  We formulate and discuss explicit computation of dynamic correlation
functions in open quadradic fermionic systems which are driven and dissipated
by the Lindblad jump processes that are linear in canonical fermionic
operators. Dynamic correlators are interpreted in terms of local quantum quench
where the pre-quench state is the non-equilibrium steady state, i.e. a fixed
point of the Liouvillian. As an example we study the XY spin 1/2 chain and the
Kitaev Majorana chains with boundary Lindblad driving, whose dynamics exhibits
asymmetric (skewed) light cone behaviour. We also numerically treat the two
dimensional XY model and the XY spin chain with additional
Dzyaloshinskii-Moriya interactions. The latter exhibits a new non-equilibrium
phase transition which can be understood in terms of bifurcations of the
quasi-particle dispersion relation. Finally, considering in some detail the
periodic Kitaev chain (fermionic ring) with dissipation at a single (arbitrary)
site, we present analytical expressions for the first order corrections (in the
strength of dissipation) to the spectrum and the non-equilibrium steady state
(NESS) correlation functions.
