  We present an experimental realization of dynamic self-trapping and
non-exponential tunneling in a multi-state system consisting of ultracold
sodium spinor gases confined in moving optical lattices. Taking advantage of
the fact that the tunneling process in the sodium spinor system is resolvable
over a broader dynamic energy scale than previously observed in rubidium scalar
gases, we demonstrate that the tunneling dynamics in the multi-state system
strongly depends on an interaction induced nonlinearity and is influenced by
the spin degree of freedom under certain conditions. We develop a rigorous
multi-state tunneling model to describe the observed dynamics. Combined with
our recent observation of spatially-manipulated spin dynamics, these results
open up prospects for alternative multi-state ramps and state transfer
protocols.
