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Chinese Physics Letters

Magic Wavelengths for the 1 S —2 S and 1 S —3 S Transitions in Hydrogen Atoms

Dong Yin, Yong-Hui Zhang, Chengbin Li, Xianzhou Zhang

Published Jul 1, 2016• 3 citations
Publisher

Abstract

The dynamic dipole polarizabilities for 1S, 2S and 3S states of the hydrogen atom are calculated using the finite B-spline basis set method, and the magic wavelengths for 1S—2S and 1S—3S transitions are identified. In comparison of the solutions from the Schrödinger and Dirac equations, the relativistic corrections on the magic wavelengths are of the order of 10 −2 nm. The laser intensities for a 300-E r -deep optical trap and the heating rates at 514 and 1371 nm are estimated. The reliable prediction of the magic wavelengths would be helpful for the experimental design on the optical trapping of the hydrogen atoms, and in turn, it would be helpful to improve the accuracy of the measurements of the hydrogen 1S—2S and 1S—3S transitions.

DOI
10.1088/0256-307x/33/7/073101

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arXiv:1804.01218•Physical Review A•Oct 5, 2018•17 citations
The dynamic dipole polarizabilities for 1S, 2S and 3S states of the hydrogen atom are calculated using the finite B-spline basis set method, and the magic wavelengths for 1S—2S and 1S—3S transitions are identified. In comparison of the solutions from the Schrödinger and Dirac equations, the relativistic corrections on the magic wavelengths are of the order of 10 −2 nm. The laser intensities for a 300-E r -deep optical trap and the heating rates at 514 and 1371 nm are estimated. The reliable prediction of the magic wavelengths would be helpful for the experimental design on the optical trapping of the hydrogen atoms, and in turn, it would be helpful to improve the accuracy of the measurements of the hydrogen 1S—2S and 1S—3S transitions.
We study the magic wavelength for two-photon 1S–nS transitions in a hydrogen and deuterium atom, as well as 2S–nS transitions, where the lower level is the metastable 2S state. At the magic wavelength, the dynamic Stark shifts of the ground and the excited state of the transition coincide, so that the transition frequency is independent of the intensity of the trapping laser field. Experimentally feasible magic wavelengths of transitions with small slopes in the atomic polarizabilities are determined; these are the most stable magic wavelengths against variations of the laser frequency. We provide data for the magic wavelengths for the 1S–nS and 2S–nS transitions in hydrogen and deuterium, with n=2,⋯,8. We also analyze the stability of the elimination of the ac Stark shift at the magic wavelength against tiny variations of the trapping laser frequency from the magic value.
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.
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 $\ensuremath{-}0.2157\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/[\mathrm{GHz}\phantom{\rule{0.28em}{0ex}}(\mathrm{kW}/{\mathrm{cm}}^{2})]$.
Abstract The tune‐out and magic wavelengths (TMWs) for the hydrogenlike and screened‐hydrogenlike atoms (Z = 1‐8) are calculated using a variational method. The first three tune‐out wavelengths for each of the nS (n = 1, 2, 3) states and the first three magic wavelengths for each of the 1S‐nS (n = 2, 3) transitions of the hydrogenlike atoms in terms of nuclear charge Z, and the first TMWs for the above mentioned states or transitions of the screened‐hydrogenlike atoms in terms of the screening parameter μ and Z are presented. All the results for the TMWs are reported for the first time in the literature except the magic wavelengths for the hydrogen atom. The magic wavelengths for hydrogen atom are in agreement with the available data. The precise estimation of the TMWs would be helpful for the design of the ion‐trapping experiment of the hydrogenic systems. Our study is expected to be useful for the measurements of the hydrogenic 1S‐2S and 1S‐3S transitions spectroscopy for the improved accuracy.
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.