  We present a method to study the time evolution of the single impurity
Anderson model which exploits a mean field decoupling of the interacting
impurity and the non-interacting bath (in form of a chain). This is achieved by
the introduction of a pair of auxiliary Majorana fermions between the impurity
and the chain. After decoupling, we obtain a self-consistent set of equations
for the impurity and chain. First, we study the behavior of the system in
equilibrium at zero temperature. We obtain a phase transition as a function of
the interaction at the impurity and the coupling between the impurity and the
chain between the Kondo regime, where the mean field parameters are zero and,
hence, we have a well-defined spin at the impurity, to a phase where mean field
parameters acquire finite values leading to a screening of the impurity spin by
conduction bath electrons. In the latter case, we observe charge and spin
fluctuations at the impurity site. Starting from this equilibrium ground state
at zero temperature we quench in the interaction strength at the impurity
and/or the hybridization strength between the impurity and the chain and study
the time evolution of the system. We find that for quenches to weak to
intermediate coupling the system converges to the equilibrium state defined by
the final set of parameters after the quench. We analyze the oscillation
frequency and as well as the thermalization rate during this quench. A quench
to a strong interaction value results in persistent oscillations and a trapping
of the system in a non-thermal state. We speculate that these two regimes of
different long-time behavior are separated by a dynamical phase transition.
