Topological input-output theory for directional amplification

TL;DR

This paper introduces a topological framework for analyzing directional amplification in photonic lattices, linking non-Hermitian coupling matrices to topological insulator models to predict and optimize quantum amplification performance.

Contribution

It develops a novel topological input-output theory for non-Hermitian photonic systems, enabling analytical characterization of amplification properties in topologically non-trivial regimes.

Findings

Directional amplification is near quantum-limited with exponential gain growth.

Noise-to-signal ratio decreases as the inverse square root of system size.

The theory applies to one-dimensional non-reciprocal photonic lattices with analytical predictions.

Abstract

We present a topological approach to the input-output relations of photonic driven-dissipative lattices 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, whose inverse matrix determines the linear input-output response of the system. In topologically non-trivial 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 non-reciprocal 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 points out to interesting applications of our theory for quantum signal amplification and single-photon detection.