  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.

Chemical Physics;Statistical Mechanics;Quantum Physics