Fundamental issues with light propagation through $\mathcal{PT}$-symmetric systems

TL;DR

This paper investigates the unphysical superluminal group velocities in $ ext{PT}$-symmetric SSH chains, showing that proper material dispersion considerations restore causality and finite group velocities, setting practical limits on system performance.

Contribution

It demonstrates that including material dispersion in $ ext{PT}$-symmetric models resolves superluminal velocity issues, clarifying the physical limits of such systems.

Findings

Superluminal group velocities are unphysical and caused by neglecting material dispersion.

Restoring causality through dispersion considerations yields finite, physically consistent group velocities.

Real part of group velocity exceeds light speed only with significant imaginary component, maintaining causality.

Abstract

We analyse the emergence of unphysical superluminal group velocities in Su--Schrieffer--Heeger (SSH) parity-time ($\mathcal{PT}$) symmetric chains, and explore the origins of such a behaviour. By comparing the band structure of an infinite loss-gain SSH chain with that of a one-dimensional Bragg stack, we first exclude insufficient coupling consideration in the tight-binding description as the cause of group-velocity divergence. We then focus on material dispersion, and show that indeed, restoring causality in the description of both the lossy and the gain components resolves the problem and recovers finite group velocities, whose real part can only exceed the speed of light in vacuum when accompanied by a significant imaginary part. Our analysis introduces thus the required practical limits in the performance of common $\mathcal{PT}$-symmetric systems.