Spectral properties of simple classical and quantum reset processes

Abstract

We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.

Run and tumble particle under resetting: a renewal approach

“…Recent variations on the resetting theme have been to consider: resetting of discretetime Lévy flights [37] and continuous-time Lévy walks [38,39], resetting for random walks in a bounded domain [40,39], resetting of extended systems such as fluctuating interfaces [41] and a reaction diffusion process in one dimension [42], Michaelis-Menten reaction schemes [43,35], the thermodynamics of resetting [44,45] and large deviations of the additive functionals of resetting processes [46,47,48], interaction-driven resetting [49], resetting with branching [50] and fractional Brownian motion with resetting [51]. Very recently, resetting dynamics in quantum systems have also been studied [52,53].…”

First passage under stochastic resetting in an interval

“…animal foraging, restart has been found to be an indispensable part of chemical reactions [44] and randomized computer algorithms [45]. Further progress has seen applications of restart mechanisms in biophysics [46][47][48], stochastic thermodynamics [49,50], and quantum mechanics [51,52]. In this paper, we have investigated the motion of a Brownian particle (subjected to resetting) confined in a box [a, b] in one dimension.…”

Dark states of quantum search cause imperfect detection

“…One of the general aims in this direction of research is to gain information on the statistics of this event, which is inherently random by the basic laws of quantum mechanics. Our approach explicitly incorporates repeated strong measurements into the definition of the first detected arrival and therein differs from other quantum search setups [1,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] and from the time-of-arrival problem [53][54][55][56][57][58][59][60][61].…”