The quantum Levy walk

TL;DR

This paper introduces the quantum Levy walk model to analyze quantum transport and decoherence, deriving a master equation and exploring properties like quantum irreversibility, mean-square displacement, and eigenenergy behavior.

Contribution

The paper develops an analytical framework for quantum Levy walks, revealing unique decoherence features and critical behaviors in long-range quantum hopping models.

Findings

Quantum mean-square displacement can be finite or infinite depending on system parameters.

Density of states exhibits complex null-sets inside the Brillouin zone when displacement is infinite.

Critical eigenenergy behavior is linked to non-differentiability and self-affinity.

Abstract

We introduce the quantum Levy walk to study transport and decoherence in a quantum random model. We have derived from second order perturbation theory the quantum master equation for a \textit{Levy-like particle}that moves along a lattice through hopping scale-free while interacting with a thermal bath of oscillators. The general evolution of the quantum Levy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation, and the probability to be in a lattice site, all them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general our approach allows to study analytically quantum fluctuations and decoherence in a long-range hopping model.