Topological input-output theory for directional amplification

We present a topological approach to the input-output relations of photonic driven-dissipative lattices acting as directional amplifiers. Our theory relies on mapping from optical non-Hermitian coupling matrix an effective insulator Hamiltonian. This is based singular value decomposition matrices, whose inverse determines linear response system. In topologically non-trivial regimes, lattice dominated by vectors with zero values that are equivalent zero-energy states in insulators, leading amplification coherent input signal. such regime, our theoretical framework allows us fully characterize properties quantum device gain, bandwidth, added noise, and noise-to-signal ratio. exemplify ideas one-dimensional non-reciprocal lattice, for which we derive analytical predictions. show near quantum-limited gain growing exponentially system size, $N$, while ratio suppressed $1/\sqrt{N}$. points out interesting applications signal single-photon detection.