Abstract The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.
Abstract The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.
The hierarchical equations of motion (HEOM) method is a powerful numerical approach to solve the dynamics and steady-state of a quantum system coupled to a non-Markovian and non-perturbative environment. Originally developed in the context of physical chemistry, it has also been extended and applied to problems in solid-state physics, optics, single-molecule electronics, and biological physics. Here we present a numerical library in Python, integrated with the powerful QuTiP platform, which implements the HEOM for both bosonic and fermionic environments. We demonstrate its utility with a series of examples. For the bosonic case, we present examples for fitting arbitrary spectral densities, modelling a Fenna-Matthews-Olsen photosynthetic complex, and simulating dynamical decoupling of a spin from its environment. For the fermionic case, we present an integrable single-impurity example, used as a benchmark of the code, and a more complex example of an impurity strongly coupled to a single vibronic mode, with applications in single-molecule electronics.
The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
The ``hierarchical equations of motion'' (HEOM) method is a powerful exact numerical approach to solve the dynamics and find the steady-state of a quantum system coupled to a non-Markovian and nonperturbative environment. Originally developed in the context of physical chemistry, it has also been extended and applied to problems in solid-state physics, optics, single-molecule electronics, and biological physics. Here we present a numerical library in Python, integrated with the powerful QuTiP platform, which implements the HEOM for both bosonic and fermionic environments. We demonstrate its utility with a series of examples consisting of benchmarks against important known results and examples demonstrating insights gained with this library for this article. For the bosonic case, our results include demonstrations of how to fit arbitrary spectral densities with different approaches, and a study of the dynamics of energy transfer in the Fenna-Matthews-Olson photosynthetic complex. For the latter, we both clarify how a suitable non-Markovian environment can protect against pure dephasing, and model recent experimental results demonstrating the suppression of electronic coherence. Importantly, we show that by combining the HEOM method with the reaction coordinate method we can observe nontrivial system-environment entanglement on timescales substantially longer than electronic coherence alone. We also demonstrate results showing how the HEOM can be used to benchmark different strategies for dynamical decoupling of a system from its environment, and show that the Uhrig pulse-spacing scheme is less optimal than equally spaced pulses when the environment's spectral density is very broad. For the fermionic case, we present an integrable single-impurity example, used as a benchmark of the code, and a more complex example of an impurity strongly coupled to a single vibronic mode, with applications to single-molecule electronics.
Abstract We report the implementation of a hierarchical equations of motion (HEOM) module within the open‐source Libra software. It includes the standard and scaled HEOM algorithms for computing the dynamics of open quantum systems interacting with a harmonic bath. The module allows the computing of the evolution of the reduced density matrix, as well as spectral lineshapes. The truncation, filtering, and “update list” schemes, as well as OpenMP parallelization, allow for further computational saving. The package is written in a mix of C++ and Python languages, delivering the best compromise between user friendliness and efficiency. The Python layer of the package takes advantage of standard Python libraries, such as h5py, which allows efficient storage and retrieval of the generated results. The package can be seamlessly used within Jupyter notebooks; its careful design shall provide the maximal convenience and intuitiveness to its users.
Several recent advancements for the hierarchical equations of motion (HEOM) approach are reported. First, we propose an a priori estimate for the optimal number of basis functions for the reservoir memory decomposition. Second, we make use of the sparsity of auxiliary density operators (ADOs) and propose two ansatzs to screen out all the intrinsic zero ADO elements. Third, we propose a new truncation scheme by utilizing the time derivatives of higher-tier ADOs. These novel techniques greatly reduce the memory cost of the HEOM approach, and thus enhance its efficiency and applicability. The improved HEOM approach is applied to simulate the coherent dynamics of Aharonov-Bohm double quantum dot interferometers. Quantitatively accurate dynamics is obtained for both noninteracting and interacting quantum dots. The crucial role of the quantum phase for the magnitude of quantum coherence and quantum entanglement is revealed.
The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.