Two-state Markovian theory of input-output frequency and phase synchronization

TL;DR

This paper develops a Markovian framework to analyze frequency and phase synchronization in systems with two metastable states, providing explicit formulas based on escape rates for classical and quantum applications.

Contribution

It introduces a novel Markovian approach to quantify frequency and phase synchronization using a discrete phase derived from switching events.

Findings

Derived explicit formulas for output frequency and phase diffusion.

Provided a framework applicable to classical and quantum two-state systems.

Facilitated analysis of stochastic synchronization phenomena.

Abstract

A Markovian dichotomic system driven by a deterministic time-periodic force is analyzed in terms of the statistical properties of the switching events between the states. The consideration of the counting process of the switching events leads us to define a discrete phase. We obtain expressions for the instantaneous output frequency and phase diffusion associated to the dichotomic process, as well as for their cycle averages. These expressions are completely determined by the rates of escape from both states. They are a convenient starting point for the study of the stochastic frequency and phase synchronization in a wide range of situations (both classical and quantum) in which two metastable states are involved.