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The theory of hierarchical equations of motion (HEOM) is one of the standardmethods to give exact evaluations of the dynamics as coupled to harmonicoscillator environments. However, the theory is numerically demanding due toits hierarchy, which is the set of auxiliary elements introduced to capture thenon-Markovian and non-perturbative effects of environments. When system-bathcoupling becomes relatively strong, the required computational resources andprecision move beyond the regime that can be currently handled. This articlepresents a new representation of HEOM theory in which the hierarchy is mappedinto a continuous space of a collective bath coordinate and several auxiliarycoordinates as the form of the quantum Fokker-Planck equation. Thisrepresentation gives a rigorous time evolution of the bath coordinatedistribution and is more stable and efficient than the original HEOM theory,particularly when there is a strong system-bath coupling. We demonstrate thesuitability of this approach to treat vibronic system models coupled toenvironments.

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