Partly invariant steady state of two interacting open quantum systems

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.