During the project, we conducted two studies in parallel of the stability of resonant modes in black holes space-times (by J.-L. Jaramillo et al.) and topological periodic structures (A. Coutant et al.). This lead to a total of 5 publications in international peer-review journals.
1) We analyzed the presence of localized modes in an acoustic networks of tubes connected on a square grid. We showed that localized modes can exist in corners of the system if a geometric quantity, the Berry phase, is non-trivial. Moreover, these modes coexist with a continuous set of radiating waves, which would suggest that they are unstable: they lose energy through radiation (a situation known as ``bound state in the continuum''). Surprisingly, we showed that these corner modes are remarkably stable, and conserve their characteristic frequency even in the presence of randomness in the background. The results were published in Physical Review B 102, 214204 (2020) (available in arXiv:2007.13217).
2) In the same system as 1), localized waves can also propagate along edges of the system. We studied the interaction of these edge waves with corner modes. We developed a method to compute the scattering of these waves across various edge defects. In particular, it was found that they can be used to detect the presence of corner modes: the phase accumulated by edge waves when bouncing off a corner is directly related to the Berry phase signaling a corner localized mode. This provides a tool to experimentally observe these modes. This work was published in Journal of Applied Physics 129 no. 12, 125108, Special topics: Acoustic Metamaterials (2021) (available in arXiv:2012.15168).
3) We performed an experimental demonstration of the presence of localized modes in a one-dimensional analogue of the system studied in 1) and 2). Th setup is an acoustic waveguide (a hollow tube) with varying cross-sections. We successfully observe localized edge modes and obtained their characteristic frequency. We then showed that this mode and its properties are robust when adding randomness in the choice of cross-section values, as predicted by the theory. The agreement between theory and experiment is very high, and required no adjustment of parameter to fit. This work was published in Physical Review B 103, 224309 (2021) (available in arXiv:2103.03859).
4) We provided a stability analysis of black hole resonances (also known as quasi-normal modes or QNM) by computing the contour level of the pseudo-spectrum. To do so, we needed to appeal to a powerful geometric tool in order to recast the problem as an non-hermitian eigenvalue problem. The method uses a hyperbolic foliation of the conformally compactified black hole space-time. Once the problem is mathematically well-posed, we could compute the pseudo-spectrum numerically. We found that most QNM are unstable to high frequency perturbations, except for the lowest one (i.e. with the longest life-time). This has crucial implications for the characterization of black holes through the detection of gravitational waves, as currently aimed by the LIGO-VIRGO collaboration. The results were published in \emph{Physical Review X 11 (3), 031003} (open access - also available in arXiv:2004.06434) in Phys. Rev. D 104, 084091 (available in arXiv:2107.09673) and in Classical and Quantum Gravity (accepted manuscript https://doi.org/10.1088/1361-6382/ac5054 - available in arxiv:2107.12865).
In the last part of the project, we gathered the expertise and tools developed in the two parts in a study of atmospheric acoustic waves. In this context, we used the pseudo-spectrum technic to characterize the stability of resonances around jets or vortices. This work is planned to be released during the first half of 2022.
