  The response of matter to fields underlies the physical sciences, from
particle physics to astrophysics, and from chemistry to biophysics. We observe
an atom's response to an electric quadrupole field to second- and higher
orders; this arises from the atom's electric quadrupole polarizability and
hyperpolarizabilities. We probe a single atomic ion which is excited to Rydberg
states and confined in the electric fields of a Paul trap. The quadrupolar
trapping fields cause atomic energy level shifts and give rise to spectral
sidebands. The observed effects are described well by theory calculations.
