  We study the out-of-equilibrium dynamics of noninteracting fermions in one
dimension and in continuum space, in the presence of a delta impurity potential
at the origin whose strength $g$ is varied at time $t=0$. The system is
prepared in its ground state with $g=g_0=+\infty$, with two different densities
and Fermi wave-vectors $k_L$ and $k_R$ on the two half-spaces $x>0$ and $x<0$
respectively. It then evolves for $t>0$ as an isolated system, with a finite
impurity strength $g$. We compute exactly the time dependent density and
current. For a fixed position $x$ and in the large time limit $t \to \infty$,
the system reaches a non-equilibrium stationary state (NESS). We obtain
analytically the correlation kernel, density, particle current, and energy
current in the NESS, and characterize their relaxation, which is algebraic in
time. In particular, in the NESS, we show that, away from the impurity, the
particle density displays oscillations which are the non-equilibrium analog of
the Friedel oscillations. In the regime of "rays", $x/t=\xi$ fixed with $x, t
\to \infty$, we compute the same quantities and observe the emergence of two
light cones, associated to the Fermi velocities $k_L$ and $k_R$ in the initial
state. Interestingly, we find non trivial quantum correlations between two
opposite rays with velocities $\xi$ and $-\xi$ which we compute explicitly. We
extend to a continuum setting and to a correlated initial state the analytical
methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the
context of a discrete fermionic chain with an impurity. We also generalize our
results to an initial state at finite temperature, recovering, via explicit
calculations, some predictions of conformal field theory in the low energy
limit.
