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arXiv:2608.20657

Symmetry Adapted Hierarchical Equations of Motion for Exact Simulations of Large Polariton Systems

M. Elious Mondal, Pengfei Huo

physics.chem-phphysics.comp-phquant-phSubmitted Aug 21, 2026• Announced Aug 24, 2026• Revised Aug 30, 2026 (v2)
v1 · Aug 21, 2026v2 · Aug 30, 2026
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Abstract

Hierarchical equations of motion(HEOM) provide exact dynamics of open quantum systems coupled to harmonic baths, but their computational cost becomes prohibitive for systems with many independent local environments. In this work, we develop a symmetry-adapted HEOM formalism to significantly reduce the computational cost for the permutationally invariant Holstein-Tavis-Cummings (HTC) model. The method removes redundant information in two stages. First, all auxiliary density operators (ADOs) related only by relabeling identical molecules and their bath channels are replaced by a single canonical occupation-pattern representative. Second, molecules with the same local hierarchy occupation produce repeated matrix elements within each representative, allowing only the distinct complex variables to be propagated instead of the full (N + 1) $\times$ (N + 1) ADO matrices. The resulting matrix-free equations are evaluated using precomputed connections and molecular multiplicities. At fixed hierarchy depth L and number of bath correlation exponentials m, the number of canonical representatives becomes independent of the ensemble size for N $\geq$ L and the number of unique variables saturates for N $\geq$ L + 2. The formulation easily extends to multiple-exponential bath decompositions, arbitrary initial density operators, static disorders, and cavity loss. Our benchmarks reproduce conventional HEOM dynamics while requiring far fewer propagated variables and substantially less memory.

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Hierarchical equations of motion(HEOM) provide exact dynamics of open quantum systems coupled to harmonic baths, but their computational cost becomes prohibitive for systems with many independent local environments. In this work, we develop a symmetry-adapted HEOM formalism to significantly reduce the computational cost for the permutationally invariant Holstein-Tavis-Cummings (HTC) model. The method removes redundant information in two stages. First, all auxiliary density operators (ADOs) related only by relabeling identical molecules and their bath channels are replaced by a single canonical occupation-pattern representative. Second, molecules with the same local hierarchy occupation produce repeated matrix elements within each representative, allowing only the distinct complex variables to be propagated instead of the full (N + 1) ×\times× (N + 1) ADO matrices. The resulting matrix-free equations are evaluated using precomputed connections and molecular multiplicities. At fixed hierarchy depth L and number of bath correlation exponentials m, the number of canonical representatives becomes independent of the ensemble size for N ≥\geq≥ L and the number of unique variables saturates for N ≥\geq≥ L + 2. The formulation easily extends to multiple-exponential bath decompositions, arbitrary initial density operators, static disorders, and cavity loss. Our benchmarks reproduce conventional HEOM dynamics while requiring far fewer propagated variables and substantially less memory.
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.
The hierarchical equations of motion (HEOM) method has recently been widely applied to many problems of quantum dynamics in condensed phase. It is now well known that the auxiliary density operators (ADOs) in the HEOM formalism contain system-bath correlations that are important in calculating various dynamical properties, yet quantitative relations to explicitly calculate such correlations from the ADOs are still scarce. This paper extends a previous study [Q. Shi et al., J. Chem. Phys. 130, 164518 (2009)] in investigating the physical meaning of ADOs to general spectral densities and lower temperature cases. Using the path-integral technique, we derive exact relations between the expectation values of the collective bath coordinate and the ADOs, which could be very useful in investigating the correlated system-bath dynamics directly with the HEOM formalism. Numerical examples concerning the evolution of the expectation values of the collective bath coordinate are also presented.
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.
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.
The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.
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.