  The dynamics of waveguide-QED systems involving coherent time-delayed
feedback give rise to a hierarchy of multi-time correlations within the
Heisenberg picture due to the induced non-Markovianity. We propose to perform a
projection onto a complete set of states in the Hilbert space to decompose the
multi-time correlations into single-time matrix elements. To illustrate the
procedure, we consider the paradigmatic example of a two-level system that
couples to a semi-infinite waveguide and interacts with quantum light pulses.
Our approach complements the range of available methods as it allows
calculating the dynamics under the inclusion of additional dissipation channels
in a numerically exact and efficient manner for multiphoton pulses of arbitrary
shape where memory requirements are known in advance.
