We consider an integrable system of two one-dimensional fermionic
chains connected by a link. The hopping constant at the link can be
different from that in the bulk. Starting from an initial state in which
the left chain is populated while the right is empty, we present
time-dependent full counting statistics and the Loschmidt echo in terms
of Fredholm determinants. Using this exact representation, we compute
the above quantities as well as the current through the link, the shot
noise and the entanglement entropy in the large time limit. We find that
the physics is strongly affected by the value of the hopping constant at
the link. If it is smaller than the hopping constant in the bulk, then a
local steady state is established at the link, while in the opposite
case all physical quantities studied experience persistent oscillations.
In the latter case the frequency of the oscillations is determined by
the energy of the bound state and, for the Loschmidt echo, by the bias
of chemical potentials.
Effect of double local quenches on the Loschmidt echo and entanglement entropy of a one-dimensional quantum system