טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentLevy Eli
SubjectTopological Properties of Quasiperiodic Chains: Structural
and Spectral Analysis
DepartmentDepartment of Physics
Supervisor Professor Eric Akkermans
Full Thesis textFull thesis text - English Version


Abstract

The topological properties of quasiperiodic chains are revisited. Chern numbers known to label the infinite set of spectral gaps and diffraction peaks for infinite chains, are shown to be related to the underlying structural palindromic symmetry of finite chains. The deviation from this symmetry forms a periodic cycle as a function of a structural degree of freedom f, driven by a series of structural events termed phason-flips. Topological classification of spectral gaps is related to the two independent phases of the scattering matrix: the scattering total phase shift describing the frequency spectrum and the scattering chiral phase carrying topological information in a winding number as a function of f. Conveniently designed edge stated of a generalized edge perform a spectral winding (as a function of f) directly related to the Chern numbers, allowing to scan these phases. A full quantitative description of these states by an effective topological Fabry-Perot cavity, and a first experimental measurement of these predictions in a cavity polariton setup are presented for the Fibonacci chain. An identical analysis was carried out in reciprocal space. Topological classification of diffraction peaks is related to a diffraction chiral phase, found to carry the topological information (labeling the diffraction peaks) in a winding number as a function of f. A splitting of diffraction peaks in a generalized edge setup performs a spectral winding (as a function of f) directly related to the Chern numbers and allows to scan this phase. A first experimental observation of this winding and its robustness against disorder, performed in a programmable optical grating setup, is presented for the Fibonacci chain. The purely structural origin of the topological labeling of spectral gap and diffraction peaks is also revisited. A two-dimensional structural map of phason-flips as a function of φ in a finite chain is shown to form a perfect torus. This property constitutes an internal periodic boundary conditions with respect to f, and leads to the definition of a quasi-Brillouin zone in reciprocal space. A quantitative formulation for the allowed number of Chern numbers, and the deviation of the spectral gaps/diffraction peaks from the infinite chain values are given. A first experimental observation of the quasi-Brillouin zone is given for the Fibonacci chain, capturing 89 different Chern numbers in a single measurement. A general discussion is presented regarding the role of true quasiperiodicity in the properties predicted and observed, and also regarding the role of the quasiperiodic chain natural lengths. This work has relied on the Fibonacci quasiperiodic chain as a leading example, but the results presented extend to a very large family of quasiperiodic chains, and expected to be generalizable to higher dimensions.