Quantum unitary evolution interspersed with repeated non-unitary interactions at random times

What happens when the unitary evolution of a generic closed
quantum system is interrupted at random times with non-unitary evolution
due to interactions with either the external environment or a measuring
apparatus? For interactions due to projective measurements at random
times, we show signatures of universal decay laws of survival probability
of an initial state irrespective of discrete or continuous time evolution.
We consider quantum random walk for the discrete and tight-binding model
for the continuous time evolution. We then adduce a general framework to
obtain the average density operator of a generic quantum system
experiencing any form of non-unitary interaction. We provide two explicit
applications in the context of the tight-binding model for two
representative forms of interactions: (i) stochastic resets and (ii)
projective measurements at random times. For the resetting case, our exact
results show how the particle is localized on the sites at long times,
leading to a time-independent mean-squared displacement. For the
projective measurement case, repeated projection to the initial state
results in an effective suppression of the temporal decay in the
probability of the particle being in the initial state. The amount of
suppression is comparable to the one in conventional Zeno effect
scenarios, where measurements are performed at regular intervals.