Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals

Abstract

The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.

Editorial for Special Issue “Fractional Dynamics: Theory and Applications”

“…Iomin, in his paper "Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals" [3], considered the quantum evolution of wave functions, which is controlled by the spectrum of Lévy random matrices. By the analytical treatment of quantum recurrences and revivals in the Hilbert space, the author showed that, for chaotic systems with a uniform mixing property, the distribution of the return probability is exponential, while in systems with nonuniform mixing, the distribution of the return probability is algebraic in large recurrence times.…”