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arXiv:1804.01218 · Physical Review A

Relativistic full-configuration-interaction calculations of magic wavelengths for the 2 3S1→2 1S0 transition of helium isotopes

Fang-Fei Wu, Sanjiang Yang, Yong-Hui Zhang, Jun-Yi Zhang, Haoxue Qiao, Ting-Yun Shi, Li-Yan Tang

Published Oct 5, 2018• 17 citations
arXivHTMLPDF

Abstract

A large-scale full-configuration-interaction calculation based on the Dirac-Coulomb-Breit (DCB) Hamiltonian is performed for the $2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ and $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}$ 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 $n$ (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 $^{4}\mathrm{He}$ 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 $^{3}\mathrm{He}$ would provide theoretical support for experimentally designing an optical dipole trap to precisely determine the nuclear charge radius of helium in the future.

DOI
10.1103/physreva.98.040501

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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.

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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 23S1\ensuremath→21S02{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}23S1​\ensuremath→21S0​ 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.
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.
A series of magic wavelengths for the 23S1 → 23PJ (J = 0, 1, 2) transitions in helium is determined with high precision using the relativistic configuration interaction combined with the radiative potential method. The calculation explicitly incorporates relativistic, finite nuclear mass, and quantum electrodynamic corrections through the inclusion of mass shift and radiative potential operators into the Dirac-Coulomb-Breit Hamiltonian. The resulting magic wavelengths lie entirely within the visible spectrum (400-700 nm) and converge to seven significant digits, making them directly accessible for experimental verification. These accurate predictions serve as benchmarks for testing atomic structure theory and support the development of state-insensitive optical trapping schemes in precision spectroscopy.
Motivated by recent direct measurement of the forbidden $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ transition in helium [Thomas et al., Phys. Rev. Lett. 125, 013002 (2020)], where the ac Stark shift is one of the main systematic uncertainties, we propose a dichroic two-photon transition measurement for $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$, which could effectively suppress the ac Stark shift by utilizing magic wavelengths: one magic wavelength is used to realize state-insensitive optical trapping; the other magic wavelength is used as one of the two lasers driving the two-photon transition. Carrying out calculations based on the no-pair Dirac-Coulomb-Breit Hamiltonian with mass shift operator included, we report the magic wavelength of 1265.615 9(4) nm for 4He^{4}\mathrm{He}4He [or 1265.683 9(2) nm for 3He]^{3}\mathrm{He}]3He] can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for 4He^{4}\mathrm{He}4He [or 934.255 4(4) nm for 3He]^{3}\mathrm{He}]3He] can be as one excitation laser in the two-photon process and the ac Stark shift can be reduced to less than 100 kHz, as long as the intensity of the other excitation laser does not exceed $1\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{3.33333pt}{0ex}}{\text{W/cm}}^{2}$. Alternatively, by selecting detuning frequencies relative to the 23P2\phantom{\rule{0.16em}{0ex}}^{3}P23P state in the region of 82--103 THz, as well as adjusting the intensity ratios of the two lasers, the ac Stark shift in the $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ two-photon transition can be canceled.
We have calculated ac polarizabilities of the 23S2{\phantom{\rule{4pt}{0ex}}}^{3}S23S and 21S2{\phantom{\rule{4pt}{0ex}}}^{1}S21S states of both 4He^{4}\mathrm{He}4He and 3He^{3}\mathrm{He}3He in the range 318 nm to 2.5 $\ensuremath{\mu}\mathrm{m}$ and determined the magic wavelengths at which these polarizabilities are equal for either isotope. The calculations, only based on available ab initio tables of level energies and Einstein A coefficients, do not require advanced theoretical techniques. The polarizability contribution of the continuum is calculated using a simple extrapolation beyond the ionization limit, yet the results agree to better than 11%1 with such advanced techniques. Several promising magic wavelengths are identified around 320 nm with sufficient accuracy to design an appropriate laser system. The extension of the calculations to 3He^{3}\mathrm{He}3He is complicated due to the additional hyperfine structure, but we show that the magic wavelength candidates around 320 nm are predominantly shifted by the isotope shift.
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 $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.
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 $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}\phantom{\rule{0.28em}{0ex}}({M}_{J}=\ifmmode\pm\else\textpm\fi{}1)$ 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 23S12{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}23S1​ state might provide a test of atomic structure theory.