Non-Hermitian topological filters

TL;DR

The paper addresses how to extract a single topological state from arbitrary, including incoherent, light by designing a non-Hermitian filter that relies on a topological zero mode with eigenvalue $\lambda_0=0$ and eigenvector $|\psi_0\rangle$. The method uses engineered dissipation to suppress non-topological modes while the zero mode remains dissipation-free, leading to a unique steady state $\rho_0=|\psi_0\rangle|\psi_0^{*}\rangle$ in Liouville space where eigenvalues are $\lambda_m-\lambda_n^{*}$. The approach is demonstrated for coherent, incoherent, and partially coherent inputs, with convergence to $\rho_0$ observed in the propagation window $z\in[4,6]$ cm, though energy yields depend on the input's coherence and spatial profile. The result is a robust, linear method for on-demand generation of topologically protected states in integrated photonics, with potential generalization to quantum regimes using multi-photon states across zero-mode lattices.

Abstract

We introduce a non-Hermitian photonic filter that harnesses dissipation to selectively isolate a desired topological state. In science and engineering, dissipation is often used to filter incoherent waves, producing a pure coherent output. Here, we apply this principle to topological states, creating a linear filter that effectively isolates a specific topological state regardless of the initial input's coherence properties. This approach creates a dissipation-free topological subspace, where the desired states are preserved and their topological protection is enhanced. Our work provides a versatile and simple method for topological state selection, opening the door to new applications in integrated topological photonics.

Figures (5)

• Figure 1: a) Destructive interference of the evanescent amplitudes of $\varphi_L$, and $-\varphi_R$ occurring in the central waveguide. Black curves represent the refractive index profile defining the waveguide trimer. $\Delta n(x)$ is the refractive-index profile of the central waveguide. The blue and red shaded areas represent the terms $\varphi_{L}^{*}(x)\Delta n$ and $\varphi_{R}^{*}(x)\Delta n(x)$, respectively. Hence, is clear that the integral defining the simultaneous coupling $\kappa_S$ in Eq. \ref{['eq:CoupledEqn']} is identically zero. b) Central section of a long-long defective SSH PTI lattice with engineered dissipation in the odd parity sublattice. The defect waveguide is located right at the center of the array $(m=10)$. Losses are here realized by sinusoidal modulations of the waveguides along the propagation direction $z$.
• Figure 2: a) Eigenvalue spectrum of a non-Hermitian SSH system comprising $M=21$ waveguides with coupling coefficients $\kappa_1 = 2\text{ cm}^{-1}$, $\kappa_2 = 1 \text{ cm}^{-1}$, and dissipation rate $\gamma=1\text{ cm}^{-1}$. b) Liouville (correlation) spectrum. Inset in b) is a close-up of the central part of the spectrum showing that the eigenvalue for the topological mode 221st eigenvalue is purely real. c) Topological zero-eigenvalue mode and its amplitudes populating the non-Hermitian SSH lattice.
• Figure 3: Coherent light dynamics in a non-Hermitian topological SSH filter. (a) Coherent single site excitation $\hat{\rho}(0)$ at the $m= 10$th waveguide. (b) Coherent topological steady state $\hat{\rho}_{0}(z)$ at the output of the array. (c) Diagonal elements of the correlation matrix $\hat{\rho}^{}(z)$ representing the evolution of the waveguide intensities $I_m(z)$. (d) Trace distance as a function of the propagation coordinate $z$. (e) Evolution of the total intensity $I_{t}(z)$.
• Figure 4: Incoherent (left) and partially coherent (right) light dynamics in a non-Hermitian topological SSH filter. (a), (f) Input correlation matrix $\hat{\rho}(0)$ representing a fully incoherent excitation of the entire waveguide system and a partially coherent excitation with Gaussian profile centered at the $m = 10$th waveguide. (b), (g) Filtered coherent topological steady state $\hat{\rho}_{0}(z)$ at the output of the system. (c), (h) Diagonal elements of the correlation matrix $\hat{\rho}^{}(z)$ representing the evolution of the waveguide intensities $I_m(z)$. The colormaps were normalized independently. (d), (i) Trace distance as a function of the propagation coordinate $z$. (e), (j) Evolution of the total intensity $I_{t}(z)$.
• Figure 5: (a) Non-Hermitian waveguide system in real space. $\kappa_{1,2}$ are the coupling coefficients and $\gamma$ is the dissipation rate. (b) Non-Hermitian Hamiltonian that governs light dynamics in the waveguide array in (a). (c) Liouville (correlation) space corresponding to the waveguide system (a). The Liouvillian for the system pictorially shown in (c) is obtained using the expression $\hat{\mathcal{L}}=\hat{H}_{\text{eff}}\otimes\hat{I}-\hat{I}\otimes\hat{H}_{\text{eff}}^{\dagger}$, where $\hat{H}_{\text{eff}}$ is the non-Hermitian Hamiltonian depicted in (b).