Quantum-Classical Correspondence for the Equilibrium Distributions of Two Interacting Spins

Abstract

We consider the quantum and classical Liouville dynamics of a non-integrable model of two coupled spins. Initially localised quantum states spread exponentially to the system dimension when the classical dynamics are chaotic. The long-time behaviour of the quantum probability distributions and, in particular, the parameter-dependent rates of relaxation to the equilibrium state are surprisingly well approximated by the classical Liouville mechanics even for small quantum numbers. As the accessible classical phase space becomes predominantly chaotic, the classical and quantum probability equilibrium configurations approach the microcanonical distribution, although the quantum equilibrium distributions exhibit characteristic `minimum' fluctuations away from the microcanonical state. The magnitudes of the quantum-classical differences arising from the equilibrium quantum fluctuations are studied for both pure and mixed (dynamically entangled) quantum states. In both cases the standard deviation of these fluctuations decreases as $(\hbar/{\mathcal J})^{1/2}$, where ${\mathcal J}$ is a measure of the system size. In conclusion, under a variety of conditions the differences between quantum and classical Liouville mechanics are shown to become vanishingly small in the classical limit (${\mathcal J}/\hbar \to \infty$) of a non-dissipative model endowed with only a few degrees of freedom.