Characteristics of quantum-classical correspondence for two interacting spins

The conditions of quantum-classical correspondence for a system of two interacting spins are investigated. Differences between quantum expectation values and classical Liouville averages are examined for both regular and chaotic dynamics well beyond the short-time regime of narrow states. We find that quantum-classical differences initially grow exponentially with a characteristic exponent consistently larger than the largest Lyapunov exponent. We provide numerical evidence that the time of the break between the quantum and classical predictions scales as $\log (\mathcal{J}/ħ),$ where $\mathcal{J}$ is a characteristic system action. However, this logarithmic break-time rule applies only while the quantum-classical deviations are smaller than $O\left(ħ\right).\mathrm{}$ We find that the quantum observables remain well approximated by classical Liouville averages over long times even for the chaotic motions of a few degree-of-freedom system. To obtain this correspondence it is not necessary to introduce the decoherence effects of a many degree-of-freedom environment.