Comparing two evidences of quantum chaos

We compute the evolution of Wigner distribution in the case of dissipation in a classical chaotic system to analyze the quantum chaos. We find that the quantum chaos can be denoted in two ways. First, we can use the evolutions of χ²(t) and h(ρ^W, t) to denote quantum chaos: in regular case, χ²(t) and h(ρ^W, t) oscillate with constant amplitude; in chaotic case, they exponentially increase to equilibrium values. Second, when the coupling of environment exists, we can also use the evolution the Renyi entropy to denote chaos: in regular case, Renyi entropy decreases linearly; in chaotic case, Renyi entropy decreases exponentially. Our results show that besides coupling, quantum coherence can also be disturbed by chaos.