Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments

TL;DR

This work introduces TTN-HEOM, a tensor-network–based reformulation of the bexcitonic hierarchical equations of motion to efficiently simulate driven open quantum dynamics in highly structured bosonic baths. By representing the extended density operator with a tree tensor network and deriving TDVP-based master equations for the TTN cores, the method curbs the exponential growth with bath features while preserving accuracy. The authors implement three propagation strategies (direct integration, PS1, PS2) in the Python package TENSO and demonstrate accurate dynamics for a two-level system coupled to a Drude–Lorentz plus eight Brownian oscillators bath, highlighting non-Markovian features and convergence to standard HEOM. The approach offers scalable, accurate modeling of chemically realistic environments and has broad implications for quantum dynamics, spectroscopy, and device design where structured baths are essential.

Abstract

We introduce an efficient method TTN-HEOM for exactly calculating the open quantum dynamics for driven quantum systems interacting with highly structured bosonic baths by combining the tree tensor network (TTN) decomposition scheme to the bexcitonic generalization of the numerically-exact hierarchical equations of motion (HEOM). The method yields a series of quantum master equations for all core tensors in the TTN that efficiently and accurately capture the open quantum dynamics for non-Markovian environments to all orders in the system-bath interaction. These master equations are constructed based on the time-dependent Dirac--Frenkel variational principle which isolates the optimal dynamics for the core tensors given the TTN ansatz. The dynamics converges to the HEOM when increasing the rank of the core tensors, a limit in which the TTN ansatz becomes exact. We introduce TENSO, Tensor Equations for Non-Markovian Structured Open systems, as a general-purpose Python code to propagate the TTN-HEOM dynamics. We implement three general propagators for the coupled master equations: Two fixed-rank methods that require a constant memory footprint during the dynamics, and one adaptive-rank method with variable memory footprint controlled by the target level of computational error. We exemplify the utility of these methods by simulating a two-level system coupled to a structured bath containing one Drude--Lorentz component and eight Brownian oscillators, which is beyond what can presently be computed using the standard HEOM. Our results show that the TTN-HEOM is capable to simulate both dephasing and relaxation dynamics of driven quantum system interacting with structured baths, even those of chemical complexity, with affordable computational cost.

Figures (22)

• Figure 1: The topological structure of (a) a tensor train with $K$ bexcitons for the EDO, (b) a perfectly balanced tensor tree with $16$ bexcitons, and (c) a balanced tensor tree with $20$ bexcitons. TTN-HEOM admits all these topologies with core tensors of variable order; the figure focuses on order-3.
• Figure 2: Bath spectral density $J(\omega)$ describing electron-nuclear interactions for thymine nucleotide in room temperature waterGustin2023 with broadenings $\gamma_b = 50~{\mathrm{cm}^{-1}}$ for each Brownian oscillator.
• Figure 3: TTN-HEOM dynamics captured by different propagation methods for the two-level system in Eq. \ref{['eq:qubit']} interacting with a highly structured thermal environment (Fig. \ref{['fig:sd']}) using the balanced tree in Fig. \ref{['fig:tn']}c. Different colors denote varying model parameters. The direct (solid line) and PS1 (crosses) integration use a rank of $60$ for all tensors. The mixed propagator (PS2 $\to$ direct, circles) starts with PS2 propagator with an initial rank $3$ and switches to direct integration when the adaptive rank grows beyond $60$. Note that the convergence and stability of the dynamics for all model parameters and integrators.
• Figure 4: Computation time of each propagation strategy in Fig. \ref{['fig:ps2']} on 8 cores of an Intel Xeon Gold 6330 Processor. Each bar shows the computation time of different propagation strategies for the initial 100 fs of dynamics (left axis). The dashes mark the computation time for the PS2 segment ( $\sim 2$ fs) of the PS2 $\to$ direct method (right axis).
• Figure 5: TTN-HEOM dynamics for varying maximum rank thresholds for the system and TTN in Fig. \ref{['fig:ps2']} using the mixed propagator (PS2 $\to$ direct). The propagator starts with PS2 with an initial rank $3$ and switches to direct when one of the adaptive ranks grows beyond $80$ (solid lines), $60$ (crosses) and $40$ (circles).

Theorems & Definitions (1)

• proof