Open Access
Add to BibSonomy
Abstract
We have identified magic wavelengths for 1S0 ↔ 3P1,2 (mJ = 0) transitions and zero-magic wavelengths for the 3P1,2 (mJ = 0) states of 200Hg atoms, analysed the robustness of the magic conditions with respect to wavelength and polarization imperfections. We show that the most experimentally feasible magic wavelength for the 1S0 ↔ 3P2 transition is 351.8 nm of π polarized light. Relevant transition wavelengths and transition strengths are calculated using the state-of-the-art Complete Active Space Self-Consistent-Field (CASSCF) method with a perturbative inclusion of spin-orbit coupling. The transition wavelengths are a posteriori corrected for the dynamical energy using the second-order perturbation theory.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The most important component of an optical frequency standard [1] is a sample of cold atoms or a single ion trapped in a coherence-preserving electromagnetic trap. In the case of ions, a Paul trap provides nearly state-independent confinement with minimal trap field perturbation [2]. An alternative to single-ions, are neutral atoms trapped in an optical lattice potential [3], which relies on spatially inhomogeneous Stark shift that depends on the relevant electronic state [4].
For a given atomic species, a special narrow optical transition (the so-called clock transition) is selected to ensure that its frequency is hardly sensitive to external fields. Nevertheless, the intense trapping light affects the clock transition frequency by the ac-Stark shift effect and lattice photons scattering on atoms decrease atomic coherence time. These effects limit both accuracy and stability of the optical atomic clock.
The so-called "magic" wavelength is a trap laser wavelength, where scalar polarizabilities of both ground and excited states of the clock transition are identical and the light shift of the clock transition is largely suppressed [5]. However, to achieve high accuracy, the residual polarization-dependent and higher-order light shifts still have to be evaluated [5–9].
The optical lattice atomic clock with ${}^{199}$Hg fermionic isotope was demonstrated [10,11] with the (6$s^2$) $^1S_0 \rightarrow (6s6p) ^3P_0$ transition. The magic wavelength for this transition was experimentally measured to be 362.51(0.16) nm [12,13], close to previously existing theoretical estimations ($\approx 363$ nm) [14,15]. The frequency of this transition is highly immune to environmental perturbations which are the sources of frequency shifts and instabilities. In particular, the low polarizability of neutral Hg results in low sensitivity of this transition to the blackbody radiation shift [15].
In principle, the (6$s^2$) $^1S_0 \rightarrow (6s6p) ^3P_0$ transition is strictly forbidden, but the presence of a non-zero nuclear spin in fermionic isotopes, and consequently the existence of hyperfine structure interactions, mixes the ${}^3P_0$ state with the ${}^3P_1$ and ${}^1P_1$ states and yields the non-zero natural linewidth $\Delta \nu _{265.6 nm} = 121$ mHz of the (6$s^2$) $^1S_0 \rightarrow (6s6p) ^3P_0$ transition [16,17].
Recently, it was pointed out that having two separate clock transitions in a single isotope can have a huge impact on reducing systematic uncertainties in a clock that would perform interleaved interrogations on two clock transitions [18]. The second clock transition, while not necessary so insensitive to external perturbations, can increase the sensitivity of an atomic clock to fine-structure variations [18–22].
Theoretical predictions show that the $^1S_0 \rightarrow ^3P_2$ transition in ${}^{199}$Hg fermionic isotope has the natural linewidth of the same order as the $^1S_0 \rightarrow ^3P_0$ [23]. This transition is, for instance, much more susceptible to external magnetic fields and can be used as an internal sensor for a real-time accuracy budget of the $^1S_0 \rightarrow ^3P_0$ transition.
Moreover, rich isotopic diversity in Hg in combination with the existence of two extremely narrow transitions provide a fascinating perspective of probing new fundamental interactions like Higgs boson couplings to the electron and the up and down quarks [24]. The practical realization of this idea is based on precise measurements of the King plot linearity [25–30].
Determining the magic wavelengths for the $^1S_0 \rightarrow ^3P_2$ transition is essential for future use of this line in an optical lattice clock with reasonable accuracy.
The same theoretical methods that are used for determining the magic wavelengths for a transition can be used for determining the "zero-magic" (or tune-out) wavelength for a state. The "zero-magic" is the wavelength when the state does not experience any light shift, and hence is insensitive to that light. The spectroscopy of zero-magic wavelengths provides information on atomic transition matrix elements, especially those that cannot be determined otherwise, serving as a benchmark of the spectroscopic accuracy that is required for the development of high-precision theoretical models [31]. The atomic transition matrix elements can verify such physical parameters as static polarizabilities, lifetimes of the states, transitions’ oscillator strengths, van der Walls potentials, or last but not least, magic wavelengths [32,33]. Furthermore, the exact knowledge of the oscillator strengths is essential in many areas of research. Examples are studies of fundamental symmetries [34,35], quantum degenerate gases [36], quantum information [37,38], plasma physics [39], and astrophysics [40,41]. The zero-magic wavelengths in alkali atoms have been used in experiments to study entropy exchange between ultra-cold quantum gases [42] and diffraction of matter waves on an ultra-cold atom crystal [43]. The zero-magic wavelengths, where the frequency-dependent polarizability of the particular state vanishes [44,45], can be used for engineering trapping potentials for neutral atoms in mixed-species experiments [42,46–48] and can be applied for state-selective atom manipulation for implementation of quantum logic operations. Applying the state-of-the-art ultra-narrow clock laser to the engineering of quantum states inside the lattice with a tunable frequency around the zero-magic wavelength opens up completely new possibilities in quantum engineering. One possible idea is to create "a quantum billiard", where the atoms are prepared in a superposition of two clock states, and tuning the trapping light can switch off the trapping force for one of the states.
We have identified magic wavelengths for $^1S_0 \leftrightarrow {^3P_{1,2} (m_J = 0)}$ transitions and and zero-magic wavelengths for the ${^3P_{1,2} (m_J = 0)}$ states of the 200Hg atom. Because of its short lifetime, the $^3P_{1}$ state is not applicable for the precise optical atomic clock realisation. The experimental verification of its polarizability dependence on the magic wavelength, however, would serve as a benchmark to the method used in this article.
Since the magic wavelength does not depend heavily on the particular isotope, as was shown for the red-detuned magic wavelength of Sr at 813 nm [49], we based our calculations on bosonic $^{200}$Hg isotope without limiting the generality of the results. Our ab initio calculations did not take into account the hyper-fine splittings and one can expect small deviations in the electronic structures of the Hg isotopes. These differences are inside the uncertainty of our approach.
In the next section, we briefly discuss the theoretical background and computational details of calculations. In section 3, we present the results of ab initio calculations and analysis of the robustness of the magic conditions with respect to wavelength and polarization imperfections. We conclude in section 4.
2. Theoretical background
2.1 Magic and zero-magic wavelengths
The magic wavelength for a transition is a wavelength where the ac Stark shifts of both the upper and lower atomic states are the same, while the zero-magic wavelength of a state is a wavelength where the ac Stark shift is zero. The ac Stark shift $U_i$ of the state $|i\rangle$ in an external monochromatic electromagnetic field can be written as [50]
The ac energy shift depends on the polarization of the light field if angular momenta of the upper and the excited states are not both zeros. In an ideally linear polarization of the field any predominance of the left-rotating wave over the right-rotating one, or vice versa, is absent. However, in a real experiment a deviation of the alignment of the magnetic field $\vec {B}$ from the polarization vector $\vec {E}_0$ and/or propagation vector $\vec {k}$ is always present. In the next section we consider the robustness of the magic condition with respect to such misalignment.
We introduce the angle $\varphi$ between $\vec {E}_0$ and $\vec {B}$: $\varphi =0$ for ideal $\pi$-polarization, and $\varphi =\pi /2$ for ideal $\sigma _{\pm }$-polarization. Polarizability for linearly polarized light at arbitrary $\varphi$ can be expressed as
Computational details of the transition wavelengths and Einstein coefficients $A_{Jki}$ from the $^1S_0, ^3P_1, ^3P_2$ states are presented in the next subsection.
2.2 Computational details of ab initio calculations
We have performed a highly reliable ab initio electronic structure calculations [51] based on the state-averaged Complete Active Space Self Consistent Field (CASSCF) [52,53] second-order Perturbation Theory (CASPT2) [54–56] approach with inclusion of spin-orbit coupling (SOC-CASPT2) in the open-source OpenMolcas software package [57,58]. In this approach we start with the Hartree–Fock method for the spin-free (relativistic) Douglas–Kroll–Hess quantum chemical Hamiltonian of the second order [59–61]. Since we work within the Born–Oppenheimer approach, the Hamiltonian does not depend on the isotope type. The presence of relativistic Hamiltonian is, however, indispensable for elements as heavy as Hg. Solutions of these equations in a given finite basis set provide us with approximate (uncorrelated) orbitals and orbital energies, which accounts for relativistic contraction of s and p orbitals as well as relativistic expansion of d and f orbitals. These orbitals are further used as a starting point for the multi-reference CASSCF calculation which reliably accounts for correlated motion of electrons in a given subspace. Larger the subspace, more reliable the wavefunction. We used what is computationally very expensive, but still feasible and correlated 12 electrons in 15 orbitals, CAS(12,15), comprising the 5d, 6s, 6p, 6d, and 7s atomic orbitals of Hg. After solving the subspace Hamiltonian, the orbitals are re-optimized in the self-consistent (SCF) manner—the above mentioned CASSCF approach. The missing weak electron correlation effects in the Hg atom, that is, coming from orbitals below 5d and above 7s, are accounted for using the second-order perturbation theory, the CASPT2 approach. These equations are solved to obtain the ground- and all possible excited-states within the subspace Hamiltonian, that is, among the 5d, 6s, 6p, 6d, and 7s atomic orbitals of Hg. Finally, we correct these energy states (and Einstein coefficients) by taking into account the the spin-orbit coupling [62]. The calculations from above scrutinized SOC-CASPT2 are used as reference in our work. All technical details indispensable for reproducibility are provided in Supplement 1. Example inputs can be found at [63].
2.2.1 Limitations of the SOC-CASPT2 approach
The SOC-CASPT2 approach is well-tested and applicable to atoms and molecules across the periodic table. Despite its superiority over standard electronic structure methods in the description of strong electron correlation effects, that is, multi-reference systems like the Hg atom, it has its own potential limitations. (a) First, we work within the finite basis set. The completeness of the basis set was checked by performing a series of calculations with an increasing basis set size. Our results are converged at the quadruple-$\zeta$ basis set and these results are used as a reference in this work. (b) Second, our ab initio calculations are limited by the number of active orbitals in the CASSCF approach. We believe that correlating the 5d, 6s, 6p, 6d, and 7s atomic orbitals of Hg represents the essential physics of its electronic excitations. Taking into account larger number of electrons and orbitals is computationally prohibited by the exponential cost of the approach. We should also note that due to the multi-reference nature of the electronic structure of Hg, there is no other more reliable quantum chemistry method to compare with. (c) Third, in our approach we do not account for small effects beyond the Born–Oppenheimer approach and the hyperfine interactions as their contributions are expected to be small. Finally, building on previous experience with the SOC-CASPT2 approach in determining electronic structures of heavy atoms [64–66], we expect that our electronic excitation energies are accurate within at least a few hundreds of cm$^{-1}$.
3. Results
3.1 Results of ab initio calculations
The calculated energies of ground states $^3P_i, ~ i = 0,1,2$ ($5d^{10} 6s^1 6p^1$) are summarized in Table 1 and results are compared with data available in Ref. [67]. $E$ is the energy of the atomic transition from the ground state $^1S_0$ ($5d^{10} 6s^2$) given in cm$^{-1}$. As one can see, there is a good agreement between the calculated energies and experimental data.
Table 1. Electronic excitation energies $E$ of the $^3P_0$, $^3P_1$, and $^3P_2$ ($5d^{10} 6s^1 6p^1$) states according to our calculations and NIST data.
The results of the spectroscopic data calculations are summarized in Table 2. It includes transitions from the states $^1S_0$ ($5d^{10} 6s^2$), $^3P_1$ ($5d^{10} 6s^1 6p^1$) and $^3P_2$ ($5d^{10} 6s^1 6p^1$) to the possible manifolds. When available, data is presented from Ref. [67] as well. $E$ is the energy of the corresponding atomic transition from one of the corresponding states $^1S_0$ ($5d^{10} 6s^2$), $^3P_1, ^3P_2$ ($5d^{10} 6s^16p^1$), given in cm$^{-1}$, the transition spontaneous rate $A_{Jki}$ is given in s$^{-1}$. Please note that for the $(5d^{10}6s^16d^1)$ energy states, the full determination of the atomic term symbol is invalid due to the strong mixing between several atomic orbitals and the Russel–Sunders approximation no longer holds. In the calculations for the term $^3P_0$, transitions $^3P_0 (5d^{10}6s^16p^1) \leftrightarrow J = 2 (5d^{10}6s^16d^1)$ with $\Delta J = +2$ appear and the Wigner 3-j symbols, and subsequently the polarizabilities, are impossible to calculate.
Table 2. Spectroscopic data of relevant transitions according to our calculations and NIST data. $E$ is the electronic excitation energy of the mentioned atomic transition, $A_{Jki}$ is the transition Einstein coefficient.
The data summarized in Table 2 demonstrates a good agreement between the theory and experiment, with exception of the energy states with the leading $5d^{10} 6s^1 7s^1$ electronic configuration. The NIST data [67] refers these values to the wavelength measurements of spectral lines in mercury pencil lamps [68].
3.2 Analysis of magic and zero-magic wavelengths
We consider two particular cases of polarization that can be attained with linearly polarized laser light, namely $\pi$-polarization and $xy$-polarization. First, we suppose that the polarization is perfect, i.e., the electric field of the trapping laser light is either parallel or perpendicular to the quantization axis. Using the data of Tables 1 and 2, the calculated ac Stark shifts dependence on wavelength for the $6s^2~{}^1S_0$, $6s6p~{}^3P_1, m_J=0$ and $6s6p~{}^3P_2, m_J=0$ states of the Hg atom corresponding to these configurations are presented in Figs. 1 and 2, respectively. In both cases, the top figures correspond to $\pi$ polarization, while the bottom ones correspond to $xy$ polarization. The black solid line is the Stark shift for the ground state $^1S_0$, while the red dashed line corresponds to $^3P_{1,2}$ states. The crossing of red and black lines correspond to magic wavelengths, shown in blue circles. At this points, the Stark shifts of the ground and excited states equal to each other. The green diamonds correspond to zero-magic wavelengths, when the Stark shift of the excited states is zero, hence these states are insensitive to the applied beam.
Fig. 1. Energy shifts for $^1S_0$ (black) and $^3P_1,m_J=0$ (red) states of the Hg atom in the running-wave laser field of $\pi$ (top) and $xy$ (bottom) polarization. Blue circles indicate magic wavelengths, where the energy shifts are equal, and green diamonds indicate zero-magic wavelengths, where the $^3P_1, m_J = 0$ state is not trapped. Intensity is taken $I = 10$~kW/cm$^2$.
Fig. 2. Energy shifts for $^1S_0$ (black) and $^3P_2,m_J=0$ (red) states of the Hg atom in the running-wave laser field of $\pi$ (top) and $xy$ (bottom) polarization. Blue circles indicate magic wavelengths, where the energy shifts are equal, and green diamonds indicate zero-magic wavelengths, where the $^3P_2, m_J = 0$ state is not trapped. Intensity is taken $I = 10$~kW/cm$^2$.
To estimate the robustness of the identified magic and zero-magic wavelengths with respect to the calculated spectroscopic data, we consider 10% of error in transition wavelengths and transition strengths and calculate the magic and zero-magic wavelengths for that case. The results are summarized in Table S1 of Supplement 1of the Supplementary Materials magic $\lambda _m$ and zero-magic $\lambda _{zm}$ wavelengths, when 10% of error is allowed for Einstein coefficients $A_{Jki}$ and atomic transition wavelengths $\lambda$ with respect to the calculated values using the data from SOC-CASPT2 approach. We observe that while the magic and zero-magic wavelengths are strongly susceptible to the SOC-CASPT2 transition energies, they are insensitive to variation of the Einstein coefficients.
To characterize the magic wavelength $\lambda _m$, the following set of parameters is used: $U_m/I$ — trap depth at magic wavelength to intensity ratio, $d U_{ge}/(U_m d\lambda )$ — sensitivity of differential energy shift $U_{ge}$ to variation of the wavelength at unit trap depth $U_m$ at magic wavelength, and $d^2 U_{ge}/(U_m d\varphi ^2)$ — sensitivity of the differential ac Stark shift $U_{ge}$ to the variation of the polarization angle $\varphi$ at unit trap depth ratio. Here the differential light shift is $U_{ge}=U_{g}-U_{e}$, where the index $g$ and $e$ correspond to the $^1S_0$-state and $^3P_1,m_J=0$ or $^3P_2,m_J=0$-states, correspondingly. Using Eqs. (2) and (3), the sensitivity of the differential energy shift $U_{ge}$ can be rewritten as
Table 3. Magic wavelengths for the $^3P_1$ and $^3P_2$ states for $\pi$- and $\sigma ^{\pm }$- polarizations, and analysis of sensitivity to wavelength and polarization fluctuations.
To characterise zero-magic wavelengths $\lambda _{zm}$, we introduce energy shift sensitivity to wavelength $d U_{e}/ d\lambda$ and to variation of polarization angle $d^2 U_{e}/d\varphi ^2$. These parameters are summarized in Tables 4 and 5, for $^3P_1$ and $^3P_2$ states, respectively.
Table 4. Zero-magic wavelength for the $^3P_1$ state for both $\pi$ and $xy$-polarizations and their sensitivity to wavelength and polarization fluctuations.
Table 5. Zero-magic wavelength for the $^3P_2$ state for both $\pi$ and $xy$-polarizations and their sensitivity to wavelength and polarization fluctuations.
In experimental realisation, e.g. of optical lattice clocks, the uncertainties related to the ac Stark shift depend on sensitivity of the magic wavelength on the wavelength and phase fluctuations, which implies that the most experimentally feasible magic wavelengths are 310.1 nm of $\sigma ^{\pm }$ polarized light for the $^1S_0 \leftrightarrow ^3P_1$ transition and 351.8 nm of $\pi$ polarized and 308.3 nm of $\sigma ^{\pm }$ polarized light for the $^1S_0 \leftrightarrow ^3P_2$ transition. The 351.8 nm magic wavelength trap can be experimentally realised with a reasonable trap depth and waist — the corresponding magic wavelength for the $^1S_0 \leftrightarrow ^3P_0$ clock transition, used in a vertical 1D optical lattice of the operating optical atomic clock, is equal to 362.53 nm [12].
On the other hand, the calculated ultraviolet zero-magic wavelengths can have limited usefulness in mixed-species experiments due to the single photon ionization barriers in other species. Before exploiting the optical traps one needs to measure the relevant photoionization cross-sections. Still, for instance it was shown that the cross section for the photoionization of Rb close to UVC is small enough to restrain the Rb magneto-optical trap losses in the case when the trap is spatially overlapped with Hg magneto-optical trap [25,69].
4. Conclusions
We have performed transition wavelengths and transition strengths calculations for ${}^{200}$Hg with the SOC-CASPT2 method. Based on the calculated spectroscopic data, magic and zero-magic wavelengths are identified. The analysis of robustness of identified wavelengths with respect to wavelength and imperfect polarization allows us to choose the most suitable wavelength for experimental realization. The choice of the ${}^{200}$Hg isotope for calculations does not limit the generality of the results. The wavelength difference between isotopes is much below the accuracy limit. For instance, the difference between 813.4 nm magic wavelength in $^{88}$Sr and $^{87}$Sr is in the order of 0.003 cm$^{-1}$ [70].
We should stress that the ANO-RCC basis set used in this work, although the highest quality, has been optimized for the ground state electronic structure of Hg and, as a consequence, might not be ideal for higher-lying excited states calculations. Moreover, the limitations on the number of active orbitals comprised in the CASSCF approach prohibits us from correlating virtual orbitals above the 7s sub-shell. Since the SOC-CASPT2-derived magic and zero-magic wavelengths have never been used to compare with experimental data, future possible measurements of magic wavelengths for $^1S_0 \leftrightarrow {^3P_{1,2} (m_J = 0)}$ transitions and zero-magic wavelengths for the ${^3P_{1,2} (m_J = 0)}$ states in Hg will serve as a benchmark of the method. Nonetheless, to the best of our knowledge, the SOC-CASPT2 calculations presented in this work account for multi-reference nature, dynamical correlation, as well as the spin-free (scalar) and spin-orbit relativistic effects present in Hg, and, thus, can be considered as the most reliable up to date quantum chemical description of electronic transitions in Hg.
Funding
The EMPIR Programme cofinanced by the Participating States and from the European Union’s Horizon 2020 Research and Innovation Programme (EMPIR 17FUN03 USOQS); H2020 Future and Emerging Technologies (No 820404, iqClock project); Narodowe Centrum Nauki (2017/25/B/ST2/00429, 2017/25/Z/ST2/03021, 2019/33/B/ST4/02114, Quantera Q-Clocks); Fundacja na rzecz Nauki Polskiej (Project TEAM/2017-4/42); Science Committee of the Republic of Armenia (project N 20TTAT-QTc004).
Acknowledgments
The UMK research was performed at the National Laboratory FAMO (KL FAMO) in Toruń, Poland, and was supported by a subsidy from the Polish Ministry of Science and Higher Education. P.T. acknowledges a scholarship for outstanding young scientists from the Ministry of Science and Higher Education.
Disclosures
The authors declare no conflicts of interest.
Supplemental document
See Supplement 1 for supporting content.
References
1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
2. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th decimal place,” Science 319(5871), 1808–1812 (2008). [CrossRef]
3. T. Ido and H. Katori, “Recoil-free spectroscopy of neutral Sr atoms in the lamb-dicke regime,” Phys. Rev. Lett. 91(5), 053001 (2003). [CrossRef]
4. R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At., Mol., Opt. Phys. 42, 95–170 (2000). [CrossRef]
5. H. Katori, M. Takamoto, V. G. Pal’chikov, and V. D. Ovsiannikov, “Ultrastable optical clock with neutral atoms in an engineered light shift trap,” Phys. Rev. Lett. 91(17), 173005 (2003). [CrossRef]
6. P. G. Westergaard, J. Lodewyck, L. Lorini, A. Lecallier, E. A. Burt, M. Zawada, J. Millo, and P. Lemonde, “Lattice-induced frequency shifts in Sr optical lattice clocks at the 10−17 level,” Phys. Rev. Lett. 106(21), 210801 (2011). [CrossRef]
7. H. Katori, V. D. Ovsiannikov, S. I. Marmo, and V. G. Palchikov, “Strategies for reducing the light shift in atomic clocks,” Phys. Rev. A 91(5), 052503 (2015). [CrossRef]
8. C. Shi, J.-L. Robyr, U. Eismann, M. Zawada, L. Lorini, R. Le Targat, and J. Lodewyck, “Polarizabilities of the 87Sr clock transition,” Phys. Rev. A 92(1), 012516 (2015). [CrossRef]
9. R. C. Brown, N. B. Phillips, K. Beloy, W. F. McGrew, M. Schioppo, R. J. Fasano, G. Milani, X. Zhang, N. Hinkley, H. Leopardi, T. H. Yoon, D. Nicolodi, T. M. Fortier, and A. D. Ludlow, “Hyperpolarizability and operational magic wavelength in an optical lattice clock,” Phys. Rev. Lett. 119(25), 253001 (2017). [CrossRef]
10. J. J. McFerran, L. Yi, S. Mejri, S. Di Manno, W. Zhang, J. Guéna, Y. Le Coq, and S. Bize, “Neutral atom frequency reference in the deep ultraviolet with fractional uncertainty =5.7 × 10−15,” Phys. Rev. Lett. 108(18), 183004 (2012). [CrossRef]
11. J. J. McFerran, D. V. M. aes, C. Mandache, J. Millo, W. Zhang, Y. L. Coq, G. Santarelli, and S. Bize, “Laser locking to the 199Hg 1S0 − 3P0 clock transition with $5.4 \times 10^{-15}/\sqrt {\tau}$ fractional frequency instability,” Opt. Lett. 37(17), 3477–3479 (2012). [CrossRef]
12. L. Yi, S. Mejri, J. J. McFerran, Y. Le Coq, and S. Bize, “Optical lattice trapping of 199Hg and determination of the magic wavelength for the ultraviolet 1S0 ↔ 3P0 clock transition,” Phys. Rev. Lett. 106(7), 073005 (2011). [CrossRef]
13. S. Mejri, J. J. McFerran, L. Yi, Y. Le Coq, and S. Bize, “Ultraviolet laser spectroscopy of neutral mercury in a one-dimensional optical lattice,” Phys. Rev. A 84(3), 032507 (2011). [CrossRef]
14. A. Ye and G. Wang, “Dipole polarizabilities of ns21S0 and nsnp3P0 states and relevant magic wavelengths of group-IIB atoms,” Phys. Rev. A 78(1), 014502 (2008). [CrossRef]
15. H. Hachisu, K. Miyagishi, S. G. Porsev, A. Derevianko, V. D. Ovsiannikov, V. G. Pal’chikov, M. Takamoto, and H. Katori, “Trapping of neutral mercury atoms and prospects for optical lattice clocks,” Phys. Rev. Lett. 100(5), 053001 (2008). [CrossRef]
16. M. C. Bigeon, “Probabilité de transition de la raie 61S0 − 63P0 du mercure,” J. Phys. 28(1), 51–60 (1967). [CrossRef]
17. S. G. Porsev, U. I. Safronova, and M. S. Safronova, “Theoretical study of the g factor and lifetime of the 6s6p3P0 state of mercury,” Phys. Rev. A 96(1), 012509 (2017). [CrossRef]
18. M. S. Safronova, S. G. Porsev, C. Sanner, and J. Ye, “Two clock transitions in neutral Yb for the highest sensitivity to variations of the fine-structure constant,” Phys. Rev. Lett. 120(17), 173001 (2018). [CrossRef]
19. P. Wcisło, P. Morzyński, M. Bober, A. Cygan, D. Lisak, R. Ciuryło, and M. Zawada, “Experimental constraint on dark matter detection with optical atomic clocks,” Nat. Astron. 1(1), 0009 (2017). [CrossRef]
20. P. Wcisło, P. Ablewski, K. Beloy, S. Bilicki, M. Bober, R. Brown, R. Fasano, R. Ciuryło, H. Hachisu, T. Ido, J. Lodewyck, A. Ludlow, W. McGrew, P. Morzyński, D. Nicolodi, M. Schioppo, M. Sekido, R. Le Targat, P. Wolf, X. Zhang, B. Zjawin, and M. Zawada, “New bounds on dark matter coupling from a global network of optical atomic clocks,” Sci. Adv. 4(12), eaau4869 (2018). [CrossRef]
21. B. M. Roberts, P. Delva, A. Al-Masoudi, A. Amy-Klein, C. Bærentsen, C. F. A. Baynham, E. Benkler, S. Bilicki, S. Bize, W. Bowden, J. Calvert, V. Cambier, E. Cantin, E. A. Curtis, S. Dörscher, M. Favier, F. Frank, P. Gill, R. M. Godun, G. Grosche, C. Guo, A. Hees, I. R. Hill, R. Hobson, N. Huntemann, J. Kronjäger, S. Koke, A. Kuhl, R. Lange, T. Legero, B. Lipphardt, C. Lisdat, J. Lodewyck, O. Lopez, H. S. Margolis, H. Álvarez-Martínez, F. Meynadier, F. Ozimek, E. Peik, P.-E. Pottie, N. Quintin, C. Sanner, L. D. Sarlo, M. Schioppo, R. Schwarz, A. Silva, U. Sterr, C. Tamm, R. L. Targat, P. Tuckey, G. Vallet, T. Waterholter, D. Xu, and P. Wolf, “Search for transient variations of the fine structure constant and dark matter using fiber-linked optical atomic clocks,” New J. Phys. 22(9), 093010 (2020). [CrossRef]
22. C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye, “Precision Metrology Meets Cosmology: Improved Constraints on Ultralight Dark Matter from Atom-Cavity Frequency Comparisons,” Phys. Rev. Lett. 125(20), 201302 (2020). [CrossRef]
23. R. H. Garstang, “Hyperfine structure and intercombination line intensities in the spectra of magnesium, zinc, cadmium, and mercury,” J. Opt. Soc. Am. 52(8), 845–851 (1962). [CrossRef]
24. C. Delaunay, R. Ozeri, G. Perez, and Y. Soreq, “Probing atomic higgs-like forces at the precision frontier,” Phys. Rev. D 96(9), 093001 (2017). [CrossRef]
25. M. Witkowski, R. Munoz-Rodriguez, A. Raczyński, J. Zaremba, B. Nagórny, P. S. Żuchowski, R. Ciuryło, and M. Zawada, “Photoionization cross sections of the 5S1/2 and 5P3/2 states of Rb in simultaneous magneto-optical trapping of Rb and Hg,” Phys. Rev. A 98(5), 053444 (2018). [CrossRef]
26. V. A. Yerokhin, R. A. Müller, A. Surzhykov, P. Micke, and P. O. Schmidt, “Nonlinear isotope-shift effects in Be-like, B-like, and C-like argon,” Phys. Rev. A 101(1), 012502 (2020). [CrossRef]
27. C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs, and M. Drewsen, “Improved isotope-shift-based bounds on bosons beyond the standard model through measurements of the 2D3/2 − 2D5/2 interval in Ca+,” Phys. Rev. Lett. 125(12), 123003 (2020). [CrossRef]
28. H. Miyake, N. C. Pisenti, P. K. Elgee, A. Sitaram, and G. K. Campbell, “Isotope-shift spectroscopy of the 1S0 → 3P1 and 1S0 → 3P0 transitions in strontium,” Phys. Rev. Res. 1(3), 033113 (2019). [CrossRef]
29. I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, “Evidence for nonlinear isotope shift in Yb+ search for new boson,” Phys. Rev. Lett. 125(12), 123002 (2020). [CrossRef]
30. V. A. Dzuba, V. V. Flambaum, and S. Schiller, “Testing physics beyond the standard model through additional clock transitions in neutral ytterbium,” Phys. Rev. A 98(2), 022501 (2018). [CrossRef]
31. M. S. Safronova, Z. Zuhrianda, U. I. Safronova, and C. W. Clark, “Extracting transition rates from zero-polarizability spectroscopy,” Phys. Rev. A 92(4), 040501 (2015). [CrossRef]
32. J. Mitroy, M. S. Safronova, and C. W. Clark, “Theory and applications of atomic and ionic polarizabilities,” J. Phys. B: At., Mol. Opt. Phys. 43(20), 202001 (2010). [CrossRef]
33. A. Derevianko and H. Katori, “Colloquium: Physics of optical lattice clocks,” Rev. Mod. Phys. 83(2), 331–347 (2011). [CrossRef]
34. S. G. Porsev, K. Beloy, and A. Derevianko, “Precision determination of electroweak coupling from atomic parity violation and implications for particle physics,” Phys. Rev. Lett. 102(18), 181601 (2009). [CrossRef]
35. A. A. Vasilyev, I. M. Savukov, M. S. Safronova, and H. G. Berry, “Measurement of the 6s − 7p transition probabilities in atomic cesium and a revised value for the weak charge QW,” Phys. Rev. A 66(2), 020101 (2002). [CrossRef]
36. M. S. Safronova, U. I. Safronova, and C. W. Clark, “Magic wavelengths for optical cooling and trapping of lithium,” Phys. Rev. A 86(4), 042505 (2012). [CrossRef]
37. M. Saffman and T. G. Walker, “Analysis of a quantum logic device based on dipole-dipole interactions of optically trapped Rydberg atoms,” Phys. Rev. A 72(2), 022347 (2005). [CrossRef]
38. A. V. Gorshkov, A. M. Rey, A. J. Daley, M. M. Boyd, J. Ye, P. Zoller, and M. D. Lukin, “Alkaline-earth-metal atoms as few-qubit quantum registers,” Phys. Rev. Lett. 102(11), 110503 (2009). [CrossRef]
39. H. P. Summers, N. R. Badnell, M. G. O’Mullane, A. D. Whiteford, R. Bingham, B. J. Kellett, J. Lang, K. H. Behringer, U. Fantz, K.-D. Zastrow, S. D. Loch, M. S. Pindzola, D. C. Griffin, and C. P. Ballance, “Atomic data for modelling fusion and astrophysical plasmas,” Plasma Phys. Controlled Fusion 44(12B), B323–B338 (2002). [CrossRef]
40. D. Verner, E. Verner, and G. Ferland, “Atomic data for permitted resonance lines of atoms and ions from H to Si, and S, Ar, Ca, and Fe,” At. Data Nucl. Data Tables 64(1), 1–180 (1996). [CrossRef]
41. J. Pickering, R. Blackwell-Whitehead, A. Thorne, M. Ruffoni, and C. Holmes, “Laboratory measurements of oscillator strengths and their astrophysical applications1this review is part of a special issue on the 10th international colloquium on atomic spectra and oscillator strengths for astrophysical and laboratory plasmas,” Can. J. Phys. 89(4), 387–393 (2011). [CrossRef]
42. J. Catani, G. Barontini, G. Lamporesi, F. Rabatti, G. Thalhammer, F. Minardi, S. Stringari, and M. Inguscio, “Entropy exchange in a mixture of ultracold atoms,” Phys. Rev. Lett. 103(14), 140401 (2009). [CrossRef]
43. B. Gadway, D. Pertot, J. Reeves, and D. Schneble, “Probing an ultracold-atom crystal with matter waves,” Nat. Phys. 8(7), 544–549 (2012). [CrossRef]
44. L. J. LeBlanc and J. H. Thywissen, “Species-specific optical lattices,” Phys. Rev. A 75(5), 053612 (2007). [CrossRef]
45. B. Arora, M. S. Safronova, and C. W. Clark, “Tune-out wavelengths of alkali-metal atoms and their applications,” Phys. Rev. A 84(4), 043401 (2011). [CrossRef]
46. D. C. McKay, C. Meldgin, D. Chen, and B. DeMarco, “Slow thermalization between a lattice and free bose gas,” Phys. Rev. Lett. 111(6), 063002 (2013). [CrossRef]
47. T. Topcu and A. Derevianko, “Tune-out wavelengths and landscape-modulated polarizabilities of alkali-metal rydberg atoms in infrared optical lattices,” Phys. Rev. A 88(5), 053406 (2013). [CrossRef]
48. V. D. Vaidya, J. Tiamsuphat, S. L. Rolston, and J. V. Porto, “Degenerate bose-fermi mixtures of rubidium and ytterbium,” Phys. Rev. A 92(4), 043604 (2015). [CrossRef]
49. T. Takano, R. Mizushima, and H. Katori, “Precise determination of the isotope shift of 88Sr-87Sr optical lattice clock by sharing perturbations,” Appl. Phys. Express 10(7), 072801 (2017). [CrossRef]
50. X. Zhou, X. Xu, X. Chen, and J. Chen, “Magic wavelengths for terahertz clock transitions,” Phys. Rev. A 81(1), 012115 (2010). [CrossRef]
51. T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, 2000).
52. B. O. Roos, P. R. Taylor, and P. E. M. Siegbahn, “A complete active space SCF method-(CASSCF) using a density matrix formulated super-CI approach,” Chem. Phys. 48(2), 157–173 (1980). [CrossRef]
53. P. E. M. Siegbahn, J. Almlöf, A. Heiberg, and B. O. Roos, “The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule,” J. Chem. Phys. 74(4), 2384–2396 (1981). [CrossRef]
54. K. Andersson, P.-Å. Malmqvist, B. O. Roos, A. J. Sadlej, and K. Woliński, “Second-order perturbation theory with a CASSCF reference function,” J. Chem. Phys. 94(14), 5483–5488 (1990). [CrossRef]
55. K. Andersson, P.-Å. Malmqvist, and B. O. Roos, “Second-order perturbation theory with a complete active space self-consistent field reference function,” J. Chem. Phys. 96(2), 1218–1226 (1992). [CrossRef]
56. J. Finley, P.-Å. Malmqvist, B. O. Roos, and L. Serrano-Andrés, “The multi-state CASPT2 method,” Chem. Phys. Lett. 288(2-4), 299–306 (1998). [CrossRef]
57. I. Fdez. Galván, M. Vacher, A. Alavi, C. Angeli, F. Aquilante, J. Autschbach, J. J. Bao, S. I. Bokarev, N. A. Bogdanov, R. K. Carlson, et al., “Openmolcas: From source code to insight,” J. Chem. Theory Comput. 15(11), 5925–5964 (2019). [CrossRef]
58. (2019). OpenMolcas https://gitlab.com/Molcas/OpenMolcas.
59. N. Douglas and N. M. Kroll, “Quantum electrodynamical corrections to fine-structure of helium,” Ann. Phys. 82(1), 89–155 (1974). [CrossRef]
60. B. A. Hess, “Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators,” Phys. Rev. A 33(6), 3742–3748 (1986). [CrossRef]
61. M. Reiher, “Relativistic Douglas–Kroll–Hess theory,” Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2(1), 139–149 (2012). [CrossRef]
62. P.-Å. Malmqvist, B. O. Roos, and B. Schimmelpfennig, “The restricted active space (RAS) state interaction approach with spin–orbit coupling,” Chem. Phys. Lett. 357(3-4), 230–240 (2002). [CrossRef]
63. A. Gogyan, P. Tecmer, and M. Zawada, “The SOC-CASPT2 calculations of the Hg atom,” (2021). RepOD http://doi.org/10.18150/7OPCLD.
64. S. Roy, R. Prasad, and P. Chandra, “Electron correlation and relativistic effects in atomic structure calculations of the thorium atom,” J. Chem. Phys. 134(23), 234302 (2011). [CrossRef]
65. P. Tecmer, K. Boguslawski, M. Borkowski, P. S. Żuchowski, and D. Kędziera, “Modeling the electronic structures of the ground and excited states of the ytterbium atom and the ytterbium dimer: A modern quantum chemistry perspective,” Int. J. Quantum Chem. 119(18), e25983 (2019). [CrossRef]
66. M. Barysz and Ł. Syrocki, “Theoretical interpretation of photoelectron spectra of the iridium neutral atom and anion,” J. Quant. Spectrosc. Radiat. Transfer 255, 107278 (2020). [CrossRef]
67. A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.7.1), [Online]. Available: https://physics.nist.gov/asd [2020, June 19]. National Institute of Standards and Technology, Gaithersburg, MD. (2019).
68. C. J. Sansonetti, M. L. Salit, and J. Reader, “Wavelengths of spectral lines in mercury pencil lamps,” Appl. Opt. 35(1), 74–77 (1996). [CrossRef]
69. M. Witkowski, B. Nagórny, R. Munoz-Rodriguez, R. Ciuryło, P. S. Żuchowski, S. Bilicki, M. Piotrowski, P. Morzyński, and M. Zawada, “Dual hg-rb magneto-optical trap,” Opt. Express 25(4), 3165–3179 (2017). [CrossRef]
70. T. Akatsuka, M. Takamoto, and H. Katori, “Three-dimensional optical lattice clock with bosonic 88Sr atoms,” Phys. Rev. A 81(2), 023402 (2010). [CrossRef]
