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We study the spin mixing dynamics of ultracold spin-1 atoms in a weaknon-uniform magnetic field with field gradient $G$, which can flip the spinfrom +1 to -1 so that the magnetization $m=\rho_{+}-\rho_{-}$ is not any more aconstant. The dynamics of $m_F=0$ Zeeman component $\rho_{0}$, as well as thesystem magnetization $m$, are illustrated for both ferromagnetic and polarinteraction cases in the mean-field theory. We find that the dynamics of systemmagnetization can be tuned between the Josephson-like oscillation similar tothe case of double well, and the interesting self-trapping regimes, i.e. thespin mixing dynamics sustains a spontaneous magnetization. Meanwhile thedynamics of $\rho_0$ may be sufficiently suppressed for initially imbalancednumber distribution in the case of polar interaction. A "beat-frequency"oscillation of the magnetization emerges in the case of balanced initialdistribution for polar interaction, which vanishes for ferromagneticinteraction.

Magnetization Oscillation of a Spinor Condensate Induced by Magnetic Field Gradient