  We set up a scattering experiment of matter against an impurity which
separates two generic one-dimensional critical quantum systems. We compute the
flux of reflected and transmitted energy, thus defining a precise measure of
the transparency of the interface between the related two-dimensional conformal
field theories. If the largest symmetry algebra is Virasoro, we find that the
reflection and transmission coefficients are independent of the details of the
initial state, and are fixed in terms of the central charges and of the
two-point function of the displacement operator. The situation is more
elaborate when extended symmetries are present. Positivity of the total energy
flux at infinity imposes bounds on the coefficient of the two-point function of
the displacement operator, which controls the free-energy cost of a small
deformation of the interface. Finally, we study out-of-equilibrium steady
states of a critical system connecting two reservoirs at different
temperatures. In the absence of extended symmetries, our result implies that
the energy flux across an impurity is proportional to the difference of the
squared temperatures and controlled by the reflection coefficient.
