Collective bath coordinate mapping of "hierarchy" in hierarchical equations of motion

TL;DR

This paper introduces a new collective bath coordinate mapping for hierarchical equations of motion (HEOM), transforming the hierarchy into a continuous space to improve computational efficiency and stability, especially under strong system-bath coupling.

Contribution

The authors develop a collective bath coordinate representation of HEOM, enabling more stable and efficient simulations of open quantum systems with strong system-bath interactions.

Findings

The new method provides a rigorous time evolution of bath coordinate distribution.

It is more stable and efficient than traditional HEOM methods.

Demonstrated effectiveness on vibronic system models.

Abstract

The theory of hierarchical equations of motion (HEOM) is one of the standard methods to give exact evaluations of the dynamics as coupled to harmonic oscillator environments. However, the theory is numerically demanding due to its hierarchy, which is the set of auxiliary elements introduced to capture the non-Markovian and non-perturbative effects of environments. When system-bath coupling becomes relatively strong, the required computational resources and precision move beyond the regime that can be currently handled. This article presents a new representation of HEOM theory in which the hierarchy is mapped into a continuous space of a collective bath coordinate and several auxiliary coordinates as the form of the quantum Fokker-Planck equation. This representation gives a rigorous time evolution of the bath coordinate distribution and is more stable and efficient than the original HEOM theory, particularly when there is a strong system-bath coupling. We demonstrate the suitability of this approach to treat vibronic system models coupled to environments.