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  V 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)PRLTAO0031-900710.1103/PhysRevLett.107.203001; A. Matveev et al., Phys. Rev. Lett. 110, 230801 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.230801] and the 1S hyperfine splitting [L. Essen et al., Nature (London) 229, 110 (1971)NATUAS0028-083610.1038/229110a0], we determine the ionization frequency of the 1S(0) ground state to be 3 288 087 922 407.2(3.7)_{stat}(1.8)_{syst}  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)SCIEAS0036-807510.1126/science.aau7807], we determine the Rydberg frequency to be cR_{∞}=3 289 841 960 204(15)_{stat}(7)_{syst}(13)_{2S-2P}  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.