Description of ion motion in a Paul trap immersed in a cold atomic gas

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