Nakajima-Zwanzig versus time-convolutionless master equation for the non-Markovian dynamics of a two-level system

TL;DR

This paper compares the exact non-Markovian dynamics of a two-level system using Nakajima-Zwanzig and time-convolutionless master equations, revealing differences in operator structures influenced by the environment.

Contribution

It provides a non-perturbative derivation of both master equations for a two-level system, highlighting their structural differences and environmental dependence.

Findings

Operator contributions in TCL are multiplied by time-dependent functions.

Memory kernels in NZ are convoluted with operator contributions.

Operator structures vary significantly with environmental states.

Abstract

We consider the exact reduced dynamics of a two-level system coupled to a bosonic reservoir, further obtaining the exact time-convolutionless and Nakajima-Zwanzig non-Markovian equations of motion. The considered system includes the damped and undamped Jaynes-Cummings model. The result is obtained by exploiting an expression of quantum maps in terms of matrices and a simple relation between the time evolution map and time-convolutionless generator as well as Nakajima-Zwanzig memory kernel. This non-perturbative treatment shows that each operator contribution in Lindblad form appearing in the exact time-convolutionless master equation is multiplied by a different time dependent function. Similarly, in the Nakajima-Zwanzig master equation each such contribution is convoluted with a different memory kernel. It appears that depending on the state of the environment the operator structures of the two set of equations of motion can exhibit important differences.