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