Continuous Time Quantum Walks and Recurrences in the Hilbert Space

Abstract: An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincaré recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.