Projection-Based Memory Kernel Coupling Theory for Quantum Dynamics: A Stable Framework for Non-Markovian Simulations

TL;DR

The paper addresses the challenge of simulating non-Markovian open quantum dynamics with stable long-time behavior. It introduces projection-based memory kernel coupling theory (PMKCT), which stabilizes the MKCT hierarchy by spectrally decomposing the MKCT generator and projecting onto the stable subspace, ensuring asymptotic stability while preserving physical kernels. The authors provide a rigorous mathematical framework based on Mori–Zwanzig projection, derive scalable numerical schemes with two scaling options, and demonstrate that PMKCT reproduces numerically exact results for the spin-boson model by DEOM, achieving accurate memory kernels and absorption spectra without empirical tuning. The work delivers a robust, efficient approach for non-Markovian quantum dynamics with potential applications to quantum materials, chemical dynamics, and biological systems, offering reliable long-time simulations and transparent physical kernel extraction.

Abstract

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.

Figures (4)

• Figure 1: Eigenvalue spectra for the spin-boson model with $\Delta = 20$, $\gamma = 0.5$, $\omega_D = 1$, $\beta = 5$, $N=40$. Power-law scaling is used with $\Lambda=100$. (a) Original matrix $M$ shows 21 eigenvalues with $\Re(\lambda) > 0$ (crosses). (b) After projection, all eigenvalues satisfy $\Re(\lambda) \leq 0$.
• Figure 2: Time evolution for the spin-boson model (parameters as in Fig. \ref{['fig:eigenvalue_comparison']}). Solid lines: PMKCT with $N=40$. Dotted lines: Reference DEOM calculation.
• Figure 3: Convergence of $K_1(t=5)$ with truncation order $N$ in PMKCT for different values: (a) $N=10$, (b) $N=20$, (c) $N=30$, (d) $N=40$.
• Figure 4: Frequency-domain results. (a) Memory kernel magnitude. (b) Memory kernel real part. (c) Memory kernel imaginary part. (d) Absorption lineshape. Parameters as in previous figures.