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Physical Review A

Tune-out wavelengths for the 2 3S1 state of the Li+ ion

Fang-Fei Wu, Ke Deng, Z. H. Lu

Published Oct 25, 2022• 7 citations
Publisher

Abstract

Configuration interaction calculations are performed to determine the dipole polarizabilities and tune-out wavelengths for the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state in ${\mathrm{Li}}^{+}$ ion. The finite nuclear mass correction is given by including the mass shift operator in the Dirac-Coulomb-Breit Hamiltonian directly, while the QED correction to dynamic dipole polarizabilities is evaluated by using the perturbation theory. The tune-out wavelengths for the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=0)$ and $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=\ifmmode\pm\else\textpm\fi{}1)$ states are 159.620 55(6) nm and 159.631 14(6) nm, respectively. And the QED correction for 159 nm tune-out wavelength is 43 ppm. In addition, we obtain the static dipole polarizability for the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=\ifmmode\pm\else\textpm\fi{}1)$ state to be 46.857 03(4) a.u., in which the contribution from QED is 70 ppm. The 159 nm tune-out wavelength and the static dipole polarizability of $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state might provide a test of atomic structure theory.

DOI
10.1103/physreva.106.042816

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Configuration interaction calculations are performed to determine the dipole polarizabilities and tune-out wavelengths for the 23S12{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}23S1​ state in Li+{\mathrm{Li}}^{+}Li+ ion. The finite nuclear mass correction is given by including the mass shift operator in the Dirac-Coulomb-Breit Hamiltonian directly, while the QED correction to dynamic dipole polarizabilities is evaluated by using the perturbation theory. The tune-out wavelengths for the 23S1(MJ=0)2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=0)23S1​(MJ​=0) and 23S1(MJ=\ifmmode±\else\textpm\fi1)2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=\ifmmode\pm\else\textpm\fi{}1)23S1​(MJ​=\ifmmode±\else\textpm\fi1) states are 159.620 55(6) nm and 159.631 14(6) nm, respectively. And the QED correction for 159 nm tune-out wavelength is 43 ppm. In addition, we obtain the static dipole polarizability for the 23S1(MJ=\ifmmode±\else\textpm\fi1)2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=\ifmmode\pm\else\textpm\fi{}1)23S1​(MJ​=\ifmmode±\else\textpm\fi1) state to be 46.857 03(4) a.u., in which the contribution from QED is 70 ppm. The 159 nm tune-out wavelength and the static dipole polarizability of 23S12{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}23S1​ state might provide a test of atomic structure theory.
The electric dipole polarizabilities and the tune-out wavelengths for the n3S1 (n = 2, 3) states of Be2+ are determined through the application of the relativistic full-configuration-interaction approach. Our calculations directly integrate the mass shift operator into the Dirac-Coulomb-Breit Hamiltonian and further assess the quantum electrodynamics (QED) correction to the dynamic dipole polarizabilities using perturbation theory. The results reveal that the static electric dipole polarizability of the 23S1 and 33S1 states, as well as the 93 nm tune-out wavelength of the 23S1 state and the 238 nm tune-out wavelength of the 33S1 state, exhibits a high sensitivity to QED correction, which exceeds 80 ppm, providing a sensitive test for atomic structure theory.
The B-spline configuration-interaction method is applied to the investigations of dynamic dipole polarizabilities for the four lowest triplet states (2 3 S, 3 3 S, 2 3 P, and 3 3 P) of the Li + ion. The accurate energies for the triplet states of n 3 S, n 3 P, and n 3 D, the dipole oscillator strengths for 2 3 S(3 3 S) → n 3 P, 2 3 P(3 3 P) → n 3 S, and 2 3 P(3 3 P) → n 3 D transitions, with the main quantum number n up to 10 are tabulated for references. The dynamic dipole polarizabilities for the four triplet states under a wide range of photon energy are also listed, which provide input data for analyzing the Stark shift of the Li + ion. Furthermore, the tune-out wavelengths in the range from 100 nm to 1.2 μm for the four triplet states, and the magic wavelengths in the range from 100 nm to 600 nm for the 2 3 S → 3 3 S, 2 3 S → 2 3 P, and 2 3 S → 3 3 P transitions are determined accurately for the experimental design of the Li + ion.
A large-scale full-configuration-interaction calculation based on the Dirac-Coulomb-Breit (DCB) Hamiltonian is performed for the 21S02{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}21S0​ and 23S12{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}23S1​ states of helium. The operators of the normal and specific mass shifts are directly included in the DCB framework to take the finite nuclear mass correction into account. High-accuracy energies and matrix elements involved nnn (the main quantum number) up to 13 are obtained from one diagonalization of Hamiltonian. The dynamic dipole polarizabilities are calculated by using the sum rule of intermediate states, and a series of magic wavelengths with QED and hyperfine effects included for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ transition of helium are identified. In addition, the high-order ac Stark shift determined by the dynamic hyperpolarizabilities at the magic wavelengths are also evaluated. Since the most promising magic wavelength for application in experiment is 319.8 nm, the high-accuracy magic wavelength of 319.815 3(6) nm of 4He^{4}\mathrm{He}4He is in good agreement with the recent measurement value of 319.815 92(15) nm [R. J. Rengelink et al., arXiv:1804.06693], and the present magic wavelength of 319.830 2(7) nm for 3He^{3}\mathrm{He}3He would provide theoretical support for experimentally designing an optical dipole trap to precisely determine the nuclear charge radius of helium in the future.
The dynamic dipole polarizabilities for the four lowest triplet states (23S,33S,23P2{\phantom{\rule{0.16em}{0ex}}}^{3}S, 3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}P23S,33S,23P, and 33P3{\phantom{\rule{0.16em}{0ex}}}^{3}P33P) of helium are calculated using the BBB-spline configuration interaction method. Present values of the static dipole polarizabilities in the length, velocity, and acceleration gauges are in good agreement with the best Hylleraas results. Also the tune-out wavelengths in the range 400 nm--4.2 $\ensuremath{\mu}\mathrm{m}$ for the four lowest triplet states are identified, and the magic wavelengths in the range 460 nm--3.5 $\ensuremath{\mu}\mathrm{m}$ for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transitions are determined. We show that the tune-out wavelength of 23S2{\phantom{\rule{0.16em}{0ex}}}^{3}S23S state is 413.03828(3)413.038\phantom{\rule{0.16em}{0ex}}28(3)413.03828(3) nm, which corroborates the value of Mitroy and Tang [Phys. Rev. A 88, 052515 (2013)], and the magic wavelength around 1066 nm for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transition can be expected for precision measurement to determine the ratio of transition matrix elements $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.
The dynamic dipole polarizabilities for the 3s3s3s state of the Li atom are computed using Hylleraas and semiempirical approaches. This enables the calculation of the 7{}^{7}7Li($2s\ensuremath{\rightarrow}3s$) Stark shift at photon wavelengths of 610.5 nm and 735.1 nm. They are calculated to be 834.7(5) and $\ensuremath{-}686.7(25)$ a03{a}_{0}^{3}a03​, respectively. The first two tune-out frequencies for the 7{}^{7}7Li(2s2s2s) ground state are determined to be 670.971626(1)670.971\phantom{\rule{0.16em}{0ex}}626(1)670.971626(1) nm and 324.192(2)324.192(2)324.192(2) nm, respectively.
A series of magic wavelengths for the forbidden transition of $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ in helium are determined by performing large-scale full relativistic configuration interaction calculations based on Dirac-Coulomb-Breit Hamiltonian with the mass shift operator included. QED corrections to magic wavelengths are taken into account by using the perturbation theory in nonrelativistic configuration interaction calculations. The 1335.55(2)-nm magic wavelength for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}(M=\ifmmode\pm\else\textpm\fi{}1)\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}(M=0)$ transition is found to be sensitive to the finite nuclear mass, relativistic, and QED effects, and its measurement might provide a test of atomic structure theory.