  We consider the quench dynamics of non-interacting fermions in one dimension
in the presence of a finite-size impurity at the origin. This impurity is
characterized by general momentum-dependent reflection and transmission
coefficients which are changed from ${\sf r}_0(k), {\sf t}_0(k)$ to ${\sf
r}(k), {\sf t}(k)$ at time $t=0$. The initial state is at equilibrium with
${\sf t}_0(k)=0$ such that the system is cut in two independent halves with
${\sf r}_0^R(k)$, ${\sf r}_0^L(k)$ respectively to the right and to the left of
the impurity. We obtain the exact large time limit of the multi-time
correlations. These correlations become time translationally invariant, and are
non-zero in two different regimes: (i) for $x=O(1)$ where the system reaches a
non-equilibrium steady state (NESS) (ii) for $x \sim t$, i.e., the ray-regime.
For a repulsive impurity these correlations are independent of ${\sf
r}_0^R(k)$, ${\sf r}_0^L(k)$, while in the presence of bound states they
oscillate and memory effects persist. We show that these nontrivial
relaxational properties can be retrieved in a simple manner from the large time
behaviour of the single particle wave functions.
