Heisenberg treatment of multiphoton pulses in waveguide QED with time-delayed feedback

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