We study the coherent nonlinear tunneling dynamics of a binary mixture ofBose-Einstein condensates in a double-well potential. We demonstrate theexistence of a new type of mode associated with the "swapping" of the twospecies in the two wells of the potential. In contrast to the symmetry breakingmacroscopic quantum self-trapping (MQST) solutions, the swapping modescorrespond to the tunneling dynamics that preserves the symmetry of the doublewell potential. As a consequence of two distinct types of broken symmetry MQSTphases where the two species localize in the different potential welils orcoexist in the same well, the corresponding symmetry restoring swapping modesresult in dynamics where the the two species either avoid or chase each other.In view of the possibility to control the interaction between the species, thebinary mixture offers a very robust system to observe these novel effects aswell as the phenomena of Josephson oscillations and pi-modes
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