Research Article

The Rydberg constant and proton size from atomic hydrogen

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Science  06 Oct 2017:
Vol. 358, Issue 6359, pp. 79-85
DOI: 10.1126/science.aah6677
  • Fig. 1 Rydberg constant R and proton RMS charge radius rp.

    Values of rp derived from this work (green diamond) and spectroscopy of μp (μp; pink bar and violet square) agree. We find a discrepancy of 3.3 and 3.7 combined standard deviations with respect to the H spectroscopy world data (12) (blue bar and blue triangle) and the CODATA 2014 global adjustment of fundamental constants (3) (gray hexagon), respectively. The H world data consist of 15 individual measurements (black circles, optical measurements; black squares, microwave measurements). In addition to H data, the CODATA adjustment includes deuterium data (nine measurements) and elastic electron scattering data. An almost identical plot arises when showing R instead of rp because of the strong correlation of these two parameters. This is indicated by the R axis shown at the bottom.

  • Fig. 2 Hydrogen 2S-4P spectroscopy.

    (A) Relevant energy levels for hydrogen 2S-4P spectroscopy are shown (not to scale). The atoms are prepared in the Embedded Image metastable state (Embedded Image) by two-photon excitation with a preparation laser at 243 nm. The spectroscopy laser at 486 nm drives the one-photon 2S-4P1/2 and 2S-4P3/2 transitions to the Embedded Image (Embedded Image) and Embedded Image (Embedded Image) states to determine the transition frequencies ν1/2 and ν3/2, respectively. These states decay rapidly, predominantly to the 1S ground state (Embedded Image) either directly through Lyman-γ fluorescence at 97 nm (Ly-γ, branching ratio 84%) or indirectly through the 3S, 3D, and 2P levels, yielding one Lyman-α photon at 121 nm (Ly-α, branching ratio 4%). The remaining 12% of the decays lead back to the 2S state through Balmer-β decay (Ba-β), with 4% decaying back to the initial Embedded Image state. Excitations from the Embedded Image to the Embedded Image and Embedded Image levels are forbidden by angular momentum conservation. (B) Typical experimental fluorescence signal from a single line scan over the 2S-4P1/2 (left) and 2S-4P3/2 (right) resonance (black diamonds). The observed line width (full width at half maximum) of ~2π × 20 MHz is larger than the natural line width Γ = 2π × 12.9 MHz because of Doppler and power broadening. The accuracy of our measurement corresponds to almost 1 part in 10,000 of the observed line width. The constant background counts are caused by the decay of 2S atoms inside the detector (17). kcts, kilocounts.

  • Fig. 3 Experimental apparatus (not to scale).

    A preparation laser at 243 nm is used to excite hydrogen atoms that emerge from the cold copper nozzle (5.8 K) from the ground state to the 2S state. The 2S-4P transition is driven with the spectroscopy laser at 486 nm. This laser is coupled to an active fiber-based retroreflector [consisting of polarization-maintaining (PM) fiber, collimator, and high-reflectivity (HR) mirror] oriented perpendicular to the atomic beam; this setup provides a large suppression of the first-order Doppler effect (36). In the dark phases of the chopper wheel, Lyman-γ fluorescence photons (γ) emitted upon the rapid 4P→1S decay are detected via photoelectrons (e) by channel electron multipliers CEM1 and CEM2. The two detectors are separated by a vertical wall along the direction of the 486-nm light propagation. The 2S-4P excitation region is shielded from stray electric fields (with dedicated meshes) and magnetic fields (with magnetic shielding, not shown), resulting in stray fields below 0.6 V/m and 1 mG, respectively (17). The blue double-sided arrow labeled Embedded Image indicates the electric field of the 486-nm spectroscopy laser with orientation θL against the horizontal.

  • Fig. 4 Observation of quantum interference.

    Shown are apparent line shifts caused by quantum interference (A and B) and their suppression (C and D). Observed line centers of the 2S-4P1/2 (A) and 2S-4P3/2 (B) transitions determined with symmetric Voigt fits show a dependence on the direction of the linear laser polarization θL with an amplitude of up to 40 kHz in our geometry. Our numerical simulation (dashed lines) reproduces this behavior very well (17). Using the Fano-Voigt line shape (Eq. 5) removes the θL dependence (C) and (D). Blue and green symbols indicate data recorded with CEM1 and CEM2, respectively (see Fig. 3). Error bars indicate the statistical uncertainty only. The red-shaded areas (17) show the weighted mean of both detectors, including the final uncertainty dominated by the Doppler shift uncertainty. The dotted lines show an estimate of possible remaining line shifts with amplitude Ares.

  • Table 1

    List of corrections Δν and uncertainties σ for the determination of the 2S-4P fine structure centroid ν2S-4P. See (17) for details.

    ContributionΔv (kHz)σ (kHz)
    First-order Doppler shift0.002.13
    Quantum interference shift0.000.21
    Light force shift–0.320.30
    Model corrections0.110.06
    Sampling bias0.440.49
    Second-order Doppler shift0.220.05
    dc-Stark shift0.000.20
    Zeeman shift0.000.22
    Pressure shift0.000.02
    Laser spectrum0.000.10
    Frequency standard (hydrogen maser)0.000.06
    Recoil shift–837.230.00
    Hyperfine structure corrections–132,552.0920.075

Supplementary Materials

  • The Rydberg constant and proton size from atomic hydrogen

    Axel Beyer, Lothar Maisenbacher, Arthur Matveev, Randolf Pohl, Ksenia Khabarova, Alexey Grinin, Tobias Lamour, Dylan C. Yost, Theodor W. Hänsch, Nikolai Kolachevsky, Thomas Udem

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Fig. S1
    • Tables S1 to S3
    • References

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