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.

Keywords

• Lévy matrix
• Hilbert space
• Poincaré recurrences
• statistics of quantum spectrum
• almost periodic functions
• quantum revivals