  Based on tensor network realizations of path integrals reducing exponential
memory scaling to polynomial efficiency and a Liouville space implementation of
a time-discrete quantum memory, we investigate a quantum system simultaneously
exposed to two structured reservoirs. For this purpose, we employ a numerically
exact quasi-2D tensor network combining both diagonal and off-diagonal
system-reservoir interactions with a twofold memory for continuous and discrete
retardation effects. As a possible example, we study the non-Markovian
dynamical interplay between discrete photonic feedback and structured acoustic
phonon modes, resulting in emerging inter-reservoir correlations and
long-living population trapping within an initially-excited two-level system.
