Topological Phase Transition in the Non-Hermitian Coupled Resonator Array

In a two-dimensional non-Hermitian topological photonic system, the physics of topological states is complicated, which brings great challenges for clarifying the topological phase transitions and achieving precise active control. Here, we prove the topological phase transition exists in a two-dimensional parity-time-symmetric coupled-resonator optical waveguide system. We reveal the inherent condition of the appearance of topological phase transition, which is described by the analytical algebraic relation of coupling strength and the quantity of gain-loss. In this framework, the system can be switched between the topological and trivial states by pumping the site rings. This work provides a new degree of freedom to control topological states and offers a scheme for studying non-Hermitian topological photonics.

Figure 1

(a) CROW system with a parity-time-symmetric lattice. Horizontally neighboring site rings are gain (red) and loss (black), while all the link rings (gray) are passive. A schematic topological edge mode is shown on the lower edge. (b) Coupling model of site rings. All link rings are replaced with dashed lines, and their couplings with site rings are abstracted into the transfer matrices, which are labeled on the model. (c) Unit cell of the lattice with field amplitudes, phase delay, and coupling coefficients labeled on it.

Figure 2

(a) and (b): Bulk and projected band structures of the real frequency with different gain-loss quantities $\kappa$ and a fixed coupling strength $t=0.2$, respectively. When $\kappa <0.761$, there are topological edge states in the band gap. Two bulk bands get closer to form a Dirac point until $\kappa =0.761$. The Dirac point opens up a new gap without any edge state in it when $\kappa >0.761$.

Figure 3

Topological phase diagram of coupling strength $\gamma$ and gain-loss quantity $\kappa$. The relation $\tan \gamma =\cosh 2\kappa$ is plotted on the diagram, which is the boundary of topological phase transition. The blue line denotes the topological phase transition process in the Hermitian system with the variation of $\gamma$. The green line denotes the topological phase transition process under the variation of $\kappa$ with a fixed $\gamma$ ($\tan \gamma =2.4$), on which four dots with different colors are highlighted, exactly corresponding to the four quantities of gain-loss in Fig. 2.

Figure 4

The electric field distribution under different gain-loss quantities $\kappa$. An $8\times 8$ CROW array is solved numerically based on the transfer matrix method with accurate coupling conditions. The input is set on the upper-left corner, and the output is set on the upper-right corner. The frequency of input light is selected close to the Dirac point—that is, $\omega =0.46$. The parameters $\kappa$ of each diagram correspond to the dots in Fig. 3.

Figure 5

(a) The transmission varies with gain-loss quantity $\kappa$ under different coupling strengths $\gamma$, where ${\gamma }_{\max }=\pi /2$. The model has the same configuration as Fig. 4, but with only five unit cells. (b)–(e): The eigenmodes of two neighboring bulk bands at $\mathrm{\Gamma }$ points before and after the topological phase transition. The modes of upper (b) and lower (c) bands reverse to those of lower (e) and upper (d) bands across the phase transition, respectively. The band inversion happens associated with the topological phase transition.