We report on the production of a $^{41}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ dual-species Bose-Einstein condensate in a hybrid trap, consisting of a magnetic quadrupole and an optical dipole potential. After loading both atomic species in the trap, we cool down
87Rb first by magnetic and then by optical evaporation, while
41K is sympathetically cooled by elastic collisions with
87Rb. We eventually produce two-component condensates with more than
105 atoms and tunable species population imbalance. We observe the immiscibility of the quantum mixture by measuring the density profile of each species after releasing them from the trap.
We show, both experimentally and theoretically, that sympathetic cooling of
87Rb atoms in the
∣F=2,mF=2〉 state by evaporatively cooled atoms in the $|{F=1,m}_{F}=\ensuremath{-}1〉$ state can be precisely controlled so as to produce a condensate in either or both states. We also study the thermalization rate between two species. Our model renders a quantitative account of the observed role of the overlap between the two clouds and verifies that sympathetic cooling becomes inefficient when the masses are very different. Our calculation also yields an analytical expression of the thermalization rate for a single species.