Light propagation in topological two-level structures

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Presentation transcript:

Light propagation in topological two-level structures Istituto dei Sistemi Complessi Light propagation in topological two-level structures Laura Pilozzi - Claudio Conti Istituto dei Sistemi Complessi – CNR Via dei Taurini 19, Roma www.isc.cnr.it 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Topological properties torus sphere ellipsoid Topological properties donut coffecup Euler characteristic Gaussian curvature 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Topological physics Quantum Hall effect edge B edge In the presence of a magnetic field B charged particles execute closed orbits inside the bulk but skip along the edge. Landau symmetry-breaking theory Topological classification Topological insulator 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Topological physics Topological indexes Chern number Euler characteristic Chern number Gaussian curvature Berry curvature Berry connection Electron Bloch function interface 3 3’ edge state edge state 2 2’ 1 1’ 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Topological physics……with light one-dimensional quasicrystals topological insulators 1D crystals with compound unit cell 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Radiative topological states in resonant photonic crystals. A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko Phys. Rev. Lett. 112, 107403 (2014) Photonic topological insulator 1D Harper model 1D chains of resonant two-level layers (blue) in an homogeneus bulk (pink) of frequency-independent dielectric function elementary cell Structure with three resonant layers A in the unit cell. z Uniform structure - US Bragg structure - BS period of the primary lattice modulation strenght resonant layer barrier A B Harper modulation 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Topological physics……with light 1D 2D Topological indexes 1D AAH model Chern numbers x 2D ancestor model genus 3 2 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

lack of inversion centre Eigenfrequencies symmetries Time inversion symmetry Even eigenfrequencies in k phase shift the structure corresponding to a particular value of is spatially inverted under the reversal elementary cell 1/2 1/4 1/5 symmetric non symmetric 1/3 asymmetric cell lack of inversion centre 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Band dispersion Single layer transfer matrix reflectivity left edge mode right edge mode B 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Uniform structure Reflectivity phase intensity edge modes gap Chern number = -1 winding number = 1 intensity Chern number = 2 winding number = - 1 edge modes gap Chern number = -1 immaginary real 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Modulated structure Reflectivity intensity edge modes phase gap phase gap winding number = - 1 winding number = 1 winding number = 0 Chern number = -1 Chern number = 2 gap 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Modes detection Maxwell-Bloch equations Finite Difference Time Domain simulations Maxwell-Bloch equations We follow the evolution of a pulse that moves in the structured region with an initial profile: polarization relaxation rate Rabi frequency population relaxation rate inversion population initial inversion population 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Self-Induced Transparency (SIT) pulse Finite Difference Time Domain simulations structured region output layer input layer pulse duration 2p pulse pulse amplitude 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Self-Induced Transparency (SIT) pulse Finite Difference Time Domain simulations 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Self-Induced Transparency (SIT) pulse Finite Difference Time Domain simulations Population inversion profiles at different simulation times. 5 10 15 20 25 30 x (mm) 1 -1 5 10 15 20 25 30 x (mm) 1 -1 5 10 15 20 25 30 x (mm) 2 1 -2 -1 5 10 15 20 25 30 x (mm) 1 -1 5 10 15 20 25 30 x (mm) 1 -1 5 10 15 20 25 30 x (mm) 1 -1 Electric field profiles at different simulation times. 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Edge modes detection Finite Difference Time Domain simulations t topological periodic x 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Edge modes detection Finite Difference Time Domain simulations 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015

Light propagation in topological two-level structures Conclusions We have demonstrated the presence of radiative topologically protected edge states in 1D resonant photonic crystals with a compound non centrosymmetric unit cell that survive despite the long-range light-induced coupling of the resonances and finite lifetime of their radiative decay. The states are manifested in the stationary reflection spectra of the structure with finite nonradiative losses as well as in the time-dependent response to short optical pulses. Trough FDTD solutions of Maxwell-Bloch equations we have shown that a direct observation of topological protected edge states can be achieved following the time evolution of the TLS population inversion. Under SIT conditions localization manifests with a transition from a travelling population inversion to a standing one in the input face layers of the structure. The realization of such RTI can be achieved considering as the active TLS rare-earth ions or quantum wells embedded in a semiconductor structure as effective two-level systems for low densities excitons with resonance close to the operating frequency. 101° Congresso Nazionale SIF - Roma, 21 - 25 Settembre 2015