Spectral properties of simple classical and quantum reset processes

TL;DR

This paper analyzes how random resets affect the spectral properties and relaxation dynamics of classical and quantum Markov processes, revealing a uniform eigenvalue shift that influences stationary states and metastability.

Contribution

It provides a unified spectral analysis of classical and quantum reset processes, deriving explicit expressions for stationary states and currents, and exploring effects on metastable systems.

Findings

Reset causes a uniform eigenvalue shift in the generator.

Resets can accelerate or induce relaxation to stationarity.

Results demonstrated on classical and quantum models, including metastable systems.

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.