Tight-binding model subject to conditional resets at random times

TL;DR

This paper studies a quantum tight-binding model with a novel conditional resetting protocol at random times, revealing how resets influence localization, spatial profiles, and the emergence of stationary states under various distributions.

Contribution

It introduces a quantum resetting scheme with probabilistic, configuration-dependent resets and analyzes its effects on localization and stationary states in a tight-binding model.

Findings

System relaxes to a localized stationary state under exponential resetting.

Reset site choice influences spatial probability distribution and asymmetry.

Stationary state emergence depends on the finiteness of the reset interval distribution's mean.

Abstract

We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask: what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbour sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a set-up that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.