Cryogenic $^9$Be$^+$ Penning trap for precision measurements with (anti-)protons

Cooling and detection schemes using laser cooling and methods of quantum logic can contribute to high precision CPT symmetry tests in the baryonic sector. This work introduces an experiment to sympathetically cool protons and antiprotons using the Coulomb interaction with a $^9$Be$^+$ ion trapped in a nearby but separate potential well. We have designed and set up an apparatus to show such coupling between two identical ions for the first time in a Penning trap. In this paper, we present evidence for successful loading and Doppler cooling of clouds and single ions. Our coupling scheme has applications in a range of high-precision measurements in Penning traps and has the potential to substantially improve motional control in these experiments.