Non-Markovian dynamics of a qubit coupled to an Ising spin bath

TL;DR

This paper compares various master equations for a qubit coupled to an Ising spin bath, showing that time-convolutionless and post-Markovian approaches outperform traditional Markovian methods, especially at long times.

Contribution

It provides an analytical comparison of master equations against an exact solution for a qubit in a spin bath, highlighting the effectiveness of non-Markovian approaches.

Findings

Time-convolutionless master equation performs well up to fourth order.

Markovian approaches are inadequate due to infinite bath correlation time.

Post-Markovian master equation with a proper memory kernel outperforms others at long times.

Abstract

We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin-baths.