Signatures of quantum chaos in phonon-polariton billiards

TL;DR

This study visualizes and analyzes hyperbolic phonon polaritons in hexagonal boron nitride billiards with different geometries, revealing signatures of quantum chaos and developing a numerical framework for level statistics.

Contribution

It introduces a novel experimental approach to image polariton modes in billiards and develops a theoretical model linking boundary conditions to chaotic behavior.

Findings

Observation of irregular mode patterns consistent with quantum scarring.

Evolution of Fourier transforms toward ring-like structures matching Berry's conjecture.

Level statistics show a transition from Poisson to Wigner-Dyson behavior with increasing boundary complexity.

Abstract

We use scanning near-field optical microscopy to image hyperbolic phonon polaritons in hexagonal boron nitride (hBN) billiards with integrable and chaotic geometries. In Sinai billiards, we observe irregular mode patterns consistent with quantum scarring, together with an unexpected sensitivity to weak probe perturbations. These random-wave features coexist with non-chaotic one-dimensional boundary modes arising from nontrivial polariton reflection at the billiard edge. As the billiard boundary becomes increasingly complex, the Fourier transforms of the measured signals evolve toward ring-like structures consistent with Berry's random-wave conjecture. We develop a numerical framework based on the Helmholtz equation with generalized boundary conditions that encode angle-dependent reflection phase shifts. The calculated level statistics exhibit a crossover from Poisson-like behavior in integrable billiards to Wigner-Dyson-like behavior in chaotic geometries, with small deviations from the canonical form arising from nonlinear boundary conditions that require self-consistent bulk-boundary analysis. Theoretical analysis based on dissipative Green's functions qualitatively reproduces the near-field data. These results establish mesoscopic van der Waals billiards as a rich platform for studying generalized chaotic dynamics of hybrid light-matter polaritons.