On the Markov evolution of the $\rho$-matrix of a subsystem

TL;DR

This paper investigates how the reduced density matrix of a subsystem in a chain of oscillators deviates from Markovian behavior, revealing dependencies on system size and coupling strength through numerical analysis.

Contribution

It provides a numerical study of the non-Markovian deviations in the evolution of a subsystem's density matrix in a chain of oscillators, highlighting the impact of system size and coupling.

Findings

Deviations depend strongly on system size and coupling.

In optimal conditions, deviations decrease to about 3%.

In other cases, deviations can exceed 30%.

Abstract

Evolution of the reduced density matrix for a subsystem is studied to determine deviations from its Markov character for a system consisting of a closed chain of $N$ oscillators with one of them serving as a subsystem. The dependence on $N$ and on the coupling of the two subsystems is investigated numerically. The found deviations strongly depend on $N$ and the coupling. In the most beneficial case with $N-1=100$ and the coupling randomized in its structure the deviations fall with the evolution time up 3\%. In other cases they remain to be of the order 30\% or even more.