Quantum walks: the first detected transition time
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 $|ψ_{\rm in}\rangle$ of the walker is orthogonal to the detected state $|ψ_{\rm d}\rangle$. 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 $|ψ_{\rm in}\rangle = |ψ_{\rm d}\rangle$, 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.