  In this paper, we present a comprehensive account of quantum dissipation
theories with the quadratic environment couplings. The theoretical development
includes the Brownian solvation mode embedded hierarchical quantum master
equations, a core-system hierarchy construction that verifies the extended
dissipaton equation of motion (DEOM) formalism [R. X. Xu et al., J. Chem. Phys.
148, 114103 (2018)]. Developed are also the quadratic imaginary-time DEOM for
equilibrium and the {\lambda} (t)-DEOM for nonequilibrium thermodynamics
problems. Both the celebrated Jarzynski equality and Crooks relation are
accurately reproduced, which in turn confirms the rigorousness of the extended
DEOM theories. While the extended DEOM is more numerically efficient, the
core-system hierarchy quantum master equation is favorable for ''visualizing''
the correlated solvation dynamics.
