The Transfer Tensor Method: an Analytical Study Case

TL;DR

This paper analyzes the transfer tensor method for open quantum systems, showing its relation to the Nakajima Zwanzig equation, and applies it to a Jaynes Cummings model to identify non-Markovian behavior.

Contribution

It provides an analytical study of the transfer tensor method, deriving exact expressions for a solvable model and exploring the conditions for Markovian versus non-Markovian dynamics.

Findings

Transfer tensors converge to the memory kernel in the continuous-time limit.

Regions of enhanced non-Markovianity are identified based on system parameters.

Certain time-step choices can render the system effectively Markovian.

Abstract

The transfer tensor method is a versatile tool for analyzing and propagating general open quantum systems. It captures in a compact manner all memory effects in a non-Markovian system through a straightforward transformation of a set of dynamical maps. Transfer tensors provide the exact convolutional propagator associated with a given time discretization over the past evolution of an open quantum system. Here we show that, for any finite time discretization, the memory kernel of the Nakajima Zwanzig equation deviates from the exact transfer tensors, although both converge in the continuous-time limit, as expected. We examine this behaviour in the context of an analytically solvable model: a two level atom resonant with a lossy cavity in the Jaynes Cummings limit. The atomic dynamics separate into two decoupled degrees of freedom -- the coherence and the population inversion. We derive exact expressions for the dynamical map, the transfer tensors and the memory kernel governing the coherence, and we relate them to their counterparts for the population inversion. As a function of the ratio between the cavity loss rate and the atom-cavity coupling strength, we identify regions of enhanced non-Markovianity in which the system can be described as fully Markovian for certain time-step choices.