  We investigate two interacting open quantum systems whose time evolutions are
governed by Markovian master equations. We show a class of coupled systems
whose interaction leaves invariant the steady state of one of the systems,
i.e., only one of the reduced steady states is sensitive to the interactions. A
detailed proof with the help of the Trotter product formula is presented. We
apply this general statement to a few models, one of which is the
optomechanical coupling model where an optical cavity is coupled to a small
mechanical oscillator.
