Quantum walks: The mean first detected transition time

Q. Liu, R. Yin, K. Ziegler, E. Barkai

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We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψin) of the walker is orthogonal to the detected state |ψd). We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameters of the model, we find simple expressions describing the blow-up of the mean transition time. Using previous results on the fluctuations of the return time, corresponding to |ψin)=|ψd), we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.

Original languageEnglish
Article number033113
Number of pages16
JournalPhysical Review Research
Issue number3
StatePublished - 21 Jul 2020

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