Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands

Abstract

We experimentally realize fractal-like photonic lattices by use of the cw-laser-writing technique, thereby observing distinct compact localized states (CLSs) associated with different flatbands in the same lattice setting. Such triangle-shaped lattices, akin to the first generation Sierpinski lattices, possess a band structure where singular non-degenerate and nonsingular degenerate flatbands coexist. By proper phase modulation of an input excitation beam, we demonstrate not only the simplest CLSs but also their superimposition into other complex mode structures. Our experimental and numerical results are corroborated by theoretical analysis. Furthermore, we show by numerical simulation a dynamical oscillation of the flatband states due to beating of the CLSs that have different eigenenergies. These results may provide inspiration for exploring fundamental phenomena arising from the interplay of fractal structure, flatband singularity, and real-space topology.

Localized States Emerging from Singular and Nonsingular Flat Bands in a Frustrated Fractal‐Like Photonic Lattice

“…[17] Only recently first efforts were made to combine the two classes of flat bands in an optically induced lattice in a photorefractive crystal. [18] However, up to now, no experimental realization has proved the coexistence of these two topological classes of Singular and nonsingular flat bands in a Sierpinski fractal-like photonic lattice are reported. It is demonstrated that the lowest two bands, being isolated and degenerate due to geometrical frustration, are nonsingular and thus can be spanned by a complete set of compact localized states.…”

“…[ 17 ] Only recently first efforts were made to combine the two classes of flat bands in an optically induced lattice in a photorefractive crystal. [ 18 ] However, up to now, no experimental realization has proved the coexistence of these two topological classes of singular and nonsingular flat bands in a single lattice by analyzing the completeness conditions of the sets of CLSs. Here, we propose and experimentally demonstrate, for the first time to our knowledge, a fractal‐like lattice that exhibits multiple flat bands fabricated by femtosecond direct laser writing on a large‐scale fused silica chip.…”

Application of the real space decimation method in determining intricate electronic phases of matter: a review

“…The applicability of the decimation method to understand the fundamental character of single particle excitations in several other lattice models, including geometric fractal networks, will also be discussed. The latter systems, that is, the fractal networks, have recently been experimentally realized 7 and thus a comprehensive study of their properties, especially with regard to the exact evaluation of the compact localized eigenstates and topological orders in fractals, 8 is very important, to our mind.…”

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

“…This works easily when the unit cell has simple geometry, but turns out to be a tedious job in cases where the unit cell has an intricate internal structure. This issue has become pertinent and needs to be addressed, in particular, after a series of fascinating experiments reported laboratory realizations of the classic Sierpinski gasket fractal (SGF) [23][24][25] and observation of CLS [26][27][28][29][30] and photonic Floquet topological phase 31 on engineered SGF substrates. The latter experiments were done on SGF structures grafted either mainly by the femtosecond laser writing techniques, or designed with evanescently coupled helical waveguides, for example.…”