  We study the dynamics of the entanglement in one dimensional critical quantum
systems after a local quench in which two independently thermalized
semi-infinite halves are joined to form a homogeneous infinite system and left
to evolve unitarily. We show that under certain conditions a nonequilibrium
steady state (NESS) is reached instantaneously as soon as the entanglement
interval is within the light-cone emanating from the contact point. In this
steady state, the exact expressions for the entanglement entropy and the
logarithmic negativity are in agreement with the steady state density matrix
being a boosted thermal state, as expected. We derive various general
identities: relating the negativity after the quench with unequal left and
right initial temperatures, with that with equal left and right temperatures;
and relating these with the negativity in equilibrium thermal states. In
certain regimes the resulting expressions can be analytically evaluated.
Immediately after the interval interesects the lightcone, we find logarithmic
growth. For a very long interval, we find that the negativity approaches a
plateau after sufficiently long times, different from its NESS value. This
provides a theoretical framework explaining recently obtained numerical
results.
