Highly accurate measurements of atomic transitions can serve as precise probes of the low-energy properties of nuclei. As the Bohr radius of hydrogen-like atoms decreases with increasing orbiting particle mass, muonic hydrogen (μp)—a hydrogen-like atom formed by a negative muon and a proton—exhibits enhanced sensitivity to proton structure. The objective of this project is to measure, for the first time, the ground-state hyperfine splitting (HFS) in μp with 1 ppm relative accuracy.
By comparing the measured HFS transition frequency with the corresponding theoretical prediction based on bound-state QED calculations, the nuclear structure contribution can be extracted with 200 ppm relative accuracy. This contribution will serve as a benchmark for chiral perturbation theory and dispersion-based predictions aiming to understand the magnetic structure of the proton. Moreover, it represents a benchmark for electron-proton scattering and lattice QCD calculations.
Combined with HFS measurements in regular hydrogen spectroscopy, this measurement paves the way for testing the HFS in regular hydrogen at the 10^-8 level of accuracy, providing a powerful means to search for physics beyond the Standard Model.
Our experimental method follows this sequence: Low-energy negative muons (μ−) are stopped in H2 gas of 1 mm length, 22 K temperature, and 0.5 bar pressure, forming μp. An on-resonance laser pulse at a wavelength of 6.8 μm excites the muonic hyperfine transition from the singlet to the triplet hyperfine sublevels. The subsequent collisionally-induced quenching from the triplet back to the singlet state imparts an average kinetic energy of 0.1 eV to the μp atom, so that it can quickly diffuse in the hydrogen gas, efficiently reaching (before muon decay) one of the target walls coated with gold. At the target wall, the muon is transferred from μp to a gold atom, forming muonic gold (μAu) in an excited state. The μAu de-excitation produces various X-rays of MeV energy, which are used as a signature of a successful laser-induced transition, so that the HFS resonance can be exposed by counting the number of μAu cascade events after laser excitation as a function of the laser frequency.
This experiment, at the intersection of particle, atomic, and nuclear physics, requires the development of cutting-edge laser technologies, especially in the thin-disk laser and mid-infrared laser domains. The mid-infrared source needed for this experiment is realized starting from a single-frequency thin-disk laser operating in the 500 mJ regime and down-converting its pulses in a cascade of nonlinear processes to produce single-frequency pulses of 5 mJ energy, tunable around a wavelength of 6.8 μm, with a bandwidth < 10 MHz within 1 μs after a laser trigger. To give a sense of the challenges faced and the technological leap needed, note that the required laser energy density at the muonic atom position for this experiment must be at least 3,000 times larger compared to the energy density used for the proton radius measurement. Additionally, this larger energy density has to be obtained at a longer mid-infrared wavelength and with a factor of 50 smaller linewidth.
Our project has primarily focused on laying the groundwork and developing the necessary laser technology for this experiment. This endeavor was significantly challenged by erroneous literature values initially used in the experiment's design.
This heightened challenge prompted us to refine every aspect of the experimental setup, from the laser system to the particle detection apparatus, resulting in significant advances that are reflected in numerous publications.
We also delved into refining theoretical predictions of energy levels, investigating muonic atom diffusion in hydrogen gas, optimizing laser excitation probabilities, and enhancing optical multi-pass cells.
