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The Journal of Chemical Physics

Stable memory kernel coupling theory for quantum dynamics: Projection-based and continued fraction methods

Wei Liu, Ruihao Bi, Y W Su, Limin Xu, Zhennan Zhou, Yue Wang, Wenjie Dou

Published May 6, 2026• 1 citation
Publisher

Abstract

We introduce two complementary formulations within memory kernel coupling theory (MKCT) for non-Markovian quantum dynamics: a projection-based method (PMKCT) in the time domain and a continued fraction representation (CF-MKCT) in the frequency domain. The PMKCT operates on the matrix representation of MKCT and enforces asymptotic stability by removing unstable spectral components through orthogonal projection. The CF-MKCT yields a rapidly convergent representation, achieving high accuracy with only N ∼ 8 moments while preserving numerical stability by construction; the inverse Fourier transformation back to the time domain naturally yields a stable solution and converges rapidly with relatively few moments. Together, these two formulations provide a stable, accurate, and versatile framework for simulating non-Markovian quantum dynamics. Benchmark calculations on the spin-boson model with Ohmic spectral densities show excellent agreement with numerically exact results.

DOI
10.1063/5.0327266

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We introduce two complementary formulations within memory kernel coupling theory (MKCT) for non-Markovian quantum dynamics: a projection-based method (PMKCT) in the time domain and a continued fraction representation (CF-MKCT) in the frequency domain. The PMKCT operates on the matrix representation of MKCT and enforces asymptotic stability by removing unstable spectral components through orthogonal projection. The CF-MKCT yields a rapidly convergent representation, achieving high accuracy with only N ∼ 8 moments while preserving numerical stability by construction; the inverse Fourier transformation back to the time domain naturally yields a stable solution and converges rapidly with relatively few moments. Together, these two formulations provide a stable, accurate, and versatile framework for simulating non-Markovian quantum dynamics. Benchmark calculations on the spin-boson model with Ohmic spectral densities show excellent agreement with numerically exact results.
We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.
The generalized quantum master equation provides a powerful framework for non-Markovian dynamics of open quantum systems. However, the accurate and efficient evaluation of the memory kernel remains a challenge. In this work, we introduce a comprehensive tensorial extension to the Memory Kernel Coupling Theory (MKCT) to overcome this bottleneck. By elevating the original scalar formalism to a tensorial framework, the extended MKCT enables the calculation of general expectation values and cross-correlation functions. We demonstrate the numerical accuracy and efficiency of this method across multiple benchmark systems: capturing transient populations and coherences in the spin-boson model, resolving the excitonic absorption spectrum of the Fenna-Matthews-Olson complex, and simulating charge mobility in one-dimensional lattice models. These successful applications establish the tensorial MKCT as a highly efficient tool for investigating complex dynamics in open quantum systems.
Dynamical observables can often be described by time correlation functions (TCFs). However, efficiently calculating TCFs for complex quantum systems is a significant challenge, which generally requires solving the full dynamics of the systems. This Letter presents the memory kernel coupling theory (MKCT), a general formalism for evaluating TCFs. The MKCT builds upon Mori's memory kernel formalism for TCFs. Our theory further decomposes the memory kernel into auxiliary kernels. Rapid decay of auxiliary kernels allows us to truncate the coupled equations of motion with high accuracy. Notably, only higher-order moments are sufficient as the input for obtaining TCFs. While this formalism is general, we carry out the numerical demonstration for typical open quantum systems-the spin-boson model and the single impurity Anderson model.
The transfer tensor method is a versatile tool for analyzing and propagating general open quantum systems. It captures in a compact manner all memory effects in a non-Markovian system through a straightforward transformation of a set of dynamical maps. Transfer tensors provide the exact convolutional propagator associated with a given time discretization over the past evolution of an open quantum system. Here we show that, for any finite time discretization, the memory kernel of the Nakajima Zwanzig equation deviates from the exact transfer tensors, although both converge in the continuous-time limit, as expected. We examine this behaviour in the context of an analytically solvable model: a two level atom resonant with a lossy cavity in the Jaynes Cummings limit. The atomic dynamics separate into two decoupled degrees of freedom -- the coherence and the population inversion. We derive exact expressions for the dynamical map, the transfer tensors and the memory kernel governing the coherence, and we relate them to their counterparts for the population inversion. As a function of the ratio between the cavity loss rate and the atom-cavity coupling strength, we identify regions of enhanced non-Markovianity in which the system can be described as fully Markovian for certain time-step choices.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
Dynamical observables can often be described by time correlation functions (TCFs). However, efficiently calculating TCFs for complex quantum systems is a significant challenge, which generally requires solving the full dynamics of the systems. This Letter presents the memory kernel coupling theory (MKCT), a general formalism for evaluating TCFs. The MKCT builds upon Mori's memory kernel formalism for TCFs. Our theory further decomposes the memory kernel into auxiliary kernels. Rapid decay of auxiliary kernels allows us to truncate the coupled equations of motion with high accuracy. Notably, only higher-order moments are sufficient as the input for obtaining TCFs. While this formalism is general, we carry out the numerical demonstration for a typical open quantum system--the spin-boson model.