Multifractality in monitored single-particle dynamics

TL;DR

This paper investigates the emergence and robustness of multifractal behavior in single-particle dynamics under repeated measurements in both quantum and classical systems, revealing how measurement protocols influence multifractality.

Contribution

It introduces a comprehensive analysis of multifractality in monitored single-particle systems, including numerical simulations and simplified models for analytical insights.

Findings

Multifractal properties appear in the ensemble of wave functions or probability distributions.

Multifractality is robust to measurement rate and protocol variations.

Generalized measurements with errors or postselection can destroy multifractality.

Abstract

We study multifractal properties in time evolution of a single particle subject to repeated measurements. For quantum systems, we consider circuit models consisting of local unitary gates and local projective measurements. For classical systems, we consider models for estimating the trajectory of a particle evolved under local transition processes by partially measuring particle occupations. In both cases, multifractal behaviors appear in the ensemble of wave functions or probability distributions conditioned on measurement outcomes after a sufficiently long time. While the nature of particle transport (diffusive or ballistic) qualitatively affects the multifractal properties, they are even quantitatively robust to the measurement rate or specific protocols. On the other hand, multifractality is generically lost by generalized measurements allowing erroneous outcomes or by postselection of the outcomes with no particle detection. We demonstrate these properties by numerical simulations and also propose several simplified models, which allow us to analytically obtain multifractal properties in the monitored single-particle systems.