Time evolution of the Kondo resonance in response to a quench

We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within TDNRG a numerically tractable expression for the retarded two-time nonequilibrium Green function $G(t+t',t)$, and its associated time-dependent spectral function, $A(\omega,t)$, for times $t$ both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times $t\to 0^{+}$, only fully develops at very long times $t\gtrsim 1/T_{\rm K}$, where $T_{\rm K}$ is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\Gamma$ during the time interval $-1/\Gamma \lesssim t \lesssim +1/\Gamma$, where $\Gamma$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\rightarrow -\infty$ and $t\rightarrow +\infty$, respectively. Our formulation of two-time nonequilibrium Green functions within TDNRG provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.