Non-Hermitian Topology and Directional Amplification in Driven-Dissipative Cavity Arrays

Abstract

Directional amplification, in which signals are selectively amplified depending on their propagation direction, has attracted much attention as a key resource for applications including quantum information processing. In this thesis, we first develop a unifying framework based on non-Hermitian topology to understand non-reciprocity and directional amplification in driven-dissipative cavity arrays. Specifically, we unveil a one-to-one correspondence between a non-zero topological invariant defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the end-to-end gain grows exponentially with the number of cavities. Furthermore, we show that non-Hermitian topological amplification is robust against disorder as long as the disorder strength does not exceed the size of the point gap—the separation of the spectrum from the origin. In another part of this thesis, we revisit the bulk-boundary correspondence for non-Hermitian topological systems. It is a fundamental phenomenon of Hermitian topological phases, however, in non-Hermitian systems, it breaks down in the conventional form due to the non-Hermitian skin effect—the localisation of a macroscopic number of eigenvectors at the boundary. Here, we show that it is possible to restore the bulk-boundary correspondence by means of the singular value decomposition: when the system is topologically non-trivial, zero singular modes emerge leading to directional amplification with their number given by the topological invariant. In the final part, we extend the notion of non-reciprocity to unidirectional bosonic transport in time-reversal symmetric systems by exploiting interference between beamsplitter and two-mode-squeezing interactions. We develop a framework to identify the phenomenon we dub quadrature non-reciprocity based on features of the particle-hole graph representing the equations of motion. In addition to unidirectionality, these networks can exhibit an even-odd pairing between collective quadratures and an exponential end-to-end gain in the case of arrays of cavities. Our work opens up new avenues for signal routing, non-Hermitian topology, and amplification in bosonic systems.