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arXiv:2112.09861 · The Journal of Chemical Physics

Collective bath coordinate mapping of “hierarchy” in hierarchical equations of motion

Tatsushi Ikeda, Akira Nakayama

Published Feb 9, 2022• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0082936

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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.
An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.
The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.
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 ultrafast phenomena. In the regular framework of the HEOM theory, the environment correlation functions are restricted to linear combinations of exponential functions. In this article, we present a new formulation of the HEOM theory including treatment 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 the supplementary material.
The hierarchical equations of motion (HEOM) approach is one of the most powerful methods avail- able to simulate the dynamics of open quantum systems. Although the HEOM approach has been shown to be stable and efficient in many parameter regimes and physical scenarios, it is also well known that the unavoidable finite truncation of the hierarchy can lead to fundamental numeri- cal instabilities for bosonic environments. In this work, we apply recently developed techniques to analyze these issues. We use an auxiliary Fock-space picture of the hierarchy to first identify the unbalanced hierarchy-raising terms responsible for the strong transient amplification associated with the non-normal algebraic structure of the HEOM generator. We then apply a non-unitary similarity transformation that converts these terms into balanced combinations of hierarchy-raising and hierarchy-lowering terms, thereby suppressing their numerical impact. We first demonstrate the efficacy of this approach via a spin-boson model with a simple Brownian oscillator spectral den- sity and subsequently extend the analysis to a structured spectral density, showing for both cases that the transformed HEOM approach allows simulations of much larger system-bath coupling than would otherwise be possible.
We develop a quantum-classical hierarchical equations of motion (QC-HEOM) approach for simulating non-Markovian open quantum systems. The method combines the ensemble-averaged classical path reference of the quantum-classical path integral formalism with a hierarchy of auxiliary quantum influence functionals. By incorporating thermal fluctuations through an ensemble average over reference trajectories, the hierarchy is required to represent only the residual quantum memory associated with the imaginary part of the bath response function. Consequently, unlike conventional hierarchical equations of motion, QC-HEOM does not require Matsubara or Padé expansions of the thermal kernel and exhibits only weak temperature dependence of the hierarchy size. Furthermore, because thermal fluctuations are supplied through reference classical trajectories, the framework naturally extends beyond harmonic baths and enables the incorporation of anharmonic and molecular environments through externally generated trajectories. We derive the formalism and demonstrate its exactness for a harmonic bath. Applications to an asymmetric spin-boson model and the seven-site Fenna--Matthews--Olson complex illustrate the accuracy of QC-HEOM. It reproduces benchmark quasi-adiabatic path integral and hierarchical equations of motion results while requiring substantially fewer auxiliary objects, particularly at low temperatures. These results establish QC-HEOM as an efficient framework for treating residual quantum memory in quantum-classical descriptions of open-system dynamics. The separation of thermal fluctuations from residual quantum memory through the use of Wigner trajectories provides an approximate route toward hierarchical treatments of complex anharmonic environments that are inaccessible to conventional HEOM approaches.
For a broad class of quantum models of practical interest, we demonstrate that the Hamiltonian of the system nonlinearly coupled to a harmonic bath through the system and bath coordinates can be equivalently mapped into the Hamiltonian of the system bilinearly coupled to the bath through the system and bath momenta. We show that the Hamiltonian with bilinear system-bath momentum coupling can be treated by the hierarchical equations-of-motion (HEOM) method and present the corresponding proof-of-principle simulations. The developed methodology creates the opportunity to scrutinize a new family of nonlinear quantum systems by the numerically accurate HEOM method.