  For multi-level open quantum system, the interaction between different levels
could pose challenge to understand the quantum system both analytically and
numerically. In this work, we study the approximation of the dynamics of the
Anderson-Holstein model, as a model of multi-level open quantum system, by
Redfield and Lindblad equations. Both equations have a desirable property that
if the density operators for different levels is diagonal initially, they
remain to be diagonal for any time. Thanks to this nice property, the
semi-classical limit of both Redfield and Lindblad equations could be derived
explicitly; the resulting classical master equations share similar structures
of transport and hopping terms. The Redfield and Lindblad equations are also
compared from the angle of time dependent perturbation theory.
