Universal Structure of Computing Moments for Exact Quantum Dynamics: Application to Arbitrary System-Bath Couplings

Abstract

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) [W. Liu, Y. Su, Y. Wang, and W. Dou, arXiv:2407.01923 (2024)]. This approach is demonstrated for arbitrary system-bath coupling that can be expressed as polynomial, $H_{SB}=\hat{V}(\hat{I}{\pm }_0+\hat{I}{\pm }_1\hat{q}+\hat{I}{\pm }_2{\hat{q}}^2+\dots )$, 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.

22 pages, 6 figures