Shift of the Bose-Einstein condensation transition in the presence of a second atomic species
Atomic interactions play an important role in the properties of ultracold atomic gases. In single component bosonic systems, its effect is already present at the critical point for the Bose-Einstein condensate phase transition by shifting it to lower temperatures as a consequence of effective repulsion between the atoms. When considering atomic bosonic mixtures, interesting effects arise from the competition between intra- and interspecies interactions such as the miscible-immiscible phase transition and the particular case of self-bounded quantum droplets. In such a scenario, it is natural to expect that these interactions will also affect the critical point of each species composing the mixture. In this paper, we obtain analytical expressions for the critical temperature shift of the phase transition to a Bose-Einstein condensate in the presence of a second species. We treat differently the cases in with the second species is above or below its own critical temperature and apply the obtained relations to the case of a $^{23}$Na-$^{39}$K bosonic mixture which can be realized in current running experimental setups. Our findings can be easily extended to other atomic mixtures trapped by arbitrary conservative traps.