Quantum walks: The mean 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 |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.