Towards Lamb shift spectroscopy of antihydrogen atoms at the GBAR $\mathrm{\bar{H}}$ beam line

A microwave (MW) spectroscopy experiment has been proposed to directly measure the resonant frequency of $2S_{1/2}-2P_{1/2}$ Lamb shift transition in antihydrogen ($\mathrm{\bar{H}}$) atoms. The spectroscopy opens up the possibility of deriving the charge radius of antiproton ($\mathrm{\bar{p}}$) using a beam of $\mathrm{\bar{H}}$ atoms with a kinetic energy of a few keV travelling under a magnetic field-free environment. The requisite spectroscopy apparatuses have been developed and installed in the $\mathrm{\bar{H}}$ beam line at the GBAR experiment, where the production of $\mathrm{\bar{H}}$ beam at 6.1 keV was demonstrated through a charge exchange reaction of a $\mathrm{\bar{p}}$ beam passing through a positronium (Ps) cloud. The spectroscopy setup is composed of a MW spectrometer and a Lyman-$\alpha$ photon detector. The MW spectrometer consists of two consecutive MW apparatuses which have a relatively large borehole of 30 mm diameter, and each MW apparatus comprises a pair of parallel plate electrodes as its inner conductor and a rectangular box as its outer conductor. Downstream to the MW spectrometer, the Lyman-$\alpha$ detector has been installed to count the $\mathrm{\bar{H}}$ atoms remaining in the $2S$ state after interacting with the MW E-field. Towards the $\mathrm{\bar{H}}$ Lamb shift spectroscopy, we present here a characterization of the MW spectrometer, an evaluation of the detection efficiency of the Lyman-$\alpha$ detector, and a precision expected in the first line shape measurement of the $\mathrm{\bar{H}}$ Lamb shift.