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We examine in terms of exact solutions of the time-dependent Schrödinger equation, the quantum tunnelling process in Bose–Einstein condensates of two interacting species trapped in a double well configuration. Based on the two series of time-dependent SU(2) gauge transformations, we diagonalize the Hamilton operator and obtain analytic time-evolution formulas of the population imbalance and the berry phase. The particle population imbalance $\langle\hat{a}^{+}_{L}\hat{a}_{L}-\hat{a}^{+}_{R}\hat{a}_{R}\rangle$ of species A between the two wells is studied analytically. [ABSTRACT FROM AUTHOR]