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We investigate the influence of an additional bosonic bath onto the real-timedynamics of a localized orbital coupled to conduction band with anenergy-dependent coupling function $\Gamma(\varepsilon) \propto|\varepsilon|^r$. Recently, a rich phase diagram has been found in thisBose-Fermi Anderson model, where the transitions between competing groundstates are governed by quantum critical points. In addition to a transitionbetween a Kondo singlet and a local moment, a localized phase has beenestablished once the coupling to a sub-ohmic bosonic bath exceeds a criticalvalue. Using the time-dependent numerical renormalization group approach, weshow that the non-equilibrium dynamics with F-type of bath exponents can befully understand within an effective single-impurity Anderson model using arenormalized local Coulomb interaction $U_{\rm ren}$. For regimes with B-typeof bath exponents, the nature of the bosonic bath and the coupling strength hasa profound impact on the electron dynamics which can only partially beunderstood using an appropriate $U_{\rm ren}$. The local expectation valuesalways reach a steady state at very long times. By a scaling analysis for$\Lambda \to 1^+$, we find thermalization of the system only in the strongcoupling regime. In the local moment and in the localized phase significantdeviations between the steady-state value and the thermal equilibrium value arefound that are related to the distance to the quantum critical point.

Influence of a bosonic environment onto the non-equilibrium dynamics of  local electronic states in a quantum impurity system close to a quantum phase  transition