Bloch-like waves in random-walk potentials based on supersymmetry

Abstract

Bloch's theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and long-range correlations for Bloch's theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Bloch-like waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here we reveal a deterministic pathway to bandgaps in random-walk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with ‘Bloch-like eigenstates'.

Interdimensional optical isospectrality inspired by graph networks

“…The fragile degeneracy guarantees the presence of a connected 1D structure that has an identical spectral response to that of a 3D disordered structure. Our results thus extend the field of correlated disorder [1,2,8,10,14,32] into network-like level statistics.Our approach utilizing the Hamiltonian transformation allows the finding of 1D isospectral structures from random networks deterministically, in contrast to numerical searching techniques, e.g. genetic algorithm.…”

“…Due to the critical needs in light harvesting [6], robust bandgaps [7,8], and ultrafast optics [9], disordered optical materials have also been intensively studied to exploit their broadband responses. Although one-(1D) [8,10], two-(2D) [1,11,12], and three-dimensional (3D) [13] disordered structures have been considered promising candidates for broadband and omnidirectional operations, the deliberate control of disordered structures [1,3,7,8,14] has been achieved only in 1D or 2D structures, owing to the difficulty of manipulating 3D randomness.To understand the role of the 'dimension' in optics, we can employ the interdisciplinary viewpoint from network theory. For the structures composed of coupled resonances, light flows inside those can be interpreted as signal transport over graph networks [14][15][16][17].…”

“…The condition of a fully connected structure with a lower dimension and preserved level statistics is then investigated, showing that a 2D/3D disordered structure even with all orders of couplings can always be transformed into a single 1D structure with nearest-neighbor coupling. Our approach enables the 1D replication of the spectral information of 3D disordered structures, allowing the simplification of broadband multilevel devices, such as multimode waveguides [21], resonators [25], and bandgaps [8], and the inter-dimensional interaction from the global phase matching condition.Consider the example of optical networks ( Fig. 1a) and its graph representation (Fig.…”

“…As the supersymmetric technique firstly enables the global phase matching for optical multimode structures [20,21,23], it is noted that the inter-dimensional isospectrality provides the global phase matching between multimodes of 2D/3D and 1D optical structures for the first time. In summary, in terms of the design technique for optical isospectral structures [8,[20][21][22][23][24], we have opened a new direction by exploiting the Hamiltonian similarity in high-degree random-walk networks. Based on previous achievements in the tunable coupling regarding its magnitude [28] or sign [31] in dielectric waveguides or photonic crystal structures, our approach linking wave physics and graph theory can provide a new type of level statistics that originates from graph networks with connected far-off vertices.…”

Absence of Anderson localization in certain random lattices

“…that the scattering potential can be described as white noise. Once correlations are introduced in the scattering potential the picture becomes much less clear, and it is possible to have frequency bands where the system is localized co-existing with frequency bands where all the eigenmodes are extended [13,14], discrete sets of extended modes in an otherwise localized spectrum [15], enhanced localization [16], or even fully extended Bloch modes in random-like potentials [17].In this article we study experimentally the case where the scattering potential is described by colored noise instead of white noise, i.e. when the power spectrum of the random potential is not flat.…”