We compute the Rydberg spectrum of a single ${\mathrm{Ca}}^{+}$ ion in a Paul trap by incorporating various internal and external coupling terms of the ion to the trap in the Hamiltonian. The coupling terms include spin-orbit coupling in ${\mathrm{Ca}}^{+}$, charge (electron and ionic core) coupling to the radio frequency and static fields, ion-electron coupling in the Paul trap, and ion center-of-mass coupling. The electronic Rydberg states are precisely described by a one-electron model potential for $e^{-}+{\mathrm{Ca}}^{2+}$, and accurate eigenenergies, quantum defect parameters, and static and tensor polarizabilities for a number of excited Rydberg states are obtained. The time-periodic rf Hamiltonian is expanded in the Floquet basis, and the trapping-field-broadened Rydberg lines are compared with recent observations of ${\mathrm{Ca}}^{+}\left(23P\right)$ and ${\mathrm{Ca}}^{+}\left(52F\right)$ Rydberg lines.

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• Atom & ion trapping & guiding
• Atomic & molecular processes in external fields
• Atomic spectra
• Oscillator strength
• Stark effect
• Rydberg atoms & molecules
• Trapped ions
• Polarizability
• First-principles calculations
• Nonperturbative methods
• Schroedinger equation