Quantum random walk and tight-binding model subject to projective measurements at random times

What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements the initial state at random times? A question of general interest is: How does survival probability $S_m$, namely, that an survives even after $m$ number measurements, behave as function $m$? We address these issues context two paradigmatic systems, one, walk evolving discrete time, and other, tight-binding model continuous with both defined on one-dimensional periodic lattice finite sites $N$. For models, we present several numerical analytical results hint curious nature measurement dynamics. In particular, unveil every continues projected component instantaneous state, average typical decay exponential for large $m$. By contrast, if leftover component, what remains has been performed, exhibits characteristic values, $m_1^\star(N) \sim N$ $m_2^\star(N) N^\delta$ $\delta >1$. These scales are such (i) satisfying $m < m_1^\star(N)$, $m^{-2}$, (ii) \ll m m_2^\star(N)$, $m^{-3/2}$, while (iii) \gg exponential. find our hold independently choice distribution times between successive have corroborated by wide range distributions. This fact hints robustness ubiquity derived results.