Exceptional points in a non-Hermitian topological pump

We investigate the effects of non-Hermiticity on topological pumping and uncover a connection between a topological edge invariant based on topological pumping and the winding numbers of exceptional points. In Hermitian lattices, it is known that the topologically nontrivial regime of the topological pump only arises in the infinite-system limit. In finite non-Hermitian lattices, however, topologically nontrivial behavior can also appear, and we show that this can be understood as the effect of encircling a pair of exceptional points during a pumping cycle. This phenomenon is observed experimentally in a non-Hermitian microwave network containing variable-gain amplifiers.

Figure 1

(a) Schematic of a periodic network of directed links and nodes, with tunable gain and loss in two of the links in each unit cell. (b) Schematic of the setup for the Laughlin-Brouwer topological pump. The network is truncated to $N$ unit cells in the $y$ direction, and twisted boundary conditions are applied in the $x$ direction with tunable twist angle $k$ (corresponding to an infinite strip with fixed wave number). (c) Plot of $\arg \left({\sigma }_n\right)$ vs $k$, where $\left\{{\sigma }_n\right\}$ are the eigenvalues of the edge scattering matrix. The model parameters are $\gamma =0$ (Hermitian network), ${\theta }_x={\theta }_y=3\pi /8$, ${\mathrm{\Phi }}_x^3={\mathrm{\Phi }}_y^3=-7\pi /10$, and $\varphi =2\pi /5$; these parameters are defined in Appendix pp1.

Figure 2

(a) The gaps between arguments of scattering matrix eigenvalues, $\mathrm{\Delta }\equiv \text{min}\left(|\arg \left({\sigma }_1\right)-\arg \left({\sigma }_2\right)|\right)$, in the Hermitian limit $\gamma =0$. Circles represent gaps for different $N$. (b) Scattering matrix eigenvalues of the non-Hermitian 2D network, at $k_{ep}\simeq 0.572\pi$ and $N=2$. The left and right axes show the arguments and amplitudes of the eigenvalues, respectively. The other model parameters are the same as in Fig. 1: ${\theta }_x={\theta }_y=3\pi /8$, ${\mathrm{\Phi }}_x^3={\mathrm{\Phi }}_y^3=-7\pi /10$, and $\varphi =2\pi /5$.

Figure 3

Relationship between topological pumping and exceptional points, calculated for a network model of width $N=2$. The underlying band structure is topologically nontrivial, with $\varphi$ in a band gap (${\theta }_x={\theta }_y=3\pi /8$, ${\mathrm{\Phi }}_x^3={\mathrm{\Phi }}_y^3=-0.7\pi$, and $\varphi =2\pi /5$). (a) Parametric loops in the 2D parameter space formed by the gain/loss parameter $\gamma$ (radial coordinate) and pumping parameter $k$ (azimuthal coordinate). Exceptional points (EPs) of $S_{\mathrm{edge}}$ are indicated by stars. (b)–(d) Complex plane trajectories of $\left\{{\sigma }_n\right\}$, the eigenvalues of $S_{\mathrm{edge}}$, corresponding to the parametric loops shown in (a). The two distinct eigenvalue trajectories are plotted as solid and dashed curves, and the unit circle is plotted as dots for comparison. The trajectories in (b) have zero winding around the origin, like the Hermitian finite-$N$ limit; the trajectories in (c) join each other under one cycle because the parametric loop encloses one exceptional point; the trajectories in (d) have nonzero windings, similar to the Hermitian large-$N$ limit. (e),(f) Plots of the gain/loss parameter $\gamma$ at the two EPs [as labeled in (a)] for different $N$.

Figure 4

Behavior of the multivalued function $\sigma \left(z\right)$ given by Eq. (4), illustrating how branch points can produce winding and nonwinding trajectories. Here, we take $\alpha =0.1$. (a) Plot of the parameter space formed by the complex variable $z$, with the branch points of $\sigma \left(z\right)$ indicated by stars, along with three parametric loops corresponding to (i) $\left|z\right|=0.5$, (ii) $\left|z\right|=1$, and (iii) $\left|z\right|=1.5$. (b) The corresponding complex plane trajectories of the two branches of $\sigma \left(z\right)$.

Figure 5

(a) Experimental setup. Each of the identical units, labeled Cell 1, Cell 2, and Cell 3, corresponds to one cell in the topological pump geometry. The twisted boundary condition is applied by tuning the phase shifter (pink boxes) in lower links. The couplers (blue rods) are depicted in the strong-coupling configuration. The weak-coupling configuration is achieved by swapping each coupler's outputs. The overall input and output amplitudes are ${\psi }_{\mathrm{in}}^{1,2}$ and ${\psi }_{\mathrm{out}}^{1,2}$. Their scattering parameters are measured with a network analyzer. (b) Each cell is composed of four links ($j=1,2,3,4$) and two couplers ($S_x,S_y$). (c) Every link in our system is exactly the same and contains one phase shifter, one isolator, one digital controlled variable-gain amplifier, one bandpass filter, and five low-loss handflex interconnect coaxial rf cables.

Figure 6

(a)–(d) Experimentally measured scattering matrix eigenvalues over one pumping cycle, with approximately zero gain/loss. Results are shown for (a),(b) $N=1$ and (c),(d) $N=3$. The weak-coupling regime corresponds to a topologically trivial phase of the underlying network band structure, while the strong-coupling regime corresponds to a topologically nontrivial phase. Directly measured data are plotted with dots and calculations using measured network-component $S$ parameters are shown as solid and dashed curves. The unit circle is indicated by dotted curves. (e) Arguments of the measured scattering matrix eigenvalues $\arg \left({\sigma }_n\right)$ vs pumping parameter $k$. The network is in a topologically nontrivial phase. Inset: behavior near a crossing point; solid curves show theoretical results calculated using the network components' measured $S$ parameters, which predict an avoided crossing for $N=1$ and a crossing for $N=3$.

Figure 7

(a) Parametric loops in the 2D parameter space formed by the logarithmic gain/loss factor $\beta$ (radial coordinate) and pumping parameter $k$ (azimuthal coordinate). The gain/loss distribution is as depicted in Fig. 5. Exceptional-point positions (stars) are calculated from the measured $S$ parameters of the network components. The system size is $N=2$. (b)–(d) Eigenvalues of $S_{\mathrm{edge}}$ in the complex plane. Directly measured data are plotted with dots and calculations using measured network-component $S$ parameters are shown as solid and dashed curves. The unit circle is indicated by dotted curves.