We calculate and measure the magic wavelengths for the $6{s}^{2}^{1}S_{0}\ensuremath{-}6s6p\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ intercombination transition of the neutral ytterbium atom. The calculation is performed with the ab initio configuration interaction
+ all-order method. The measurement is done with laser spectroscopy on cold atoms in an optical dipole trap. The magic wavelengths are determined to be 1035.68(4) nm for the $\ensuremath{\pi}$ transition $(\mathrm{\ensuremath{\Delta}}m=0)$ and 1036.12(3) nm for the $\ensuremath{\sigma}$ transitions $(|\mathrm{\ensuremath{\Delta}}m|=1)$ in agreement with the calculated values. Laser cooling on the narrow intercombination transition could achieve better results for atoms in an optical dipole trap when the trap wavelength is tuned to near the magic wavelength.
We carry out relativistic many-body calculations of the static and dynamic dipole polarizabilities of the ground 6s 2 1 S 0 and the first excited 6s6p 3 P o 0 states of Yb. With these polarizabilities, we compute several properties of Yb relevant to optical lattice clocks operating on the 6s 2 1 S 0 –6s6p 3 P o 0 transition. We determine (i) the first four magic wavelengths of the laser field for which the frequency of the clock transition is insensitive to the laser intensity. While the first magic wavelength is known, we predict the second, the third and the fourth magic wavelengths to be 551 nm, 465 nm and 413 nm. (ii) We re-evaluate the effect of black-body radiation on the frequency of the clock transition, the resulting clock shift at T = 300 K being −1.41(17) Hz. (iii) We compute long-range interatomic van der Waals coefficients (in a.u.) C 6 (6s 2 1 S 0 + 6s 2 1 S 0 ) = 1909(160), C 6 (6s 2 1 S 0 + 6s6p 3 P 0 ) = 2709(338) and C 6 (6s6p 3 P 0 + 6s6p 3 P 0 ) = 3886(360). Finally, we determine the atom-wall interaction coefficients (in a.u.), C 3 (6s 2 1 S 0 ) = 3.34 and C 3 (6s6p 3 P 0 ) = 3.68. We also address and resolve a disagreement between previous calculations of the static polarizability of the ground state.
The wave functions, energy levels and matrix elements of Yb+ ions are calculated using the relativistic configuration interaction plus core polarization (RCICP) method. The static and dynamic electric dipole polarizabilities of the ground state and low-lying excited states are determined. Then, the magic wavelengths of the magnetic sublevel 6s1/2,m = 1/2 → 5d3/2,m = ±3/2,±1/2 and 6s1/2,m = 1/2 → 5d5/2,m = ±5/2,±3/2,±1/2 transitions in the linearly, right-handed, and left-handed polarized light are further determined. The dependence of the magic wavelengths upon the angle between the direction of magnetic field and the direction of laser polarization is analyzed.
The static and dynamic electric dipole polarizabilities of the
5s2 1S0 and
5s5p 3P0,2 states of Sr atoms are calculated using the relativistic configuration interaction plus the many-body perturbation theory (RCI+MBPT) method. Magic wavelengths are determined for the transitions
5s2 1S0→5s5p 3P0,
5s2 1S0→5s5p 3P2, and
5s5p 3P0→5s5p 3P2. A comprehensive study is conducted on the dependence of magic wavelengths on the angle between the laser polarization and the magnetic field. Furthermore, the conditions for realizing triple magic trapping at 813.4~nm for the
5s2 1S0,
5s5p 3P0 and
5s5p 3P2 states are investigated. In the case of linearly polarized light, when the angle (
θp) between the laser polarization direction and the magnetic field is
79.1(0.7)∘, triple magic trapping for the
5s2 1S0,
5s5p 3P0, and
5s5p 3P2 M=0 states can be achieved. This result agrees well with the recent experimental measurement (78.49(3)
∘)[Phys. Rev. Lett. 135, 143401 (2025)]. Meanwhile, triple magic trapping involving the
5s5p 3P2 M=2 state can be achieved when
θp=37.4(0.3)∘. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.
Magic wavelengths (${\ensuremath{\lambda}}_{\mathrm{magic}}$) for the $7{S}_{1/2}\ensuremath{-}7{P}_{1/2,3/2}$ transitions (D lines) in Fr were reported by Dammalapati et al. [U. Dammalapati, K. Harada, and Y. Sakemi, Phys. Rev. A 93, 043407 (2016)]. These ${\ensuremath{\lambda}}_{\mathrm{magic}}$ were determined by plotting dynamic polarizabilities ($\ensuremath{\alpha}$) of the involved states with the above transitions against a desired range of wavelengths. Electric dipole (E1) matrix elements listed in [J. E. Sansonetti, J. Phys. Chem. Ref. Data 36, 497 (2007)], from the measured lifetimes of the
7P1/2,3/2 states and from the calculations considering core-polarization effects in the relativistic Hartree-Fock (HFR) method, were used to determine $\ensuremath{\alpha}$. However, contributions from core correlation effects and from the E1 matrix elements of the $7P\ensuremath{-}7S, 7P\ensuremath{-}8S$, and $7P\ensuremath{-}6D$ transitions to $\ensuremath{\alpha}$ of the
7P states were ignored. In this work, we demonstrate importance of these contributions and improve accuracies of $\ensuremath{\alpha}$ further by replacing the E1 matrix elements taken from the HFR method by the values obtained employing relativistic coupled-cluster theory. Our static $\ensuremath{\alpha}$ are found to be in excellent agreement with the other available theoretical results, whereas substituting the E1 matrix elements used by Dammalapati et al. gives very small $\ensuremath{\alpha}$ values for the
7P states. Owing to this, we find disagreement in ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported by Dammalapati et al. for linearly polarized light, especially at wavelengths close to the D lines and in the infrared region. As a consequence, a ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported at 797.75 nm which was seen supporting a blue detuned trap in their work is now estimated at 771.03 nm and is supporting a red detuned trap. Also, none of our results match with the earlier results for circularly polarized light. Moreover, our static values of $\ensuremath{\alpha}$ will be very useful for guiding experiments to carry out their measurements.