  Time-resolved photoemission spectroscopy is the key technique to probe the
real-time non-equilibrium dynamics of electronic states. Theoretical
predictions of the time dependent spectral function for realistic systems is
however, a challenge. Employing the Kadanoff-Baym equations to find this
quantity results in a cubic scaling in the total number of time steps, quickly
becoming prohibitive and often fail quantitatively and even qualitatively. In
comparison, mean-field methods have more favorable numerical scaling both in
the number of time steps and in the complexity associated with the cost of
evolving for a single time step, however they miss key spectral properties such
as emergent spectral features. Here we present a scheme that allows for the
inclusion of dynamical correlations to the spectral function while maintaining
the same scaling in the number of time steps as for mean-field approaches,
while capturing the emergent physics. Further, the scheme can be efficiently
implemented on top of equilibrium real-time many-body perturbation theory
schemes and codes. We see excellent agreement with exact results for test
systems. Furthermore we exemplify the method on a periodic system and
demonstrate clear evidence that our proposed scheme produces complex spectral
features including excitonic band replicas, features that are not observed
using static mean-field approaches.
