  We study the quantum dynamics of an ion confined in a radiofrequency trap in
interaction with either a Bose or spin-polarized Fermi gas. To this end, we
derive quantum optical master equations in the limit of weak coupling and the
Lamb-Dicke approximations. For the bosonic bath, we also include the so-called
"Lamb-shift" correction to the ion trap due to the coupling to the quantum gas
as well as the extended Fr\"ohlich interaction within the Bogolyubov
approximation that have been not considered in previous studies. We calculate
the ion kinetic energy for various atom-ion scattering lengths as well as gas
temperatures by considering the intrinsic micromotion and we analyse the
damping of the ion motion in the gas as a function of the gas temperature. We
find that the ion's dynamics depends on the quantum statistics of the gas and
that a fermionic bath enables to attain lower ionic energies.
