Quantum walks: the first detected transition time

• URL: http://arxiv.org/abs/2001.00231v1
• Date: Wed, 1 Jan 2020 16:07:46 GMT
• Title: Quantum walks: the first detected transition time
• Authors: Q. Liu, R. Yin, K. Ziegler, and E. Barkai
• Abstract summary: We consider the quantum first detection problem for a particle evolving on a graph with fixed rate $1/tau$. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional space. 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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate $1/\tau$. 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 $|\psi_{\rm in}\rangle$ of the walker is orthogonal to the detected state $|\psi_{\rm d}\rangle$. 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 parameter of the model, which exhibits a blow-up of the mean transition time, we get simple expressions for the mean transition time. Using previous results on the fluctuations of the return time, corresponding to $|\psi_{\rm in}\rangle = |\psi_{\rm d}\rangle$, 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.

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