Quantum Two-State Dynamics Driven by Stationary Non-Markovian Discrete Noise: Exact Results

TL;DR

This paper derives exact analytical and numerical results for the averaged quantum two-state relaxation dynamics driven by non-Markovian discrete noise, extending previous Markovian models to more general non-Markovian cases with complex residence time distributions.

Contribution

It provides the first explicit exact solutions for quantum relaxation under non-Markovian discrete noise with arbitrary residence time distributions, including non-exponential and infinite-range autocorrelation cases.

Findings

Exact Laplace-transformed relaxation dynamics derived for non-Markovian noise.

Recovered known Markovian results as a special case.

Numerical solutions for biexponential and long-range autocorrelation noise cases.

Abstract

We consider the problem of stochastic averaging of a quantum two-state dynamics driven by non-Markovian, discrete noises of the continuous time random walk type (multistate renewal processes). The emphasis is put on the proper averaging over the stationary noise realizations corresponding, e.g., to a stationary environment. A two state non-Markovian process with an arbitrary non-exponential distribution of residence times (RTDs) in its states with a finite mean residence time provides a paradigm. For the case of a two-state quantum relaxation caused by such a classical stochastic field we obtain the explicit exact, analytical expression for the averaged Laplace-transformed relaxation dynamics. In the limit of Markovian noise (implying an exponential RTD), all previously known results are recovered. We exemplify new more general results for the case of non-Markovian noise with a biexponential RTD. The averaged, real-time relaxation dynamics is obtained in this case by numerically exact solving of a resulting algebraic polynomial problem. Moreover, the case of manifest non-Markovian noise with an infinite range of temporal autocorrelation (which in principle is not accessible to any kind of perturbative treatment) is studied, both analytically (asymptotic long-time dynamics) and numerically (by a precise numerical inversion of the Laplace-transformed averaged quantum relaxation).