Ground states of a mixture of two species of spinor Bose gases with interspecies spin exchange

We consider a mixture of two species of spin-1 atoms with interspecies spin exchange, which may cooperate or compete with intraspecies spin exchanges and thus dramatically affect the ground state. It represents a new class of bosonic gases differing from single-species spinor gases. We determine the exact ground states in several parameter regimes and study the composite structures using the generating function method generalized here to be applicable to a mixture of two species of spinor gases. The most interesting phase is the so-called entangled Bose-Einstein condensation (BEC), which is fragmented BEC with quantum entanglement between the two species and with both interspecies and intraspecies singlet pairs. For comparison, we also apply the generating function method to a mixture of two species of pseudospin-$\frac{1}{2}$ atoms, for which the total spin quantum number of each species is fixed as half of the atom number, in contrast with the case of spin $1$, for which it is a variable determined by energetics. Consequently, singlet pairs in entangled BEC of a pseudospin-$\frac{1}{2}$ mixture are all interspecies. Interspecies spin exchange leads to novel features beyond those of spinor BEC of a single species of atoms as well as mixtures without interspecies spin exchange.