Return to origin problem for particle on a one-dimensional lattice with quasi-Zeno dynamics

In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. The dynamics is non-unitary and, in recent work, it has been shown that it can be effectively described by two different non-Hermitian Hamiltonians. Here we explore this connection by considering the problem of detecting a free quantum particle moving on a one-dimensional lattice, where the detector is placed at the origin and the particle is initially located at some specified lattice point. We find that results for distribution times for the first detection probability, obtained from the non-Hermitian Hamiltonians, are in excellent agreement with known exact results as well as exact numerics. Interesting finite-size effects are discussed. We also study the first detection problem for the example of a particle moving in a quasi-periodic potential, an example where the unperturbed particle's motion can be ballistic, localized or diffusive.