Asymptotic TCL4 Generator for the Spin-Boson Model: Analytical Derivation and Benchmarking

TL;DR

This paper derives an analytical fourth-order TCL generator for the spin-boson model, improving accuracy over second-order methods and providing benchmarks against exact techniques, especially relevant for systems with moderate system-environment coupling.

Contribution

The work presents the first complete analytical derivation of the 4th-order TCL generator for the spin-boson model under specific spectral density conditions.

Findings

Fourth-order TCL generator improves accuracy over second-order approximations.

Second-order TCL overestimates non-Markovianity in certain regimes.

Benchmarking shows good agreement with exact hierarchical equations of motion.

Abstract

The spin-boson model is a widely used model for understanding the properties of a two-level open quantum system. Accurately describing its dynamics often requires going beyond the weak system-environment coupling approximation. However, calculating the higher-order generators of such a dynamics, with a system-environment coupling that is not too weak, has been known to be challenging, both numerically and analytically. This work presents the analytical derivation of the complete fourth-order time-convolutionless (TCL) generator for a generic spin-boson model, accurate up to 4th order in the system-environment coupling parameter, under the assumption that the environmental spectral density is an odd function of frequency. In the case of a semiconductor double-quantum-dot system, our results reveal corrections to the dynamics that may become physically significant in some parameter regimes. Furthermore, we report that the widely used second-order TCL master equation tends to overestimate the non-Markovianity of a dynamics over a large parameter regime. Within the regime of its applicability, our results provide a computational advantage over numerically exact techniques. The accuracy of the fourth-order TCL generator is rigorously benchmarked against specialized analytical calculations done for the Ohmic spectral density with Drude cutoff and against the numerically exact Hierarchical Equations of Motion technique.