  We consider the effective spin Hamiltonian describing a mixture of two
species of pseudo-spin-1/2 Bose gases with interspecies spin exchange. First we
analyze the stability of the fixed points of the corresponding classical
dynamics, of which the signature is found in quantum dynamics with a
disentangled initial state. Focusing on the case without an external potential,
we find all the ground states by taking into account quantum fluctuations
around the classical ground state in each parameter regime. The nature of
entanglement and its relation with classical bifurcation is investigated. When
the total spins of the two species are unequal, the maximal entanglement at the
parameter point of classical bifurcation is possessed by the excited state
corresponding to the classical fixed point which bifurcates, rather than by the
ground state.
