Quantum and classical dynamics of atoms in a magneto-optical lattice

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. We have analyzed the distinct predictions of classical and quantum theory in this system in order to study quantum-classical correspondence and the quantum to classical transition. The classical version of the Hamiltonian corresponds to a magnetic moment moving in a spatially inhomogeneous magnetic field. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincare surfaces of section. We present a physical picture of the primary resonances and quantify the chaos by calculating the Lyapunov exponents. A joint Husimi distribution over position and momentum as well as the Bloch sphere is used to compare quantum and classical phase space dynamics. We show that for the initial state prepared in current experiments [D. J. Haycock et al ., Phys. Rev. Lett. 85, 3365 (2000)], classical and quantum expectation values show a disagreement after a finite time, and the observed experimental dynamics is consistent with quantum mechanical predictions. Furthermore, we show that the motion corresponds to tunnelling 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 quantum dynamics to classical motion. We study the role of a continuous position measurement in the quantum to classical transition for our system of internal (spin) and external (motional) degrees of freedom. We show that even when the measured motional degree of freedom can be treated classically, entanglement between spin and motion causes strong measurement back action on the quantum spin subsystem so that classical trajectories are not recovered in this mixed quantum-classical regime. The measurement can extract localized quantum trajectories that behave classically only when the internal action also becomes large relative to h. (Abstract shortened by UMI.)