Wigner function statistics of quantum maps

Martin Horvat

Faculty of Mathematics and Physics
University of Ljubljana, Slovenia

We studied statistical properties of Wigner functions $W(x)$ of 1D quantum maps on compact phase space of finite area $V$. For this purpose we defined a Wigner function probability distribution $P(w) = 1/V \int \delta (w-W(x)) dx$, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of $W(x)$, starting from a coherent state.

We have shown that for a classically chaotic quantum counterpart the distribution $P(w)$ becomes a Gaussian distribution that is determinated by the first two moments. The numerical studies were done on the quantum sawtooth map and the quantized kicked top. In a quantum system with Hilbert space dimension $N(\sim 1/\hbar)$ the transition of $P(w)$ to a Gaussian distribution was observed at times $t\propto \log N$. In addition, it has been shown, that the statistics of Wigner functions of propagator eigenstates is a Gaussian as well in classically fully chaotic regime.