A simple improved low temperature correction for the hierarchical equations of motion

TL;DR

This paper introduces a simple, effective low temperature correction scheme for the hierarchical equations of motion (HEOM) that enhances convergence and consistency with weak-coupling quantum master equations, demonstrated on various model systems.

Contribution

A new low temperature correction scheme for HEOM based on Zwanzig projection, improving convergence and consistency with minimal hierarchy.

Findings

Improved convergence of HEOM beyond weak-coupling limit

Restores consistency with weak-coupling quantum master equation

Easy to implement in existing HEOM codes

Abstract

The study of open system quantum dynamics has been transformed by the hierarchical equations of motion (HEOM) method, which gives the exact dynamics for a system coupled to a harmonic bath at arbitrary temperature and system-bath coupling strength. However in its standard form the method is only consistent with the weak-coupling quantum master equation at all temperatures when many auxiliary density operators are included in the hierarchy, even when low temperature corrections are included. Here we propose a new low temperature correction scheme for the termination of the hierarchy based on Zwanzig projection which alleviates this problem, and restores consistency with the weak-coupling master equation with a minimal hierarchy. The utility of the new correction scheme is demonstrated on a range of model systems, including the Fenna-Metthews-Olson complex. The new closure is found to improve convergence of the HEOM even beyond the weak-coupling limit and is very straightforward to implement in existing HEOM codes.