  We have identified ultra-cold atoms in magneto-optical double-well potentials
as a very clean setting in which to study the quantum and classical dynamics of
a nonlinear system with multiple degrees of freedom. In this system,
entanglement at the quantum level and chaos at the classical level arise from
nonseparable couplings between the atomic spin and its center of mass motion.
The main features of the chaotic dynamics are analyzed using action-angle
variables and Poincare surfaces of section. We show that for the initial state
prepared in current experiments [D. J. Haycock et al., Phys. Rev. Lett. 85,
3365 (2000)], the classical and quantum dynamics diverge, and the observed
experimental dynamics are best described by quantum mechanics. Furthermore, the
motion corresponds to tunneling through a dynamical potential barrier. The
coupling between the spin and the motional subsystems, which are very different
in nature from one another, leads to new questions regarding the transition
from regular quantum dynamics to chaotic classical motion.
