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arXiv:1410.3242 · Physical Review A

Magic wavelengths for the 23S→21S transition in helium

Remy Notermans, Robert J. Rengelink, K. A. H. van Leeuwen, W. Vassen

Published Nov 13, 2014• 25 citations
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Abstract

We have calculated ac polarizabilities of the $2{\phantom{\rule{4pt}{0ex}}}^{3}S$ and $2{\phantom{\rule{4pt}{0ex}}}^{1}S$ states of both $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ 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 $1%$ 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 $^{3}\mathrm{He}$ 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.

DOI
10.1103/physreva.90.052508

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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 \ensuremathμm\ensuremath{\mu}\mathrm{m}\ensuremathμm 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.
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 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 $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.
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.
The frequency dependent polarizabilities of the francium atom are calculated from the available data of energy levels and transition rates. Magic wavelengths for the state insensitive optical dipole trapping are identified from the calculated light shifts of the 7s7s7s 2S1/2^{2}{S}_{1/2}2S1/2​, 7p7p7p 2P1/2,3/2^{2}{P}_{1/2,3/2}2P1/2,3/2​, and 8s8s8s 2S1/2^{2}{S}_{1/2}2S1/2​ levels of the 7s7s7s $^{2}{S}_{1/2}\ensuremath{-}7p$ 2P1/2,3/2^{2}{P}_{1/2,3/2}2P1/2,3/2​ and 7s7s7s $^{2}{S}_{1/2}\ensuremath{-}8s$ 2S1/2^{2}{S}_{1/2}2S1/2​ transitions, respectively. Wavelengths in the ultraviolet, visible, and near infrared region are identified that are suitable for cooling and trapping. Magic wavelengths between the 600--700 nm and 700--1000 nm regions, which are blue and red detuned with the $7s\ensuremath{-}7p$ and $7s\ensuremath{-}8s$ transitions, are feasible to implement as lasers with sufficient power are available. In addition, we calculated the tune-out wavelengths where the ac polarizability of the ground 7s7s7s 2S1/2^{2}{S}_{1/2}2S1/2​ state in francium is zero. These results are beneficial as laser cooled and trapped francium has been in use for fundamental symmetry investigations like searches for an electron permanent electric dipole moment in an atom and for atomic parity nonconservation.