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The hierarchical equations of motion (HEOM) approach can describe the reduceddynamics of a system simultaneously coupled to multiple bosonic and fermionicenvironments. The complexity of exactly describing the system-environmentinteraction with the HEOM method usually results in time-consuming calculationsand a large memory cost. Here, we introduce an open-source software packagecalled HierarchicalEOM.jl: a Julia framework integrating the HEOM approach.HierarchicalEOM.jl features a collection of methods to compute bosonic andfermionic spectra, stationary states, and the full dynamics in the extendedspace of all auxiliary density operators (ADOs). The required handling of theADOs multi-indexes is achieved through a user-friendly interface. We exemplifythe functionalities of the package by analyzing a single impurity Andersonmodel, and an ultra-strongly coupled charge-cavity system interacting withbosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significantspeedup with respect to the corresponding method in the Quantum Toolbox inPython (QuTiP), upon which this package is founded.

HierarchicalEOM.jl: An efficient Julia framework for hierarchical equations of motion in open quantum systems