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arXiv:1507.02466 · Physical Review A

Magic wavelength for the hydrogen 1S−2S transition

Akio Kawasaki

Published Oct 12, 2015• 10 citations
arXivHTMLPDF

Abstract

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 $\mathrm{\ensuremath{\Delta}}E=\ensuremath{-}119$ $\text{Hz}/(\text{kW}/{\mathrm{cm}}^{2})$ and the slope $d\mathrm{\ensuremath{\Delta}}E/d\ensuremath{\nu}=\ensuremath{-}2.77$ $\text{Hz}/(\text{GHz} \text{kW}/{\mathrm{cm}}^{2})$ 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.

DOI
10.1103/physreva.92.042507

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The magic wavelength for an optical lattice for hydrogen atoms that cancels the lowest order ac Stark shift of the 1S\ensuremath-2S1S\text{\ensuremath{-}}2S1S\ensuremath-2S transition is calculated to be 513 nm. The magnitudes of the ac Stark shift \ensuremathΔE=\ensuremath−119\mathrm{\ensuremath{\Delta}}E=\ensuremath{-}119\ensuremathΔE=\ensuremath−119 Hz/(kW/cm2)\text{Hz}/(\text{kW}/{\mathrm{cm}}^{2})Hz/(kW/cm2) and the slope d\ensuremathΔE/d\ensuremathν=\ensuremath−2.77d\mathrm{\ensuremath{\Delta}}E/d\ensuremath{\nu}=\ensuremath{-}2.77d\ensuremathΔE/d\ensuremathν=\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\ensuremath-2S1S\text{\ensuremath{-}}2S1S\ensuremath-2S transition.
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])].
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 $\ensuremath{-}0.2157\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/[\mathrm{GHz}\phantom{\rule{0.28em}{0ex}}(\mathrm{kW}/{\mathrm{cm}}^{2})]$.
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.
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.
Wavelength-dependent AC Stark shifts and magic wavelengths of the terahertz clock transitions between the metastable triplet states 6s5d 3 D 1 and 6s5d 3 D 2 are investigated with considering the optical lattice trapping of barium atoms with the linearly polarized laser. The trap depths and the slopes of light shift difference with distinct magic wavelengths of the optical lattices are also discussed in detail. Several potentially suitable working points for the optical lattice trapping laser are recommended and selected from these magic wavelengths.
We demonstrate the cancellation of the differential ac Stark shift of the microwave hyperfine clock transition in trapped 87 Rb atoms. Recent progress in metrology exploits so-called “magic wavelengths,” whereby an atomic ensemble can be trapped with laser light whose wavelength is chosen so that both levels of an optical atomic transition experience identical ac Stark shifts. Similar magic-wavelength techniques are not possible for the microwave hyperfine transitions in the alkalis, due to their simple electronic structure. We show, however, that ac Stark shift cancellation is indeed achievable for certain values of wavelength, polarization, and magnetic field. The cancellation comes at the expense of a small magnetic-field sensitivity. The technique demonstrated here has implications for experiments involving the precise control of optically-trapped neutral atoms.