Quantized topological transport mediated by the long-range couplings

Abstract

Certain topological systems with time-varying Hamiltonian enable quantized and disorder-robust transport of excitations. Here, we introduce the modification of the celebrated Thouless pump when the on-site energies remain fixed, while the nearest and next-nearest neighbor couplings vary in time. We demonstrate quantized transport of excitations and propose an experimental implementation using an array of evanescently coupled optical waveguides.

Figures (4)

• Figure 1: (a) Schematic of the studied 1D lattice. Synchronized variation of nearest and next-nearest-neighbor couplings enables quantized topological transport. Dashed rectangle shows the unit cell including the sites of A and B sublattices. (b) Evolution of the couplings during one pumping cycle. Parameters: $J_0 = 1$ , $A = 1$, $B=0.35$.
• Figure 2: Instantaneous spectrum and Wannier centers of the driven system. (a) Evolution of the instantaneous spectrum calculated for a finite system consisting of $N=14$ unit cells with open boundary conditions. Left- and right-localized edge states are highlighted by red and blue, respectively. (b) Evolution of the instantaneous Wannier centers for the same system with periodic boundary conditions
• Figure 3: (a) Diffraction of the point-like wavepacket projected onto the lower Bloch band for the 14 unit cell lattice. (b) Displacement of the wavepacket center of mass by $1.005$ unit cell over one cycle $T = 7$ (black) and corresponding Wannier center trajectory (red dashed).
• Figure 4: (a) Chosen geometry of the lattice and its spatial modulation. Insets show electric field profiles (amplitude) of symmetric and antisymmetric modes in vertical and detuned horizontal waveguides, respectively. Insets illustrate the choice of the unit cell with identical spacings and with $p$-mode photonic molecule connectors (a2), the adiabatic modulation of waveguides along their propagation coordinate $z$ for one unit cell (a3) and a $5$-unit-cell lattice (a4). (b) Modulation of the couplings $J_{1,2}$, $t_{1,2}$ and detunings $u_{1,2}$ extracted from numerically simulated splitting of the dimer eigenmodes. (c) Instantaneous spectrum of the propagation constants computed for a finite lattice with $14$ unit cells. Left- and right-localized edge states are highlighted by red and blue, respectively. (d) Tight-binding simulation of the real-space discrete diffraction pattern for the initial point-like excitation projected onto the lowest Bloch band of a $14$ unit cell lattice and modulation period $L = 50$ cm. (e) Time dependence of the center of mass of the intensity distribution (black solid line), displaced by $1.0068$ unit cells over one modulation period. The trajectory of the Wannier center is shown by the dashed red curve for comparison.