Introduction

The hydrogen atom is unique since physical theories can be applied to it “without” approximations. Any discrepancy between theoretical prediction and experimental measurement which may be unveiled at any increase of theoretical and experimental accuracy thus holds the potential for new fundamental insights.

Nothing can hide in hydrogen, not even the proton at its center. In fact, measurements with hydrogen beams by Stern in 1933 revealed that the magnetic moment of the proton deviated from the prediction of the Dirac relativistic theory. This was the first indication that the proton - contrary to the electron - has a structure. In 1947 measurements of the 2S-2P (Lamb shift) and 1S-hyperfine splitting in hydrogen deviated from those predicted by the Dirac equation. This was the initiation for the development of quantum electrodynamics (QED). In the last four decades, the goal to measure hydrogen energy levels with greater accuracy has lead to advances in high resolution spectroscopy and metrology. This peaked with the invention of the frequency comb laser by Hänsch in the late 90ies. The high accuracy obtained with such techniques provided cornerstones to test bound-state QED, to determine the Rydberg constant and the proton radius (assuming the correctness of the theory), and to search for slow time variations of fundamental constants.

Hydrogen energy levels are slightly modified by the fact that in contrast to the electron the proton has a size. Hence, to precisely predict these energy levels an accurate knowledge of the root-mean-square charge radius of the proton is necessary. The historical method of determining the proton radius was based upon scattering electrons on protons, in effect by scattering an electron beam on a liquid hydrogen target. The uncertainty related to the knowledge of the proton radius extracted from electron-proton scattering limited the prediction accuracy of the hydrogen energy levels, and consequently it was limiting the comparison between theory and measurements. Therefore to advance the check (comparison between prediction and measurement) of bound-state QED describing the hydrogen energy levels it was necessary to have a more precise determination of the proton radius. This was one of the main motivations for our experiment: to measure the 2S-2P energy difference in muonic hydrogen (µp), an exotic atom composed by a negative muon and a proton. The single electron of a hydrogen atom is replaced by a negative muon which has a lifetime of only 2 microseconds and is 200 times heavier than the electron. According to the laws of quantum mechanics the muon wave functions in S-states overlap therefore more with the proton and the corresponding µp energy levels are sensitive to the proton size. By measuring the 2S-2P transition frequency in muonic hydrogen it is thus possible to extract with great accuracy the proton radius, assuming that the main QED contributions to the 2S-2P splitting are correctly predicted by theory.

Method and measurements

Our experiment is based upon laser spectroscopy of muonic hydrogen. The main components which had to be developed for this experiment are a low energy muon beam, an infrared laser system used to drive 2S-2P transitions in µp, and detectors for 2 keV X-rays emitted from 2P-1S transitions. More details are given in Ref. [1]. Muonic hydrogen is produced by stopping negative muons in hydrogen gas. Only at the Paul Scherrer Institut (PSI), Switzerland, is there a suffciently strong low energy muon beam suited for such an experiment. The µp atoms are produced at highly excited states (around n=14). Most of these de-excite quickly to the 1S-ground state, but ~1% populate the long-lived 2S-state (Fig. 2 (a)). A short laser pulse tunable to a wavelength around λ ≈ 6 µm (corresponding to the 2S-2P energy splitting) illuminates the muonic atom. 2S → 2P transitions are induced by the laser light (Fig. 2 (b)), immediately followed by 2P → 1S de-excitation via emission of a 2 keV X-ray (lifetime t 2P = 8.5 ps). The transition from the 2S to the 2P state and the subsequent emission of X-ray only occurs if the laser frequency corresponds to the energy difference between 2S and 2P levels.

Fig. 3 shows the resulting resonance curve obtained by plotting the number of 2 keV X-rays at different laser wavelengths that occur in time-coincidence with the laser pulse. The centroid position is determined with a statistical uncertainty of 700 MHz. The linewidth is in good agreement with the theoretical prediction. The laser frequency is known over the whole region with 300 MHz accuracy and was determined with two independent methods. The systematics are completely dominated by the laser frequency calibration. Effects like Zeeman shift, Doppler shift, AC- and DC-Stark shifts and pressure shift are smaller than 50 MHz.