  We study matter wave bistability in a spin-1 Bose-Einstein condensate
dispersively coupled to a unidirectional ring cavity. A unique feature is that
the population exchange among different modes of matter fields are accomplished
via the spin-exchange collisions. We show that the interplay between the atomic
spin mixing and the cavity light field can lead to a strong matter wave
nonlinearity, making matter wave bistability in a cavity at the single-photon
level achievable.
