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arXiv:2412.13793 · Journal of Chemical Theory and Computation

Discretization of Structured Bosonic Environments at Finite Temperature by Interpolative Decomposition: Theory and Application

H. Takahashi, Raffaele Borrelli

Published Feb 17, 2025• 6 citations
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Abstract

2024, 161, 151101]. The approach leverages a low-rank decomposition of the Fourier-transform relation connecting the bath correlation function to its spectral density. By capturing the time, frequency, and temperature dependencies encoded in the spectral density-autocorrelation function relation, our method significantly reduces the degrees of freedom required for simulating open quantum system dynamics. We benchmark our approach against existing methods and demonstrate its efficacy through applications to both simple models and a realistic electron transfer process in biological systems. Additionally, we show that this new approach can be effectively combined with the tensor-train formalism to investigate the quantum dynamics of systems interacting with complex non-Markovian environments. Finally, we provide a perspective on the selection and application of various spectral density discretization techniques.

DOI
10.1021/acs.jctc.4c01728

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The accurate description of the interaction of a quantum system with its environment is a challenging problem ubiquitous across all areas of physics and lies at the foundation of quantum mechanics theory. Here, we pioneer a new strategy to create discrete low-rank models of the system-environment interaction, by exploiting the frequency and time domain information encoded in the fluctuation-dissipation relation connecting the system-bath correlation function and the spectral density. We demonstrate the effectiveness of our methodology by combining it with tensor-network methodologies and simulating the quantum dynamics of complex excitonic systems in a highly structured bosonic environment. The new modeling framework sets the basis for a leap in the analysis of open quantum systems, providing controlled accuracy at significantly reduced computational costs, with benefits in all connected research areas.
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We present a streamlined implementation of a computational framework for constructing and analyzing reduced dynamical maps for complex system--bath models at finite temperature. The methodology is based on three established ingredients of quantum dynamics: the Choi--Jamiołkowski isomorphism for the representation of quantum channels, thermofield (TFD) purification of thermal environments, and tensor-train (TT) propagation of the resulting enlarged pure state. The reduced map is obtained from a single unitary propagation in a thermofield-doubled Hilbert space and represented in matrix form through the Choi--Jamiołkowski isomorphism. The TFD evolution is implemented in the TT representation, enabling efficient propagation of high-dimensional purified thermal states. We illustrate the methodology for exciton transfer in the Fenna--Matthews--Olson complex with site-dependent structured spectral densities represented by discretized bosonic environments. The resulting maps are used to analyze decoherence, relaxation, and finite-memory effects, and to assess the crossover to an effectively time-local description. The proposed approach provides a route to compute reduced propagators and to post-process them into memory kernels, transfer tensors, and effective kinetic rate descriptions for complex molecular systems.
This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.
Two different numerically exact methods for open quantum system dynamics, the hierarchy of pure states (HOPS) method, and the multi-Davydov-Ansatz are discussed. We focus on the suitability of the underlying representations of bath correlations. While in the HOPS case the correct description of the bath correlation function (BCF) in the time domain is decisive, it turns out that a windowed Fourier transform of the BCF is an appropriate indicator of the quality of the discretization in the multi-Davydov-Ansatz. For the spin-boson model with sub-Ohmic spectral density considered here, a discretization of the spectral density based on an exponential distribution, used previously, turns out to be most favorable.
In open-system approaches with non-Markovian environments, the process of inserting an individual mode (denoted as ``pseudomode'') into the bath or extracting it from the bath is widely employed. This procedure, however, is typically performed on basis of the spectral density and does not incorporate temperature. Here, we show how the (temperature-dependent) bath correlation function (BCF) transforms in such a process. We present analytic formulas for the transformed BCF and numerically study the differences between factorizing initial state and global thermal (correlated) initial state of mode and bath, respectively. We find that in the regime of strong coupling of the mode to both system and bath, the differences in the BCFs give rise to pronounced differences in the dynamics of the system.