), that has a very large uncertainty.^[13] There are also a number of calculations on these quantities, which have been carried out by employing the variants of many-body methods.^[14–18] However, most of these calculations are not consistent with each other due to large electron correlation effects associated with this atom.

In this paper, we calculate the static α values of the 6s²¹S₀, 6s6p , and 6s6p states of Yb by using the relativistic many-body methods. First, the energies and the electric dipole (E1) matrix elements are determined by employing a combined configuration interaction (CI) method and a lower-order many-body perturbation theory (MBPT) in the relativistic theory framework. We use these quantities later to estimate the dominant valence correlation contributions to α in the above states. The core contributions are estimated by using a relativistic coupled-cluster (RCC) method from the DIRAC package^[19] in the finite field approach. These theories are described briefly in the next section. Throughout the paper, we use atomic units (a.u.) unless otherwise stated.

We use the CI + MBPT method that is implemented in the package^[20] in our calculations. This theory and implementation technique have been documented in Refs. [15] and [21]. The calculation method is based on a combination of a conventional configuration interaction (CI) method and many-body perturbation theory (MBPT). The former explicitly contains the interaction between valence electrons, while the latter includes core–core and core–valence correlations.^[20] In the CI + MBPT method, the effective Hamiltonian for two valence electrons is written as

Generally, the static polarizability α of the atomic state can be expressed as

The contribution vanishes and the contributions are obtained through using the finite-field approach. First, we calculate energies considering the Dirac–Coulumn-Hartee (H_DC) Hamiltonian. Then, we include the interaction potential energies induced by the external electric field and its gradient with H_DC and treat them perturbatively to estimate the α^E1 values, respectively. The energy of an atomic state |γJM_J⟩ in the presence of these interactions can be expressed as

First we give the results of the energies obtained by using the CI + MBPT method and compare our results with the previously available data. In Table 1, we present energies of the 17 even-parity 6s², 6s7s, 6s8s, 5d6s, 6s6d, and 6p², and 15 odd-parity 6s6p, 6s7p, and 6s8p low-lying states of Yb. We can find that our CI + MBPT results show good agreement with the experimental values listed in the National Institute of Standards and Technology (NIST) database^[16] within an error of 1% except for the 5d6s, 6s7p, and 6p² states, where the discrepancies are around 2%–5%. The reason for such a large discrepancy may be that the electron correlation arising from these states is not implemented sufficiently due to our limited configuration-state-function spaces.

Table 1.

Table 1.

Table 1. The excited energies (in unit cm−1) of the low lying excited states of Yb, obtained by using the CI + MBPT method. The absolute difference in percentage between our CI + MBPT values and the NIST data[25] is denoted as “Diff.”. . State CI + MBPT NIST Diff./% 6s21S0 0 0 6s6p 17446 17288.439 0.9 6s6p 18135 17992.007 0.8 6s6p 19858 19710.388 0.8 5d6s3D1 25229 24489.102 3.0 5d6s3D2 25479 24751.948 2.9 6s6p 25698 25068.222 2.5 5d6s3D3 25990 25270.902 2.8 5d6s1D2 28310 27677.665 2.3 6s7s3S1 32701 32694.692 0.02 6s7s1S0 34287 34350.65 0.2 6s7p 38063 38090.71 0.04 6s7p 38133 38174.17 0.1 6s7p 38509 38551.93 0.1 6s6d3D1 39849 39808.72 0.1 6s6d3D2 39882 39838.04 0.1 6s6d3D3 40004 39966.09 0.1 6s6d1D2 40132 40061.51 0.2 6s7p 38894 40563.97 4.1 6s8s3S1 41563 41615.04 0.1 6s8s1S0 41965 41939.90 0.06 6p23P0 43183 42436.91 1.8 6s8p 43612 43614.27 <0.01 6s8p 43618 43659.38 0.1 6p23P1 44471 43805.42 1.5 6s8p 43797 43806.69 0.02 6s8p 43861 44017.60 0.4 6s7d3D1 44611 44311.38 0.7 6s7d3D2 44496 44313.05 0.4 6s7d1D2 44583 44357.60 0.5 6s7d3D3 44569 44380.82 0.4 6p23P2 45768 44760.37 2.3

Table 1. The excited energies (in unit cm−1) of the low lying excited states of Yb, obtained by using the CI + MBPT method. The absolute difference in percentage between our CI + MBPT values and the NIST data[25] is denoted as “Diff.”. .

Next, in Table 2 we also give reduced E1 matrix elements (RME) of all the possible transitions among these states. We compare our CI + MBPT data of E1 matrix elements with the literature data. We find that our data are consistent with previously reported CI + all-order values^[16] within an error of 2%. But, the large difference is found for the 6s²¹S₀–6s6p transition. From the lifetime measurement of the 6s6p state, the RME of 6s²¹S₀–6s6p can be deduced very precisely. The experimental value is reported to be 4.184(2) a.u.,^[16,17] while our CI + MBPT and also the CI + all-order values^[16] are obtained to be about 4.78 a.u.–4.79 a.u. This difference is about 16% and has also been noted earlier.^[15–17] The 4f¹³nln′l′n″l″ configurations are missed in the configuration-state-function space due to the limit of the computation model. The importance of the 4f¹³nln′l′n″l″ configurations has been discussed by Dzuba et al.,^[15] who suggested that the large theory–experiment disagreement for the E1 matrix elements of the 6s²¹S₀–6s6p transition is due to the high mixing of the 6s6p state with the core-excited state 4f¹³5d6s² . The 4f¹³nln′l′n″l″ configurations have been ruled out from the computational model space of the divalence system that is used in our CI + MBPT method and the previous CI+all-order calculations. The considered 6s²¹S₀–6s6p transition contributes about 90% to the polarizability of the 6s²¹S₀ state, thus we decide to use the experimental value of the E1 matrix element for the 6s²¹S₀–6s6p transition instead of the results obtained by the CI + MBPT method, and at the same time we include the contribution of the 4f¹³5d6s² states. The RME of 6s²¹S₀–4f¹³5d6s² is estimated in a similar way to that in Ref. [27] where the weighted mean of the E1 matrix element contribution to the lifetimes of five states^[28] is determined.

Table 2.

Table 2.

Table 2. Reduced electric-dipole transition matrix elements involving the 6s21S0, 6s6p , and 6s6p states (in unit a.u.) from the CI + MBPT method. Values from the CI+all-order calculation[16] and values deduced from the experimental measurements of lifetimes of atomic states are given wherever available. . Transition CI + MBPT CI + all-order Exp. 6s21S0–6s6p 0.58 0.571[16] 0.543(11)[26] 6s21S0–6s6p 4.79 4.78[16] 4.148(2)[1] 6s21S0–6s7p 0.08 6s21S0–6s7p 0.67 0.65[16] 6s21S0–6s8p 0.01 6s21S0–6s8p 0.21 6s21S0–4f135d6s2 2.04(6)[27] 6s6p –5d6s3D1 2.96 2.89[16] 6s6p –6s6d3D1 1.79 1.84[16] 6s6p –6s7d3D1 2.10 6s6p –6s7s3S1 1.95 1.95[16] 6s6p –6s8s3S1 0.62 6s6p –6p23P1 1.97 6s6p –5d6s3D1 2.57 2.51[34] 6s6p –5d6s3D2 4.45 4.35[34] 6s6p –5d6s1D2 0.46 0.453[34] 6s6p –6s6d3D1 1.57 1.62[34] 6s6p –6s6d3D2 2.72 2.78[34] 6s6p –6s6d1D2 0.53 6s6p –6s7d3D1 2.47 6s6p –6s7d3D2 1.67 6s6p –6s7d1D2 0.86 6s6p –6s7s3S1 3.47 3.46[34] 6s6p –6s7s1S0 0.24 0.243[34] 6s6p –6s8s3S1 1.00 6s6p –6s8s1S0 0.30 6s6p –6p23P0 2.59 6s6p –6p23P1 0.18 6s6p –6p23P2 2.92

Table 2. Reduced electric-dipole transition matrix elements involving the 6s21S0, 6s6p , and 6s6p states (in unit a.u.) from the CI + MBPT method. Values from the CI+all-order calculation[16] and values deduced from the experimental measurements of lifetimes of atomic states are given wherever available. .

In Tables 3, 4, and 5, we give the contributions from the intermediate states, respectively, to the static polarizabilities of the 6s²¹S₀, 6s6p , and 6s6p states of the Yb atom. The energies are cited from the NIST data. The values of RMEs are cited from our CI + MBPT calculation except the RMEs of 6s²¹S₀–6s6p , 6s²¹S₀–6s6p , and 6s²¹S₀–4f¹³5d6s² are replaced with the experimental values as cited. Each contribution is given under “α_i” for the dominant valence states. The contribution from the core is given by “α_core”. Again as argued by Dzuba et al., due to the missing of the 4f¹³nln′l′n″l″ configurations discussed above it is necessary to include a more accurate core contribution than the random phase approximation (RPA) as in this case the excitation from the 4f_7/2 to 5d_5/2 states is important. Therefore, instead of the RPA result for 6.39 a.u.^[15] for α_c, we have taken it as 7.27±0.04 a.u. from our finite field calculation of the polarizability in Yb²⁺. The RCC theory automatically captures the correlation contribution for 4f_7/2 → 5d_5/2. Our RCC value of α_c is about 15% larger than the previously reported RPA value, but it is in agreement with another RCC calculation obtained from the perturbative formulation.^[33]

Table 3.

Table 3.

Table 3. Breakdown of the contributions of static scalar polarizability αS for 6s21S0 state of Yb atom. . Polariz. Contrib. |⟨g‖D‖m⟩| αi αS(1S0) 6s6p 4.148(2)[26] 100.4(1) 6s6p 0.543(11)[1] 2.4(1) 6s7p 0.67 1.62 6s7p 0.08 0.021 6s8p 0.21 0.15 6s8p 0.014 0.0007 4f135d6s2 2.04(6)[27] 21.1(1.2) All others 2.04 αcore 7.27(4) Total 135(3) Refs. 118(45),[17] 144.59,[18] 141(6),[15] 139(15),[29] 141(2),[16] 134.4 ± 1.0 < αS < 144.2 ± 1.0[27]

Table 3. Breakdown of the contributions of static scalar polarizability αS for 6s21S0 state of Yb atom. .

Table 4.

Table 4.

Table 4. Breakdown of the contributions of static scalar polarizabilities αS for 6s6p state of Yb atom. . Polariz. Contrib. |⟨g‖D‖m⟩| αi 5d6s3D1 2.96 177.95 6s6d3D1 1.79 20.83 6s7d3D1 2.10 23.86 6s7s3S1 1.95 36.21 6s8s3S1 0.62 2.28 6p23P1 1.97 21.33 All others 15.23 αcore 7.27(4) Total 305(19) Refs. 302(14),[15] 293(10),[16] 280.1 ± 1.0 < αS<289.9 ± 1.0[27]

Table 4. Breakdown of the contributions of static scalar polarizabilities αS for 6s6p state of Yb atom. .

Table 5.

Table 5.

Table 5. Breakdown of the contributions of static scalar polarizability αS and tensor polarizability αT for 6s6p state of Yb atom. . Polariz. Contrib. |⟨g‖D‖m⟩| αi 6s21S0 0.543(11)[1] -0.80 5d6s3D1 2.57 49.44 5d6s3D2 4.45 142.96 5d6s1D2 0.46 1.04 6s6d3D1 1.57 5.54 6s6d3D2 2.72 16.46 6s6d1D2 0.53 0.62 6s7d3D1 2.47 11.33 6s7d3D2 1.67 5.14 6s7d1D2 0.86 1.36 6s7s3S1 3.47 39.91 6s7s1S0 0.24 0.17 6s8s3S1 1.00 2.06 6s8s1S0 0.29 0.18 6p23P0 2.59 13.43 6p23P1 0.18 0.06 6p23P2 2.92 15.59 all others 16.87 αcore 7.27(4) total 329(20) Refs. 278(15),[15] 315(11)[34] 24.0(1.5) Refs. 24.3(1.5),[17] 24.06(1.37),[30] 24.26(84),[31] 23.33(52)[32]

Table 5. Breakdown of the contributions of static scalar polarizability αS and tensor polarizability αT for 6s6p state of Yb atom. .

In our calculations, the main uncertainty arises from the contributions from the neglected high-lying excited states and citations in the above table as “all others”, which is estimated based on the known contributions of all other terms except the dominant ones, as given in previous CI + all-order calculations.^[16,34] We assign the error caused by “all others” to be 50%. Such an assignment should be reasonable because the difference in the E1 matrix element between the CI + MBPT and CI + all-order methods is less than 2% for the excited states we calculated. Besides, through the comparison of our calculated E1 matrix element values with previous CI + all-order ones,^[16,34] we include the additional 2% errors to the E1 matrix elements of the 6s²¹S₀–6s6p , 6s²¹S₀–6s6p , 6s²¹S₀–4f¹³5d6s² transitions. The errors in the “all others” contributions and our calculated E1 matrix elements are translated into the net uncertainties of the α^S values.

Finally, the values of α^S for the 6s²¹S₀, 6s6p , and 6s6p states are determined to be 135(3) a.u., 305(19) a.u., and 329(20) a.u. respectively. Similarly, the tensor polarizability α^T for the 6s6p state is determined to be 24.0(1.5) a.u. The ground state result agrees quite well with the other recent RCC calculation by using the perturbative approach.^[7] It can be noted that the discussion about the static polarizability of the ground state of Yb is whether the electronic correlation arising from the 4f¹³nln′l′n″l″ configurations is important. In this work, our calculation with the contribution of 6s²¹S₀–4f¹³5d6s² transition added is just consistent with the RCC result.^[7] This proves the importance of the electronic correlation from the 4f¹³nln′l′n″l″ configurations.

We have determined the scalar and tensor dipole polarizabilities of the 6s²¹S₀, 6s6p , and 6s6p states of the ytterbium atom by a sum-over-states approach. These results will be quite useful for the theoretical and experimental studies using this atom. Accuracies of these results can be tested when the corresponding experimental results are available. We have estimated these quantities by the sum-over-states approach through dividing the total contributions into three major categories, i.e., valence–, core–, and valence–core correlation contributions. The dominant valence correlation contributions are evaluated using the accurate values of E1 matrix elements and experimental energies, and the core contributions are estimated by the finite field approach in the relativistic coupled-cluster theory. Our ground state result is in good agreement with other recent relativistic coupled-cluster theory calculations in the perturbative framework. We also compare our results for the excited states with previous theoretical results and find that they are in reasonable agreement with each other.