Fast and length-independent transport time supported by topological edge states in finite-size Su-Schrieffer-Heeger chains

Abstract

In order to transport information with topological protection, we explore experimentally the fast transport time using edge states in one-dimensional Su-Schrieffer-Heeger (SSH) chains. The transport time is investigated in both one- and two-dimensional models with topological non-trivial band structures. The fast transport is inherited with the wavefunction localization, giving a stronger effective coupling strength between the mode and the measurement leads. Also the transport time in one-dimension is independent of the system size. To verify the asertion, we implement a chain of split-ring resonators and their complementary ones with controllable hopping strengths. By performing the measurements on the group delay of non-trivially topological edge states with pulse excitations, the transport time between two edge states is directly observed with the chain length up to $20$. Along the route to harness topology to protect optical information, our experimental demonstrations provide a crucial guideline for utilizing photonic topological devices.

Figures (4)

• Figure 1: (a) The SSH model with chain length $N=20$ and $v/\omega_0, w/\omega_0 = -0.01597, 0.07256$. (b) The corresponding edge modes $\Psi_9$ supported in this finite-sized SSH model. (c) Calculated delay time $\tau_d$ as a function of frequency with intrinisc loss $\kappa_i/\omega_0=10^{-4}$ and the coupling loss $\kappa_{c,0}/\omega_0=5\times 10^{-3}$. (d) The Haldane mode with $N= 11 \times 6$ and coupling strengths $t_1/\omega_0, t_2/\omega_0 = 0.05, 0.02j$. (e) The corresponding edge mode $\Psi_{33}$ supported in a finite-size Haldane mode. (f) Calculated $\tau_d$ as a function of frequency with $\kappa_i/\omega_0=10^{-5}$ and $\kappa_{c,0}/\omega_0=5\times 10^{-3}$.
• Figure 2: SSH model realized in a finite-size chain by combining SRRs and CSRRs. The two leads in the front and end nodes are implemented by two transmission lines, denoted with the input port $1$, reflection port $2$, and transmissions ports $3$ and $4$, respectively.
• Figure 3: (a) The measured reflection amplitude for a chain of 20 superconducting resonators. (b) The measured transmission amplitude. (c) The maximum advanced time(red) and delay times(blue) as a function of frequency. (d) The simulated reflection amplitude. (e) The simulated transmission amplitude. (f) The maximum advanced time(red) and delay times(blue) as a function of frequency.
• Figure 4: (a) The delay time as a function of chain length $N$ for edge modes(red) and bulk mode(blue) determined by calculated transmission coefficient. The intrinsic loss and coupling rates are $\kappa_i/\omega_0=10^{-5}$ and $\kappa_{c0}/\omega_0=5\times10^{-3}$. (b) The delay time as a function of chain length $N$ with a higher intrinsic loss and coupling rates $\kappa_i/\omega_0=10^{-4}$ and $\kappa_{c0}/\omega_0=5\times10^{-3}$. (c) The delay times deduced from the $S_{31}$ data obtained by finite-element simulations. (d) The advance time deduced from the $S_{21}$ data obtained by finite-element simulations. (e) The delay times deduced from the $S_{31}$ data from the $N=10$ and $N=20$ normal conductor samples. Vertical arrows indicate the results for edge modes. (f) The advance time deduced from the $S_{21}$ data from the $N=10$ and $N=20$ normal conductor samples.