  We analyze the multipole excitation of atoms with twisted light, i.e., by a
vortex light field that carries orbital angular momentum. A single trapped
$^{40}$Ca$^+$ ion serves as a localized and positioned probe of the exciting
field. We drive the $S_{1/2} \to D_{5/2}$ transition and observe the relative
strengths of different transitions, depending on the ion's transversal position
with respect to the center of the vortex light field. On the other hand,
transition amplitudes are calculated for a twisted light field in form of a
Bessel beam, a Bessel-Gauss and a Gauss-Laguerre mode. Analyzing experimental
obtained transition amplitudes we find agreement with the theoretical
predictions at a level of better than 3\%. Finally, we propose measurement
schemes with two-ion crystals to enhance the sensing accuracy of vortex modes
in future experiments.
