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 (${\cal M}$), 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 ${\cal M}$ as in current experiments.