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arXiv · 1603.02046

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

H. Friedman·D. A. Kessler·E. Barkai
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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 Schrödinger 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.

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H. Friedman, D. A. Kessler, E. Barkai. 2016-03-07. Quantum Renewal Equation for the first detection time of a quantum walk. https://doi.org/10.1088/1751-8121%2Faa5191

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