  We investigate the problem of a single ion in a radio-frequency trap and
immersed in an ultracold Bose gas either in a condensed or a non-condensed
phase. We develop master equation formalism describing the sympathetic cooling
and we determine the cooling rates of ions. We show that cold atomic reservoir
modifies the stability diagram of the ion in the Paul trap creating the regions
where the ion is either cooled or heated due to the energy quanta exchanged
with the time-dependent potential.
