The focus of this internship is the improvement of theoretical methods to describe quantum electrodynamics (QED) in the vicinity of the Coulomb field created by an atomic nucleus with large nuclear charge number Z. In contrast to low-Z ions, the coupling of the quantum vacuum to the nucleus has to be described non-perturbatively in the high-Z regime because of the strong Coulomb field. This non-perturbative description creates additional challenges for theory compared to the perturbative regime, but is necessary to provide accurate theoretical predictions for many ongoing and future experiments that aim to test QED in strong fields. One of the main tasks will be adapting established methods for the description of electronic ions to exotic systems, where one or more electrons have been replaced by a heavier exotic particle, like a muon or an antiproton. Due to their higher mass, these particles orbit significantly closer to the nucleus than an electron in the same state and probe electric fields that are orders of magnitude larger. State-of-the-art experiments with exotic atoms, such as the PAX (antiProtonic Atom X-ray spectroscopy) experiment currently under development at LKB, can test the Standard Model in a regime that is otherwise inaccessible in a laboratory and need improved theoretical predictions to exploit their full potential as precision tests of strong-field QED.