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

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.