Fingerprints of classical instability in open quantum dynamics

The dynamics near a hyperbolic fixed point in phase space is modeled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate ${\kappa }^{\prime }$ depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that ${\kappa }^{\prime }$ determines the rate of the von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalizes earlier results that are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.