Exact master equations for the non-Markovian decay of a qubit

TL;DR

This paper derives exact master equations for a qubit's non-Markovian decay into a structured reservoir, providing analytical tools to compare different approaches and highlighting limitations of phenomenological models.

Contribution

It introduces exact analytical expressions for memory kernels and generators of master equations, enabling detailed analysis of non-Markovian quantum dynamics.

Findings

Exact solutions for memory kernels and generators are obtained.

Phenomenological master equations may be incompatible with microscopic models.

Comparison of perturbation expansions reveals convergence behaviors.

Abstract

Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. Employing the exact solution for the model we determine analytical expressions for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows a detailed investigation and comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, we find that the structure of widely used phenomenological master equations with memory kernel may be incompatible with a non-perturbative treatment of the underlying microscopic model. We discuss several physical implications of our results on the microscopic analysis and the phenomenological modelling of non-Markovian quantum dynamics of open systems.