Hierarchical dynamics for system-bath coherence correlation spectrum

We propose a quasi-particle description for the hierarchical equations of motion formalism for quantum dissipative dynamics systems. Not only it provides an alternative mathematical means to the existing formalism, the new protocol clarifies also explicitly the physical meanings of the auxiliary density operators and their relations to full statistics on solvation bath variables. Combining with the standard linear response theory, we construct further the hierarchical dynamics formalism for correlated spectrum of system--bath coherence. We evaluate the spectrum matrix for a demonstrative spin-boson system-bath model. While the individual diagonal element of the spectrum matrix describes the system or the solvation bath correlation, the off-diagonal elements characterize the correlation between system and bath solvation dynamics.