  High-precision laser spectroscopy of atomic hydrogen has led to an impressive
accuracy in tests of bound-state quantum electrodynamics (QED). At the current
level of accuracy many systematics have to be studied very carefully and only
independent measurements provide the ultimate cross-check. This has been proven
recently by measurements in muonic hydrogen, eventually leading to a
significant shift of the CODATA recommended values of the proton charge radius
and the Rydberg constant. We aim to contribute to tests of fundamental physics
by measuring the 1S-2S transition in the He$^+$ ion for the first time.
Combined with measurements in muonic helium ions this can probe the value of
the Rydberg constant, test higher-order QED terms, or set benchmarks for ab
initio nuclear polarizability calculations. We extend the Ramsey-comb
spectroscopy method to the XUV using high-harmonic generation in order to
excite a single, trapped He$^+$ ion.

Atomic Physics