Measuring Topological Invariants in Photonic Systems

Motivated by the recent theoretical and experimental progress in implementing topological orders with photons, we analyze photonic systems with different topologies and present a scheme to probe their topological features. Specifically, we propose a scheme to modify the boundary phases to manipulate edge state dynamics. Such a scheme allows one to measure the winding number of the edge states. Furthermore, we discuss the effect of loss and disorder on the validity of our approach.

Figure 1

(a) Coupled resonators in a ring configuration: Photons hop between resonators with the rate $J$, except for one link where they hop with $J{e}^{i\varphi }$. The inset shows that the boundary can be modified by changing the index of refraction of the connecting resonators. (b) Dispersion relation when the boundary phase changes from zero to $2\pi$, for a ring of 10 resonators.

Figure 2

Probing the state transfer in a ring using transmission spectroscopy. The simulation is performed for 10 sites and ${\kappa }_{ex}/J=0.1$. The input and output probes are separated by two sites, as shown in Fig. 1.

Figure 3

(a) Cylinder or annulus topology, the system is under a uniform magnetic flux per plaquette $\alpha$ and the hole is threaded with a magnetic flux $\varphi$. (b) shows the dispersion relation for $\varphi =0$. The bulk gap indices are shown. (c),(d) The state transfer for inner and outer edges, respectively, where the threading flux is changed from zero to $2\pi$. The simulation is performed for $(N_x,N_y)=(20,10)$, $\alpha =1/4$, and ${\kappa }_{ex}/J=0.1$. Blue shades represent the bulk states. The dashed lines are guides to the eye to highlight the number of edge states transferred during a single flux insertion.

Figure 4

(a) Dispersion relation of the fourth and the fifth bands for $\alpha =1/6$ for a $70\times 20$ lattice. (b) Transmission spectrum as a function of the twist angle $\varphi$ for a $20\times 14$ lattice. The dashed lines are guides to the eye to highlight the number of edge states transferred during a single flux insertion.

Figure 5

(a) Transmission spectrum for different values of intrinsic loss ${\kappa }_{\text{in}}/J=0$, 0.01, 0.05, 0.1. (b) Transmission spectrum in the presence of intrinsic loss (${\kappa }_{\text{in}}=0.01$) and a Gaussian disorder [$\sigma (U)/J=0.1$]. All the other parameters are the same as Fig. 3.