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arXiv:2106.15149 · Physical Review A

Proposal for suppressing the ac Stark shift in the He(2 3S1→ 3 3S1 ) two-photon transition using magic wavelengths

Yong-Hui Zhang, Li-Yan Tang, Ting-Yun Shi

Published Oct 25, 2021• 4 citations
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Abstract

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 $^{4}\mathrm{He}$ [or 1265.683 9(2) nm for $^{3}\mathrm{He}]$ can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for $^{4}\mathrm{He}$ [or 934.255 4(4) nm for $^{3}\mathrm{He}]$ 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 $2\phantom{\rule{0.16em}{0ex}}^{3}P$ 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.

DOI
10.1103/physreva.104.042817

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Motivated by recent direct measurement of the forbidden 23S1\ensuremath→33S12\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}23S1​\ensuremath→33S1​ 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 23S1\ensuremath→33S12\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}23S1​\ensuremath→33S1​, 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×\else\texttimes\fi104W/cm21\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{3.33333pt}{0ex}}{\text{W/cm}}^{2}1\ifmmode×\else\texttimes\fi104W/cm2. 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 23S1\ensuremath→33S12\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}23S1​\ensuremath→33S1​ two-photon transition can be canceled.
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 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.
Recently, we studied the magic wavelength for the atomic hydrogen $1S\text{\ensuremath{-}}2S$ transition [A. Kawasaki, Phys. Rev. A 92, 042507 (2015)]. An explicit summation over virtual atomic states of the discrete part of the hydrogen spectrum was performed to evaluate the atomic polarizability. In this paper, we supplement the contribution of the continuum part of the spectrum and add the reduced-mass correction. The magic wavelength, at which the lowest-order ac Stark shifts of the 1S1S1S and 2S2S2S states are equal, is found to be 514.6 nm. The ac Stark shift at the magic wavelength is $\ensuremath{-}221.6\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/(\mathrm{kW}/{\mathrm{cm}}^{2})$, and the slope of the ac Stark shift at the magic wavelength under a change of the driving laser frequency is $\ensuremath{-}0.2157\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/[\mathrm{GHz}\phantom{\rule{0.28em}{0ex}}(\mathrm{kW}/{\mathrm{cm}}^{2})]$.
Recently, we studied the magic wavelength for the atomic hydrogen 1S−2S transition [A. Kawasaki, Phys. Rev. A 92, 042507 (2015)]. An explicit summation over virtual atomic states of the discrete part of the hydrogen spectrum was performed to evaluate the atomic polarizability. In this paper, we supplement the contribution of the continuum part of the spectrum and add the reduced-mass correction. The magic wavelength, at which the lowest-order ac Stark shifts of the 1S and 2S states are equal, is found to be 514.6 nm. The ac Stark shift at the magic wavelength is −221.6Hz/(kW/cm[superscript 2]), and the slope of the ac Stark shift at the magic wavelength under a change of the driving laser frequency is −0.2157Hz/[GHz(kW/cm[superscipt 2])].
The magic wavelength for an optical lattice for hydrogen atoms that cancels the lowest order ac Stark shift of the $1S\text{\ensuremath{-}}2S$ transition is calculated to be 513 nm. The magnitudes of the ac Stark shift $\mathrm{\ensuremath{\Delta}}E=\ensuremath{-}119$ Hz/(kW/cm2)\text{Hz}/(\text{kW}/{\mathrm{cm}}^{2})Hz/(kW/cm2) and the slope $d\mathrm{\ensuremath{\Delta}}E/d\ensuremath{\nu}=\ensuremath{-}2.77$ Hz/(GHzkW/cm2)\text{Hz}/(\text{GHz} \text{kW}/{\mathrm{cm}}^{2})Hz/(GHzkW/cm2) at the magic wavelength suggest that a stable and narrow-line-width trapping laser is necessary to achieve a deep enough optical lattice to confine hydrogen atoms in a way that gives a small enough light shift for the precision spectroscopy of the $1S\text{\ensuremath{-}}2S$ transition.
Using first-principles calculations, we identify magic wavelengths ${\ensuremath{\lambda}}_{\mathrm{magic}}$ for the 2s2s2s-2p2p2p and 2s2s2s-3p3p3p transitions in lithium. The nsnsns and npnpnp atomic levels have the same ac Stark shifts at the corresponding magic wavelength, which facilitates state-insensitive optical cooling and trapping. Tune-out wavelengths for which the ground-state frequency-dependent polarizability vanishes are also calculated. Differences of these wavelengths between 6{}^{6}6Li and 7{}^{7}7Li are reported. Our approach uses high-precision, relativistic all-order methods in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of Li electric-dipole matrix elements. Static polarizabilities for the 2s2s2s, 2p2p2p, 3s3s3s, 3p3p3p, and 3d3d3d levels are compared with other theory and experiment where available. Uncertainties of all recommended values are estimated. The magic wavelengths for the uv 2s2s2s-3p3p3p transition are of particular interest for the production of a quantum gas of lithium [Duarte et al., Phys. Rev. A 84, 061406R (2011)].