Resonant Rydberg lines of a single trapped Rydberg ion: coupling to trap RF potential and fabircation-induced electric fields

An accurate numerical scheme is presented to calculate and analyze the spectral line shape and broadening of a single highly excited Rydberg ion in a Paul trap. The energy spectra of a free highly excited $Ca^+$ are accurately calculated for \textit{s}, \textit{p}, \textit{d}, \textit{f}, and \textit{g} states (up to $n=64$), using the parametric one-electron valence potential with spin-orbit coupling. The coupling of the ion and Rydberg electron to the trap potential is implemented within a Floquet formalism. The alternating \textit{axial} electric field noise due to the applied RF, and the oscillating \textit{radial} electric field emanating due to the trap fabrication are incorporated into the Floquet calculations. Detailed comparison with available observation (Feldker et. al, PRL 2015) is made.