  Whether a small quantum mechanical system is able to equilibrate with its
environment once an external local perturbation drives it out of thermal
equilibrium is a central question which cuts across many different fields of
science. Here we consider such a problem for a correlated quantum impurity
coupled to a fermionic reservoir and driven out of equilibrium by local quantum
quenches such as those recently realized in optical absorption experiments on
single quantum dots. We argue that equilibration in this problem is deeply
connected to the occurrence of Kondo Effect at low energy and that a highly non
trivial dynamical behavior may emerge whenever a local quantum critical point
intrudes between a conventional Kondo screened phase and a Kondo unscreened
one. We discuss this issue in the context of the Anderson Impurity model
coupled to a pseudo-gap reservoir by using a correlated time dependent
variational wave function that is able to qualitatively describe this physics.
