Periodic Reporting for period 1 - WMBH (Waves, mean flows and black holes)
Periodo di rendicontazione: 2020-04-01 al 2022-03-31
The problem of waves propagating in an inhomogeneous background is ubiquitous in physics: from gravitational waves around black holes, to ocean surface waves in mean flows or acoustic waves in media with a complex microstructure. Such a background often induces resonances: a discrete set of frequencies that strongly respond to a given excitation. A large class of applications of resonances consists in identifying them to obtain information about the background. For instance, the spectrum of gravitational waves can be used to obtain the mass and angular momentum of a black hole, or one can send acoustic waves to image defects inside a given material.
In mathematics, this kind of imaging technique is called an inverse problem: from the knowledge of the wave resonance frequencies, one would like to reconstruct the structure of the background. In this context, a key question is that of the stability of these resonance frequencies: how strongly are they changed when the background varies? The objective of this project is to develop general tools to address this problem in the seemingly unrelated contexts it arises. For this, we used a combination of differential geometric and spectral theoretic methods.
This project tackled this question in two distinct contexts: gravitational waves around black holes, and acoustic waves in periodic structures. In the former case, geometry is used to describe the black hole space-time (i.e. its gravitational field). In the latter, the geometry is that of the reciprocal space, that is, after a Fourier transform the background is characterize by a curvature called the Berry curvature.
In mathematics, this kind of imaging technique is called an inverse problem: from the knowledge of the wave resonance frequencies, one would like to reconstruct the structure of the background. In this context, a key question is that of the stability of these resonance frequencies: how strongly are they changed when the background varies? The objective of this project is to develop general tools to address this problem in the seemingly unrelated contexts it arises. For this, we used a combination of differential geometric and spectral theoretic methods.
This project tackled this question in two distinct contexts: gravitational waves around black holes, and acoustic waves in periodic structures. In the former case, geometry is used to describe the black hole space-time (i.e. its gravitational field). In the latter, the geometry is that of the reciprocal space, that is, after a Fourier transform the background is characterize by a curvature called the Berry curvature.
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.
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.
This project has significantly contributed to the field both at the experimental and theoretical level. We provided new tools to analyze localized resonances of waves in various complex backgrounds. These were also showed to be relevant for experimental detection of localized modes in the context of guided acoustics. Our results offer new techniques that are promised to a large range of application, whether it is in the field of light or sound in artificial metamaterials, or in the analysis of gravitational wave emission from compact astrophysical objects.