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Fractal and Fractional

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

Alexander Iomin

Published Oct 18, 2021• 1 citation
Publisher

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.

DOI
10.3390/fractalfract5040171

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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.
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 Poincare 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.
Recurrence of a random walk is described by the Pólya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full revival of its quantum state. Localization for two-dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show, on the example of the 2D Grover walk, that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a four-state quantum walk. Stationary states and revivals result from interference, which has no counterpart in classical random walks.
Recurrence in the classical random walk is well known and described by the Pólya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on cycles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
We study a quantum-mechanical system, prepared, at t=0, in a model state, that subsequently decays into a sea of other states whose energy levels form a discrete spectrum with given statistical properties. An important quantity is the survival probability P(t), defined as the probability, at time t, to find the system in the original model state. Our main purpose is to analyze the influence of the discreteness and statistical properties of the spectrum on the behavior of P(t). Since P(t) itself is a statistical quantity, we restrict our attention to its ensemble average 〈P(t)〉, which is calculated analytically using random-matrix techniques, within certain approximations discussed in the text. We find, for 〈P(t)〉, an exponential decay, followed by a revival, governed by the two-point structure function of the statistical spectrum, thus giving a nonzero asymptotic value for large t's. The analytic result compares well with a number of computer simulations, over a time range discussed in the text.