  In this work we construct a novel dissipaton-equation-of-motion (DEOM) theory
in quadratic bath coupling environment, based an extended algebraic statistical
quasi-particle approach. To validate the new ingredient of the underlying
dissipaton algebra, we derive an extended Zusman equation via a totally
different approach. We prove that the new theory, if it starts with the
identical setup, constitutes the dynamical resolutions to the extended Zusman
equation. Thus, we verify the generalized (non-Gaussian) Wick's theorem with
dissipatons-pair added. This new algebraic ingredient enables the dissipaton
approach being naturally extended to nonlinear coupling environments. Moreover,
it is noticed that, unlike the linear bath coupling case, the influence of a
non-Gaussian environment cannot be completely characterized with the linear
response theory. The new theory has to take this fact into account. The
developed DEOM theory manifests the dynamical interplay between dissipatons and
nonlinear bath coupling descriptors that will be specified. Numerical
demonstrations will be given with the optical line shapes in quadratic coupling
environment.
