  We study the quantum dynamics of a binary mixture of Bose-Einstein
condensates (BEC) in a double-well potential starting from a two-mode
Bose-Hubbard Hamiltonian. Focussing on the regime where the number of atoms is
very large, a mapping onto a SU(2) spin problem together with a
Holstein-Primakoff transformation is performed. The quantum evolution of the
number difference of bosons between the two wells is investigated for different
initial conditions, which range from the case of a small imbalance between the
two wells to a coherent spin state. The results show an instability towards a
phase-separation above a critical positive value of the interspecies
interaction while the system evolves towards a coherent tunneling regime for
negative interspecies interactions. A comparison with a semiclassical approach
is discussed together with some implications on the experimental realization of
phase separation with cold atoms.
