Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

Abstract

In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics. In one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, but which owing to spontaneous quantum fluctuations tunnels between nearest-neighbour sites. We consider two representative forms of interactions: stochastic reset of quantum dynamics, in which the density operator is at random times reset to its initial form, and projective measurements performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement of the particle about its initial location. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which however does not require performing measurements at exactly regular intervals that are hallmarks of such scenarios.