Design of a microwave spectrometer for high-precision Lamb shift spectroscopy of antihydrogen atoms.

We have developed a microwave spectrometer for a measurement of the Lamb shift of antihydrogen atoms towards the determination of the antiproton charge radius. The spectrometer consists of two consecutive apparatuses, of which the first apparatus, (HFS), filters out hyperfine states and pre-selects the state, and the second apparatus, (MWS), sweeps the frequency around the target transition to obtain the spectrum. We optimized the geometry of the apparatuses by evaluating the S-parameter that represents the ratio of the reflected microwave signal over the input, utilizing microwave simulations based on the finite element method. The HFS was designed to obtain a resonant property at 1.1 GHz for an efficient removal of the hyperfine states, and the MWS was designed to realize weak frequency-dependency in the signal reflection. Also, the spatial distributions of microwave electric field were simulated. We report the design of the spectrometer and discuss an expected precision of the first measurement.