Quantum-classical correspondence for the equilibrium distributions of two interacting spins - art. no. 026217

We consider the quantum and classical Liouville dynamics of a nonintegrable model of two coupled spins, Initially localized quantum states spread expo nentially to the system size when the classical dynamics are chaotic. The l ong-time behavior of the quantum probability distributions and, in particul ar, the parameter-dependent rates of relaxation to the equilibrium state ar e 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 co nfigurations approach the microcanonical distribution, although the quantum equilibrium distributions exhibit characteristic "minimum" fluctuations aw ay from the microcanonical state. The magnitudes of the quantum-classical d ifferences arising from the equilibrium quantum fluctuations are studied fo r both pure and mixed (dynamically entangled) quantum states. In both cases the standard deviation of these fluctuations decreases as ((h) over bar /J )(1/2), where J is a measure of the system size. In conclusion, under a var iety of conditions the, differences between quantum and classical Liouville mechanics are shown to become vanishingly small in the classical limit (J/ (h) over bar --> infinity) a nondissipative model endowed with only a few d egrees of freedom.