For the last line of the above equation, we have used the fact that [H k ( R)] w = H k (R) (see Eq. 11b or Eq. 12 for its expression) and [H 0 ( R, P )] w = H 0 (R, P ) because the Wigner transform of a function of position operator is the same function, and H 0 ( R, P ) only contains P up to the quadratic order. The state-dependent Liouvillian is expressed as