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Physical Review A

Magic wavelengths for the helium 23S1→21P1 forbidden transition

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

Published Mar 9, 2021• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.103.032810

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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 $7{P}_{1/2,3/2}$ 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.

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A series of magic wavelengths for the forbidden transition of 23S1\ensuremath→21P12{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}23S1​\ensuremath→21P1​ 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 23S1(M=\ifmmode±\else\textpm\fi1)\ensuremath→21P1(M=0)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)23S1​(M=\ifmmode±\else\textpm\fi1)\ensuremath→21P1​(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.
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.
The six longest tune-out wavelengths for the He(1s2s3S1e1s2s{\phantom{\rule{4pt}{0ex}}}^{3}{S}_{1}^{e}1s2s3S1e​) metastable state are determined by explicit calculation. The tune-out wavelength at 413.02 nm is expected to be sensitive to finite mass, relativistic, and quantum electrodynamic effects upon the transition matrix elements and its measurement would provide a nonenergy test of fundamental atomic structure theory.
Predicting magic wavelengths accurately requires precise knowledge of electric dipole matrix elements of nearby atomic transitions. As a result, measurements of magic wavelengths allow us to test theoretical predictions for the matrix elements that frequently cannot be probed by any other methods. Here, we calculate and measure a magic wavelength near 473 nm of the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}{S}_{0}\ensuremath{-}5s5p\phantom{\rule{0.16em}{0ex}}^{3}{P}_{1}$ intercombination transition of 88Sr^{88}\mathrm{Sr}88Sr. Experimentally, we find 473.371(6) nm for $\mathrm{\ensuremath{\Delta}}m=0$ ($\ensuremath{\pi}$ transition) and 473.117(15) nm for $\mathrm{\ensuremath{\Delta}}m=\ensuremath{-}1$ (${\ensuremath{\sigma}}^{\ensuremath{-}}$ transition). Theoretical calculations yield 473.375(22) and 473.145(20) nm, respectively. Determining magic wavelengths nearby the dominant 461-nm probe transition of 88Sr^{88}\mathrm{Sr}88Sr holds promise for state insensitive interfacing of strontium atoms and nanophotonic devices with a narrow band of operating wavelengths. Furthermore, the 3P1^{3}{P}_{1}3P1​ polarizability is dominated by the contributions to the 5p23P5{p}^{2}\phantom{\rule{0.16em}{0ex}}^{3}P5p23P levels, and excellent agreement of theory and experiment validates both theoretical values of these matrix elements and estimates of their uncertainties.
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 7P1/2,3/27{P}_{1/2,3/2}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 7P7P7P 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 7P7P7P 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.