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

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.