  Trapped Rydberg ions are a promising new system for quantum information
processing. They have the potential to join the precise quantum operations of
trapped ions and the strong, long-range interactions between Rydberg atoms.
Technically, the ion trap will need to stay active while exciting the ions into
the Rydberg state, else the strong Coulomb repulsion will quickly push the ions
apart. Thus, a thorough understanding of the trap effects on Rydberg ions is
essential for future applications. Here we report the observation of two
fundamental trap effects. First, we investigate the interaction of the Rydberg
electron with the quadrupolar electric trapping field. This effect leads to
Floquet sidebands in the spectroscopy of Rydberg D-states whereas Rydberg
S-states are unaffected due to their symmetry. Second, we report on the
modified trapping potential in the Rydberg state compared to the ground state
which results from the strong polarizability of the Rydberg ion. We observe the
resultant energy shifts as a line broadening which can be suppressed by cooling
the ion to the motional ground state in the directions orthogonal to the
excitation laser.
