The quantum first-detection problem

We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrödinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, a tight binding model on a ring exhibits critical sampling times manifested by diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like  with a superimposed oscillation, critical behavior for specific choices of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.