Quantum phase diffusions of a spinor condensate

We discuss the quantum phases and their diffusion in a spinor-1 atomic Bose-Einstein condensate. For ferromagnetic interactions, we obtain the exact ground state distribution of the phase fluctuations corresponding to the total atom number (N), the magnetization (M), and the alignment (or hypercharge) (Y) of the system. The mean-field ground state is shown to be stable against these fluctuations, which dynamically recover the two continuous symmetries associated with the conservation of N and M as in current experiments.