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For multi-level open quantum system, the interaction between different levelscould pose challenge to understand the quantum system both analytically andnumerically. In this work, we study the approximation of the dynamics of theAnderson-Holstein model, as a model of multi-level open quantum system, byRedfield and Lindblad equations. Both equations have a desirable property thatif the density operators for different levels is diagonal initially, theyremain to be diagonal for any time. Thanks to this nice property, thesemi-classical limit of both Redfield and Lindblad equations could be derivedexplicitly; the resulting classical master equations share similar structuresof transport and hopping terms. The Redfield and Lindblad equations are alsocompared from the angle of time dependent perturbation theory.

Lindblad equation and its semi-classical limit of the Anderson-Holstein model