  Usually the influence of the quadratic Stark effect on an ion's trapping
potential is minuscule and only needs to be considered in atomic clock
experiments. In this work we excite a trapped ion to a Rydberg state with
polarizability $\sim$~eight orders of magnitude higher than a low-lying
electronic state; we find that the highly-polarizable ion experiences a vastly
different trapping potential owing to the Stark effect. We observe changes in
trap stiffness, equilibrium position and minimum potential, which can be tuned
using the trapping electric fields. These effects lie at the heart of proposals
to shape motional mode spectra, simulate quantum magnetism and coherently drive
structural phase transitions; in addition we propose using these effects to
simulate cosmological particle creation, study quantum fluctuations of work and
minimize ion micromotion. Mitigation of Stark effects is important for coherent
control of Rydberg ions; we illustrate this by carrying out the first Rabi
oscillations between a low-lying electronic state and a Rydberg state of an
ion.
