  We present a method for measuring the polarizability of Rydberg ions confined
in the harmonic potential of a Paul trap. For a highly excited electronic
state, the coupling between the electronic wave function and the trapping field
modifies the excitation probability depending on the motional state of the ion.
This interaction strongly depends on the polarizability of the excited state
and manifests itself in the state-dependent secular frequencies of the ion. We
initialize a single trapped $^{40}$Ca$^+$ ion from the motional ground state
into coherent states with $|\alpha|$ up to 12 using electric voltages on the
trap segments. The internal state, firstly initialised in the long-lived
3D$_{5/2}$ state, is excited to a Rydberg S$_{1/2}$-state via the 5P$_{3/2}$
state in a two-photon process. We probe the depletion of the 3D$_{5/2}$ state
owing to the Rydberg excitation followed by a decay into the internal ground
4S$_{1/2}$ state. By analysing the obtained spectra we extract the
polarizability of Rydberg states which agree with numerical calculations. The
method is easy-to-implement and applicable to different Rydberg states
regardless of their principal or angular quantum numbers. An accurate value of
the state polarizability is needed for quantum gate operations with Rydberg ion
crystals.
