Quantum Renewal Equation for the first detection time of a quantum walk

TL;DR

This paper develops a quantum renewal equation to analyze first detection times in quantum walks, revealing critical sampling times, diverging mean detection times, and quantum Zeno effects, advancing understanding of quantum measurement dynamics.

Contribution

It introduces a quantum renewal equation linking wave function evolution and detection statistics, uncovering critical phenomena and optimal measurement strategies in quantum walks.

Findings

Discovery of critical sampling times in quantum walks

Divergence of mean first detection time at certain sampling rates

Identification of quantum Zeno effect at small sampling intervals

Abstract

We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schr\"odinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like $(\mbox{time})^{-3}$ with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.