Topological directed amplification

TL;DR

This paper reveals a topological phenomenon where certain stable lattice systems exhibit transient edge-state amplification due to nonnormal dynamics, with implications for laser array design.

Contribution

It introduces the concept of topological directed amplification in skin-effect lattices, analyzing nonnormal operators and providing tools to optimize initial conditions for maximum amplification.

Findings

Transient edge-state amplification occurs despite overall stability.

Pseudospectrum and Kreiss constant effectively describe the phenomenon.

Nonlinear effects can enhance laser emission stability and power.

Abstract

A phenomenon of topological directed amplification of certain initial perturbations is revealed theoretically to emerge in a class of asymptotically stable skin-effect lattices described by nonnormal Toeplitz operators $H_g$ with positive ``numerical ordinate" $\omega(H_g)>0$. Nonnormal temporal evolution, even in the presence of global dissipation, is shown to manifest a counterintuitive transient phase of edge-state amplification -- a behavior, drastically different from the asymptote, that spectral analysis of $H_g$ fails to directly reveal. A consistent description of the effect is provided by the general tool of ``pseudospectrum", and a quantitative estimation of the maximum power amplification is provided by the {\it Kreiss constant}. A recipe to determine an optimal initial condition that will attain maximum amplification power is given by singular value decomposition of the propagator $e^{-i H_g t}$. It is further predicted that the interplay between nonnormality and nonlinearity in a skin-effect laser array can facilitate narrow-emission spectra with scalable stable-output power.