  This paper considers a class of open quantum systems with an algebraic
structure of dynamic variables, including the Pauli matrices for finite-level
systems as a particular case. The Hamiltonian and the operators of coupling of
the system to the external bosonic fields depend linearly on the system
variables. The fields are represented by quantum Wiener processes which drive
the system dynamics in the form of a quasilinear Hudson-Parthasarathy quantum
stochastic differential equation whose drift vector and dispersion matrix are
affine and linear functions of the system variables. This quasilinearity leads
to a tractable evolution of the two-point commutator matrix of the system
variables (and their multi-point mixed moments in the case of vacuum input
fields) involving time-ordered operator exponentials. The resulting exponential
decay in the two-point commutation relations is a manifestation of quantum
decoherence, caused by the dissipative system-field interaction and making the
system lose specific unitary dynamics features which it would have in isolation
from the environment. We quantify the decoherence in terms of the rate of the
commutation relations decay and apply system theoretic and matrix analytic
techniques, such as algebraic Lyapunov inequalities and spectrum perturbation
results, to the study of the asymptotic behaviour of the related Lyapunov
exponents in the presence of a small scaling parameter in the system-field
coupling. These findings are illustrated for finite-level quantum systems (and
their interconnections through a direct energy coupling) with multichannel
external fields and the Pauli matrices as internal variables.
