Precision-Spectroscopic Determination of the Binding Energy of a Two-Body Quantum System: The Hydrogen Atom and the Proton-Size Puzzle

Precision measurements in Rydberg states of H with principal quantum number $n$ in the range between 20 and 30 are reported. In the presence of homogeneous electric fields with strengths below $2\mathrm{V}{\mathrm{cm}}^{-1}$, these Rydberg states are subject to a linear Stark effect with accurately calculable Stark shifts. From the spectral positions of field-independent and field-dependent Rydberg-Stark states, we derive the $n=20$ and 24 Bohr energies, and the ionization energy with respect to the $2{}^2{S}_{1/2}(f=0,1)$ [short $2S(0,1)$] metastable states. Combining these results with the $2S(1)-1S(1)$ transition frequency [C. G. Parthey et al., Phys. Rev. Lett. 107, 203001 (2011); A. Matveev et al., Phys. Rev. Lett. 110, 230801 (2013)] and the $1S$ hyperfine splitting [L. Essen et al., Nature (London) 229, 110 (1971)], we determine the ionization frequency of the $1S(0)$ ground state to be $3288087922407.2(3.7{)}_{\mathrm{stat}}(1.8{)}_{\mathrm{syst}}\mathrm{kHz}$, which is the most precise value ever determined for the binding energy of a two-body quantum system. Using the $2S(0)-2P_{1/2}(1)$ interval [N. Bezginov et al., Science 365, 1007 (2019)], we determine the Rydberg frequency to be $c{R}_{\infty }=3289841960204(15{)}_{\mathrm{stat}}(7{)}_{\mathrm{syst}}(13{)}_{2S-2P}\mathrm{kHz}$ in a procedure that is insensitive to the value of the proton charge radius. These new results are discussed in the context of the proton-size puzzle.

Figure 1

Energy-level diagram of the $j=1/2$ states of the $n=1$ and 2 manifolds of H and schematic structure of the high-$n$ Rydberg states. The field-dependent energy shift $\delta {\nu }_{\text{Stark}}^{(n)}(\mathcal{F})$ of the $n=20$ and 24, $k=0$, $|m_{\ell }|=1$ Stark states are depicted as orange lines. The field strength is given in $\mathrm{V}/\mathrm{cm}$. The inset shows their substructure.

Figure 2

Schematic representation of the experimental setup with the vacuum chamber comprising the supersonic-beam source and the photoexcitation region (left) and the main components of the laser system (right) (SHG, second harmonic generation; l, lens assembly; m, retroreflecting mirror; MCP, microchannel-plate detector; HV, high-voltage).

Figure 3

Spectrum of the transitions from the $2S(1)$ state of H to the $n=24$, $k=\pm 2,0$, $|m_{\ell }|=1$ Stark states (black dots) and spectrum calculated with the line-shape parameters obtained from a weighted least-squares fit (blue line). The width of the orange area corresponds to $\delta {\nu }_{\text{Stark}}^{(24)}(0.4\mathrm{V}{\mathrm{cm}}^{-1})$, which bridges the observed transition frequency to the Bohr energy at 0 MHz.

Figure 4

Scatter plot of ($R_{\infty }$, $r_{\mathrm{p}}$) values from transition frequencies in H [20, 21, 22, 23, 24] since 2010 relative to the values reported in Tiesinga et al. [2], in units of the CODATA 2018 uncertainties. The covariance ellipses with the $1\sigma$, $2\sigma$, and $3\sigma$ intervals of the CODATA 2018 and 2010 adjustments [2, 3] are in red. When only $R_{\infty }$ or $r_{\mathrm{p}}$ are reported the data are represented as vertical or horizontal lines with uncertainties given by shaded areas for $r_{\mathrm{p}}$ or $R_{\infty }$, respectively.

Figure 5

(a) Ionization frequencies ${\nu }_{\mathrm{i}}^{2S(1)}$ obtained from the frequencies $\nu (20\leftarrow 2{}^2{S}_{1/2}^0)$ (blue), $\nu (20\leftarrow 2{}^2{S}_{1/2}^1)$ (orange), and $(24\leftarrow 2{}^2{S}_{1/2}^1)$ (green). (b),(c) Dependence of the ionization frequency on the electric field strength $\mathcal{F}$ (b) and on the Doppler shift ${\nu }_{\mathrm{D}}$ (c) (see text for details).