Generalization of the hierarchical equations of motion theory for efficient calculations with arbitrary correlation functions

The hierarchical equations of motion (HEOM) theory is one of the standard methods to rigorously describe open quantum dynamics coupled to harmonic environments. Such a model is used to capture non-Markovian and non-perturbative effects of environments appearing in ultra-fast phenomena. In the regular framework of the HEOM theory, the environment correlation functions are restricted into linear combinations of exponential functions. In this article, we present a new formulation of the HEOM theory including treatments of non-exponential correlation functions, which enables us to describe general environmental effects more efficiently and stably than the original theory and other generalizations. The library and its Python binding we developed to perform simulations based on our approach, named LibHEOM and PyHEOM respectively, are provided as supplementary material.