Magic wavelengths for the transition in ytterbium atom
Abstract
The static and dynamic electricdipole polarizabilities of the and states of Yb are calculated by using the relativistic ab initio method. Focusing on the red detuning region to the transition, we find two magic wavelengths at 1035.7(2) nm and 612.9(2) nm for the transition and three magic wavelengthes at 1517.68(6) nm, 1036.0(3) nm and 858(12) nm for the transitions. Such magic wavelengths are of particular interest for attaining the stateinsensitive cooling, trapping, and quantum manipulation of neutral Yb atom.

December 2017
1 Introduction
Ytterbium has rich variety of isotopes, a ground state, longlived metastable 6s6p and states and a number of transitions at wavelengths easily accessible by lasers for cooling and trapping. The optical trapped Ytterbium atoms provide a promising tool to study degenerate quantum gases [1], optical atomic clock [2], quantum information processing [3], and search for the CPviolating [4], etc. For neutral atoms, the optical trapping potentials cause spatially inhomogeneous energy shifts of the electronic states. A carefully designed optical trap with a magic wavelength that shifts the energies of the selected states equally provides a solution to this problem. As proposed by Katori et al., magic wavelength has been demonstrated for the atomic clock transition of Sr and Yb [5, 6]. Stateinsensitive trapping that is ascribed with a magic wavelength has also been enabled for Cs and Sr atoms [7, 8, 9].
The intercombination transition of Yb has a narrow linewidth around 181kHz that could cool atoms down to the photon recoil temperature of 4.4 K [10]. In order to gain a highdensity trapping, a faroff resonant trap (FORT) for Yb atom was then adopted that is overlapped on a magnetooptical trap (MOT) using the to transition [11, 12]. The highdensity trapping and the following achievement of BoseEinstein condensation (BEC) of Yb atoms have been demonstrated by using FORT, which is expected to be an important step for a future investigation of new quantum phenomena [1]. By tuning the laser wavelength to a magic wavelength (red detuning is required for FORT), at which the and states have the same AcStark shifts, which guarantees that the FORT is compatible with the Doppler cooling then enables high loading efficiency of magnetooptically trapped atoms. The magic wavelength in Yb and the other alkali metal atoms has been studied widely in past years [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], however most of them are concentrated on the atomicclock transition, , whereas data of the magic wavelength of the transition is lacking, despite that the stateinsensitive trapping of Yb is pursued eagerly in experiments [27].
In this paper, we calculate the static and dynamic electricdipole polarizabilities of the and states in Yb by using the relativistic manybody calculation. We compute the magic wavelengths of the transition in Yb, considering its potential application in FORT for BEC experimental of Yb atoms. Yb has zero nuclear spin and then no hyperfine structure. We focus on the magic wavelengths that are larger than the resonant wavelength (=556nm) of the transition. We determine two magic wavelengths for the transition to be 612.9(2) nm and 1035.7(2) nm and three magic wavelengths for the transition to be 1517.68(6) nm, 1036.0(3) nm, and 858(12) nm. Throughout the paper, we use atomic units (a.u.) for all energies, polarizabilities unless stated explicitly. The numerical values of the elementary charge , the reduced Planck constant =, and the electron mass are set equal to 1 in atomic units. The atomic unit of is equal to about CmJ.
2 Method of calculation
The ground and low lying excited states of Yb are calculated by the configuration interaction plus manybody perturbation (CI+MBPT) method that is implemented by using package [28]. The computational theory and technique of the CI+MBPT method has been documented in Refs. [29, 19]. In the CI+MBPT method, the effective Hamiltonian for two valence electrons is written as
(1) 
where and are the singleelectron and twoelectron parts of the relativistic Hamiltonian. In the implementation of the correlation operator , in addition to corresponds to the standard CI method, a singleelectron operator , representing a correlation interaction of a particular valence electron with the atomic core, and a two electron operator , representing screening of the Coulomb interaction between the two valence electrons by the core electrons are taken into account. In the CI+MBPT package, and are calculated in the second order of the MBPT. The basis set is constructed by using “auto” generation regime that is provided in the CI+MBPT package [28]. The oneelectron basis set includes , , , , and orbitals, where the core and , , orbitals are DiracHartreeFock (DHF) ones, while all the rest orbitals are virtual ones. The potential is used in the DHF calculation and the virtual orbitals is yielded numerically by using a recurrent relationship [28].
Following the sumoverstate mythology, the dynamic polarizability is written as three parts,
(2) 
where , , and represent the valence, core, and valencecore contributions, respectively, and is the optical frequency. When =0, is reduced to the static polarizabilities . The part is formulated as
(3) 
where and are the scalar and tensor polarizabilities. For a state , the and can be written as
(4) 
(5)  
where is the reduced matrix element of the electric dipole transition. The corevalence term is generally small, therefore it is neglected at our level of accuracy in the present work.
The core polarizability is calculated by using finitefield approach. The energy of the ground state of an atom in the presence of an external weak electric field of strength can be expressed in the perturbation theory as [30, 31]
(6) 
where is the energy of the state in the absence of the electric field and is known as the dipole polarizability of the state. It is obvious from the above expression that can be determined by evaluating the secondorder differentiation of with a small magnitude of electric field as
(7) 
This procedure is known as finitefield approach for evaluating which involves calculations of after including the interaction Hamiltonian with the atomic Hamiltonian. This approach has been adopted for calculation of electric dipole polarizabilities of a few atoms and ions [32, 33]. For achieving numerical stability in the result, it would be necessary to repeat the calculations by considering a number of values. This is accomplished by using the relativistic couple cluster (RCC) method that is provided in the relativistic ab initio package DIRAC [34]. We regard Yb as a closed shell system of 68 electrons. We have verified the electron correlation arising from the internal core electrons has negligible effect on , and therefore the atomic core are frozen in the RCC calculation, while the orbitals are correlated. We use the Dyall’s uncontracted correlated consistent double, triple, quadruple GTO basis sets, which are referred to as , where =2, 3, and 4, respectively, [35]. Each shell is augmented by two additional diffuse functions (daug) and the exponential coefficient of the augmented function is calculated based on the following formula
(8) 
where and are the two most diffuse exponents for the respective atomicshells in the original GTOs. The convergence of the results with the progressively larger basis set is checked, as presented in Table 1. The final value is taken to be the obtained by using the basis of . The error bar is estimated to be two times of difference of the values obtained by using the basis of and .
3 Results and discussion
Table 2 presents the energies for 17 evenparity states of , , , , , and configurations and 15 oddparity states of , , and of Yb. Our CI+MBPT results show excellent agreement with the National Institute of Standards and Technology (NIST) data for energies. Table 3 presents the reduced matrix elements (RME) of the electric dipole transition among these states. While our RME values are consistent with the previously reported CI+allorder values [21], a big discrepancy is found in comparison of our RME value with the experimental value for the transition. The RME value for this transition has been determined experimentally to be 4.184(2) [37, 22], while our CI+MBPT and also the CI+allorder values [21] are around 4.784.79, being deviated from such experimental value 16%. This discrepancy has been noted in the previous calculations [16, 19, 21]. The reason was attributed to the missing of the configurations in the computation model. The importance of the configurations has been discussed by Dzuba, et al., [19], which suggests that the large theoryexperiment disagreement for the RME value of the transition is due to mixing of the state with the coreexcited state , whereas the latter is out of the computational model space of divalence system that is used in our CI+MBPT and the previous CI+allorder calculations. Since the transition contributes about 90% to the polarizability of , we replace the CI+MBPT value of RME for the transition by its experimental value, and include the contribution of that is taken from an evaluation that was made by Beloy [22] based on the experimental values of five lifetime results compiled in Ref. [38]. Besides, as argued by Dzuba, et al., such a substitution cannot be justified unless a similar correction is done for the calculation of core polarizability, , i.e., the excitation from to has also been included in the calculation of . Therefore, instead of the RPA result 6.39 [19], the value is taken to 7.270.04 in terms of our finite field calculation for the polarizability of Yb. The excitation of is taken into account in our RCC calculation. Our RCC value of is larger than the previously reported RPA value by about 15%.
In Table 4, we give a breakdown of the main contributions from the intermediate states to the static polarizabilities of the and states. The main uncertainty is arising from the contributions of all higher lying excited states that are beyond the calculated states in our present CI+MBPT calculation that is given under “All others”. The contribution of “All others” is estimated based on the known knowledge of contributions of all other terms except the dominant ones, as given in the previously CI+allorder calculations [21, 23]. We assign the error bar ascribed with “All others” to be 50%. Such assignment should be reasonable because the errors of the CI+allorder values we quote and our calculated RME results both are less 5%. Besides, through comparison of our calculated RME values with the previous CI+allorder ones [21, 23], we determined a 2% error in our calculated RMEs, while the error bar in RMEs of , , are quoted from their references. The errors in “All others” and our calculated RMEs are translated to the uncertainty of . Finally, the values of for the and states are determined to be 135(3) and 328(19). Similarly, the tensor polarizabilities for the state is determined be 24.1(1.5).
The dynamic polarizabilities of the and states are calculated in terms of Eqs. (2)(5) by summing over all the intermediate states for nonzero values of . The ionic core polarizability depends weakly on and therefore is approximated by their static value. The “All others” contributions, as considers the high lying excited states beyond the states listed in Table 5, are taken from their static values. The role of the high lying excited states is not important for the frequencies treated here since their contributions are not resonant for the low frequencies. The total polarizability of the state depends upon its projection and then the magic wavelengths need to be determined separately for the cases with and owing to the presence of the tensor contribution to the total polarizability of the state. The magic wavelengths are found at the crossing of the curves for the and states. In Fig. 1, we can identify two magic wavelengths for the case of . They occur at nm and 612.9(2) nm close to the and resonances, respectively. For the case of , three magic wavelengths, nm, 1036.0(3) nm, 858(12) nm are found. The first locates at a very small energy interval between the and resonances, the second locates near the resonance, and the third locates the region between the and resonances. Because the and resonances make no contribution to the total polarizabilities owing to the exact cancelation of the scalar and tensor components for the state, the magic wavelengths near the two resonances are absent for the transitions. The magic wavelength near resonance is absent for the transitions due to the similar reason. The magic wavelength at =858(12) nm is red detuned from the and resonance with a large detuning about 302 nm that can be used for the magic wavelength ascribed with FORT. Another magic wavelength at =612.9 nm has a detuning about 57 nm.
Table 5 gives the breakdown of contributions of individual transitions to the dynamic polarizabilities at the magic wavelengths . The uncertainty in the value is determined by variation of position of the magic wavelength due to variation of for the and states. The first uncertainty in stems from error of our evaluation of “All others” term in , as denoted by Uncer.I. The second one comes from error of RMEs used in calculation of . Then, the error bar is given for each magic wavelength in Table 5. The error bar of the magic wavelength at =858 nm is obviously larger than the other magic wavelengths, being 12 nm, and as our analysis, such large error bar is predominately caused by error in RME of .
4 Conclusion
In summary, we have calculated the static and dynamic electric dipole polarizabilities of the and states by using relativistic ab initio method. Five reddetuned magic wavelength are identified, which locate at 1035.7(2) nm and 612.9(2) nm for the case of the transition and 1517.68(6) nm, 1036.0(3) nm, and 858(12) nm for the transition. Magic wavelength at =858(12) nm is of particular interest for the faroff FORT, and the laser frequency is also readily available, for example using a Ti:sapphire laser or a tapered amplifier.
Acknowledgments
The authors thank Prof. Mikhail G. Kozlov for the helpful suggestions about the use of CIMBPT package. One of the authors, Z. M. Tang, is grateful to Dr. Jiguang Li and Dr. Zhan Bin Chen for the valuable discussions. The work was supported by the National Natural Science Foundation of China, Grants No. 91536106 and No. U1332206, and the CAS XDB21030300, and the NKRD Program of China (2016YFA0302104).
References
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Basis  DHF  RCCSD 

6.26  7.30  
6.36  7.29  
6.38  7.27  
Final value  7.270.04 
State  CI+MBPT  NIST  Diff. 

0  0  
17446  17288.439  0.9  
18135  17992.007  0.8  
19858  19710.388  0.8  
25229  24489.102  3.0  
25479  24751.948  2.9  
25698  25068.222  2.5  
25990  25270.902  2.8  
28310  27677.665  2.3  
32701  32694.692  0.02  
34287  34350.65  0.2  
38063  38090.71  0.04  
38133  38174.17  0.1  
38509  38551.93  0.1  
39849  39808.72  0.1  
39882  39838.04  0.1  
40004  39966.09  0.1  
40132  40061.51  0.2  
38894  40563.97  4.1  
41563  41615.04  0.1  
41965  41939.90  0.06  
43183  42436.91  1.8  
43612  43614.27  0.01  
43618  43659.38  0.1  
44471  43805.42  1.5  
43797  43806.69  0.02  
43861  44017.60  0.4  
44611  44311.38  0.7  
44496  44313.05  0.4  
44583  44357.60  0.5  
44569  44380.82  0.4  
45768  44760.37  2.3 
Transition  CI+MBPT  CI+allorder  Exp. 

0.58  0.571 [21]  0.543(11) [37]  
4.79  4.78 [21]  4.148(2) [1]  
0.08  
0.67  0.65 [21]  
0.01  
0.21  
2.04(6) [22]  
2.57  2.51 [23]  
4.45  4.35 [23]  
0.46  0.453 [23]  
1.57  1.62 [23]  
2.72  2.78 [23]  
0.53  
2.47  
1.67  
0.86  
3.47  3.46 [23]  
0.24  0.243 [23]  
1.00  
0.30  
2.59  
0.18  
2.92 

This RME is derived from Beloy’s evaluation who determined contribution of to polarizability of by taking the weighted mean of five state lifetimes [22].
Polariz.  Contrib.  
4.148(2) [37]  100.4(1)  
0.543(11) [1]  2.4(1)  
0.67  1.62  
0.08  0.021  
0.21  0.15  
0.014  0.0007  
2.04(6) [22]  21.1(1.2)  
All others  2.04  
7.27(4)  
Total  135(3)  
Refs.  118(45) [16],141(6) [19],  
144.59 [18],139(15) [20],  
141(2) [21],  
[22]  
0.58  0.91  
2.57  49.44  
4.45  142.96  
0.46  1.04  
1.57  5.54  
2.72  16.46  
0.53  0.62  
2.47  11.33  
1.67  5.14  
0.86  1.36  
3.47  39.91  
0.24  0.17  
1.00  2.06  
0.29  0.18  
2.59  13.43  
0.18  0.06  
2.92  15.59  
All others  16.87  
7.27(4)  
Total  328(19)  
Refs.  278(15) [16],315(11) [23]  
24.1(1.5)  
Refs.  24.3(1.5) [16],24.06(1.37) [39],  
24.26(84) [40],23.33(52) [41] 

This value is estimated by subtract contributions of and from the value of “All others” given in Ref. [21].

This value is estimated by subtract contributions of , and from the value of “All others” given in Ref. [23].
1035.7(2)  612.9(2)  1517.68(6)  1036.0(3)  858(12)  
Uncer.I  0.10  0.10  0.042  0.20  8.2 
Uncer.II  0.06  0.10  0.017  0.09  4.2 
117.918  174.215  107.879  117.906  128.128  
3.36757  13.4883  2.76919  3.36672  4.13213  
1.71532  1.93041  1.66200  1.71526  1.76378  
0.16130  0.17799  0.15706  0.16130  0.16512  
0.02285  0.02616  0.02204  0.02284  0.02358  
0.00069  0.00077  0.00068  0.00069  0.00071  
23.7609  31.0141  22.2616  23.7591  25.2151  
All others  2.04  2.04  2.04  2.04  2.04 
7.27  7.27  7.27  7.27  7.27  
Total  156  230  144  156  169 
3.84505  15.4008  0  0  0  
0  0  2603.08  61.4402  33.4244  
164.954  35.5567  2576.42  123.866  65.2025  
198.075  0.68009  1.74439  135.320  2.09038  
0  0  9.14158  10.3296  11.6266  
24.5451  44.6557  16.2955  18.4061  20.7086  
0.91309  1.62798  0.60794  0.68472  0.76798  
0  0  74.9143  105.212  161.165  
0.80542  97.6243  0  0  0  
0  0  3.34266  3.70036  4.07467  
0.63413  0.99097  0  0  0  
47.7388  72.6530  0  0  0  
0  0  0.09332  0.10142  0.10959  
21.5007  29.7570  14.9324  16.1240  17.3099  
0  0  18.1288  19.6331  21.1388  
7.12690  10.0160  4.93522  5.34466  5.75448  
1.88309  2.64228  1.30439  1.41218  1.52001  
All others  14.57  14.57  18.01  18.01  18.01 
7.27  7.27  7.27  7.27  7.27  
Total  156  230  144  156  169 