  Non-reciprocal couplings or drivings are known to induce steady-state,
directional, amplification in driven-dissipative bosonic lattices. This
amplification phenomena has been recently linked to the existence of a non-zero
topological invariant defined with the system's dynamical matrix, and thus, it
depends critically on the couplings' structure. In this work, we demonstrate
the emergence of unconventional, non-reciprocal, long-range dissipative
couplings induced by the interaction of the bosonic chain with a chiral,
multi-mode channel, and then study their impact on topological amplification
phenomena. We show that these couplings can lead to topological invariant
values greater than one which induce topological, multi-mode amplification and
metastability behaviour not predicted in other setups. Besides, we also show
how these couplings can also stabilize topological amplifying phases in the
presence of local parametric drivings. Finally, we conclude by showing how such
phenomena can be naturally obtained in two-dimensional topological insulators
hosting multiple edge modes.
