TY - JOUR

T1 - Quantum walks

T2 - The mean first detected transition time

AU - Liu, Q.

AU - Yin, R.

AU - Ziegler, K.

AU - Barkai, E.

N1 - Publisher Copyright:
© 2020 authors.

PY - 2020/7

Y1 - 2020/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85089874840&partnerID=8YFLogxK

U2 - 10.1103/PhysRevResearch.2.033113

DO - 10.1103/PhysRevResearch.2.033113

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AN - SCOPUS:85089874840

SN - 2643-1564

VL - 2

JO - Physical Review Research

JF - Physical Review Research

IS - 3

M1 - 033113

ER -