Simulating Exceptional Non-Hermitian Metals with Single-Photon Interferometry

We experimentally simulate in a photonic setting non-Hermitian (NH) metals characterized by the topological properties of their nodal band structures. Implementing nonunitary time evolution in reciprocal space followed by interferometric measurements, we probe the complex eigenenergies of the corresponding NH Bloch Hamiltonians, and study in detail the topology of their exceptional lines (ELs), the NH counterpart of nodal lines in Hermitian systems. We focus on two distinct types of NH metals: two-dimensional systems with symmetry-protected ELs, and three-dimensional systems possessing symmetry-independent topological ELs in the form of knots. While both types feature open Fermi surfaces, we experimentally observe their distinctions by analyzing the impact of symmetry-breaking perturbations on the topology of ELs.

Figure 1

Experimental setup. The polarization of the signal photon generated by the standard spontaneous parametric down-conversion is projected into the polarization state $|{\psi }_{\pm }⟩$ with a polarizing beam splitter (PBS), a half-wave plate (HWP), and a quarter-wave plate (QWP), and then goes through the interferometric network. After passing through a 50:50 beam splitter (BS), the photon is either transmitted or reflected, and thus separated into different paths. Subsequently, a nonunitary operation $U^{\prime }$ realized via sets of wave plates and two beam displacers (BDs) is performed on its polarization state in the transmitted mode which acquires a complex phase shift, corresponding to the eigenenergies of $H^{\prime }$. Finally, the photon is detected by avalanche photodiodes (APDs) resulting in a “click click” in coincidence detection event involving another trigger photon. We measure the eigenenergies in momentum space via interferometric measurements.

Figure 2

Observation of symmetry-protected ELs. (a) The real (left column) and imaginary (middle column) parts of the spectral gaps $\mathrm{\Delta }E$ as a function of momentum for the symmetry-protected NH metal $H_1$. Experimental data are shown as black lines and theoretical results are shown as colored contour plots. Right column: the real and imaginary parts of the energy gap $\mathrm{\Delta }E$ as a function of the parameter $d_x$. Theoretical predictions are represented by lines, and the experimental results by symbols. Error bars are obtained by assuming Poisson statistics in the photon-number fluctuations, indicating the statistical uncertainty. Effects due to the symmetry-preserving perturbation $i(\pi /20){\sigma }_y$ and the symmetry-broken perturbation $i(\pi /20){\sigma }_x$ are shown in (b) and (c), respectively.

Figure 3

Observation of symmetry-independent knotted ELs. (a) Blue and green surfaces correspond to $\mathrm{Re}(E^2)=0$ and $\mathrm{Im}(E^2)=0$, respectively. The solid red curve is trefoil knotted EL, i.e., the intersection of two surfaces. Two gray planes correspond to the surfaces with $k_z=0.65$ and 0, respectively, which are chosen in our experiment. $\sqrt{|\mathrm{Re}(E^2)|}$ and $\sqrt{|\mathrm{Im}(E^2)|}$ with fixed $k_z=0.65$ (b) and $k_z=0$ (c) as functions of $k_x$ and $k_y$. Solid black curves are the intersections between the surfaces $\mathrm{Im}(E^2)=0$ and the gray planes. Blue solid curves are the intersections between the surfaces $\mathrm{Re}(E^2)=0$ and the gray planes. EPs correspond to intersections of black and blue curves. Experimental data are shown as the black, blue, and red dots, and theoretical results are shown as the colored curves and the colored contour plots. (d)–(f) Effects of perturbations ${\sum }_{i=x,y,z}{\delta }_i{\sigma }_i$, where ${\delta }_i\in [0,0.4]$ are chosen randomly. In our experiment, we have ${\delta }_x=0.3179$, ${\delta }_y=0.3590$, and ${\delta }_z=0.2211$.

Figure 4

Observation of linked ELs. (a) Blue and green surfaces correspond to $\mathrm{Re}(E^2)=0$ and $\mathrm{Im}(E^2)=0$, respectively. The solid red curves correspond to the Hopf linked ELs, i.e., the intersection of two surfaces. The gray planes are the surfaces with $k_z=k_x,-k_x$, respectively, which are chosen in our experiment. $\sqrt{|\mathrm{Re}(E^2)|}$ and $\sqrt{|\mathrm{Im}(E^2)|}$ with fixed $k_z=k_x$ (b) and $k_z=-k_x$ (c) as functions of $k_x$ and $k_y$. Solid black curves are the intersections between the surfaces $\mathrm{Im}(E^2)=0$ and the gray planes. Blue solid curves are the intersections between the surfaces $\mathrm{Re}(E^2)=0$ and the gray planes. Red solid curves and the intersections of black and blue curves are ELs and EPs corresponding to the intersections between the Hopf link and the gray planes. The experimental data are shown as the black, blue, and red dots, and theoretical results are shown as the colored curves and the colored contour plots.