Universal structure of computing moments for exact quantum dynamics: Application to arbitrary system-bath couplings

We introduce a general procedure for computing higher-order moments of correlation functions in open quantum systems, extending the scope of our recent work on Memory Kernel Coupling Theory (MKCT) [Liu et al., arXiv:2407.01923 (2024)]. This approach is demonstrated for arbitrary system-bath coupling that can be expressed as a polynomial, HSB=V̂(α0+α1q̂+α2q̂2+⋯) , where we show that the recursive commutators of a system operator obey a universal hierarchy. Exploiting this structure, the higher-order moments are obtained by evaluating the expectation values of the system and bath operators separately, with bath expectation values derived from the derivatives of a generating function. We further apply MKCT to compute the dipole autocorrelation function for the spin-boson model with both linear and quadratic coupling, achieving agreement with the hierarchical equations of motion approach. Our findings suggest a promising path toward accurate dynamics for complex open quantum systems.