Non-Hermitian Exceptional Topology on a Klein Bottle Photonic Circuit

Abstract

Non-Hermitian physics has unlocked a wealth of unconventional wave phenomena beyond the reach of Hermitian systems, with exceptional points (EPs) driving enhanced sensitivity, nonreciprocal transport, and topological behavior unique to non-Hermitian degeneracies. Here, we present a scalable and reconfigurable silicon photonic integrated circuit capable of emulating arbitrary non-Hermitian time evolution with high precision. Using this programmable platform, we implement a two-band non-Hermitian Hamiltonian defined on a Klein-bottle topology a nonorientable parameter space that enables exceptional phases forbidden on orientable manifolds. Through an on-chip amplitude-and-phase reconstruction protocol, we retrieve the full complex Hamiltonian at multiple points in parameter space and experimentally map the associated Fermi arc where the imaginary eigenvalue gap closes. The orientation of the measured Fermi arc reveals a nontrivial exceptional topology: it implies the presence of same-charge EPs (or an EP monopole) that cannot annihilate locally on the Klein bottle. Our results demonstrate the first photonic realization of exceptional topology on a nonorientable manifold and establish a versatile platform for exploring exotic non-Hermitian and topological models relevant to classical and quantum photonics.

Figures (4)

• Figure 1: Realization of non-Hermitian model and topology.(a)Exceptional phases on nonorientable manifolds. Under the shown boundary identifications, the free parameter space rectangle transforms into a Klein bottle, the nonorientable momentum manifold on which the non-Hermitian Hamiltonian is defined. (b) The matrices $U_1$, $U_2$, $U_{E1}$, and $U_{E2}$ form a unitary $4 \times 4$ matrix that simulates the dynamics of a non-Hermitian $2 \times 2$ evolution operator $T$ , with coupling to the environment implemented through ancilla modes 1 and 2. The components $U_1$ and $U_2$ correspond to tunable beam splitters and phase shifters acting within the system, while $U_{E1}$ and $U_{E2}$ couple the system to the environment, represented here by the ancilla modes. (c) Schematic of a single Mach-Zehnder interferometer (MZI) unit, serving as the fundamental computational element of the programmable photonic circuit. The brown boxes represent 3-dB MMIs, and the circles with arrows denote the external and internal phase shifters controlling $\phi$ and $\theta$, respectively.(d) An illustration of the experimental setup is shown, where the sizes of individual components have been adjusted for clarity. A 1550 nm pulsed diode laser (PDL 800-B) is employed as the light source. The polarization of the transmitted light is controlled to be in the TE mode using a three-paddle polarization controller. Two $1\times12$ MEMS optical switches are used in the system (Use eight of the channels). The input light is coupled into the first optical switch, allowing any of the eight operational modes or four additional test modes to be selected under computer control.The core of the system is an $8\times8$ silicon PIC comprising an array of 28 Mach-Zehnder Interferometers (MZIs) and 56 heating-electrodes. The PIC is controlled by one of eight current driver modules (Q8iv, Qontrol Ltd.) interfaced with the host computer running the GUI based on $Python$ 3.11. The output light is coupled into the second optical switch and its intensity is measured using a Thorlabs PM400 power meter. During the measurement process, the second optical switch rapidly cycles through output modes 1 to 8 at each experimental step to acquire the full set of output intensities.
• Figure 2: Simulation (left) vs experiment (right) at five parameter points. Each subpanel shows normalized intensities of mode 1 (red) and mode 2 (blue) versus normalized time; left column: numerical simulation, right column: experimental data. Parameters: (a)$c=1$, $p=0.5\pi$, $q=0.51\pi$; (b)$c=3$, $p=0.2\pi$, $q=0.986\pi$; (c)$c=6$, $p=\pi$, $q=0.5\pi$; (d)$c=7$, $p=\pi$, $q=0.5\pi$; (e)$c=12$, $p=\pi$, $q=0.5\pi$. The Fidelity is shown as an inset. The measured traces closely match the simulations in oscillation and decay, confirming faithful implementation of the non-Hermitian dynamics at each $(c,p,q)$ point.
• Figure 3: Simulated and experimentally sampled Fermi arc with phase- and amplitude-resolved measurements.(a) Numerically computed bifurcation of the imaginary parts of the eigenvalues over the $(p,q)$ parameter space for $c=1$, together with the experimentally sampled parameter points. White markers indicate points that lie on the Fermi arc (imaginary-gap closure), while red markers denote points in gapped regions. (b–g) For each labeled parameter point, we measure the amplitudes and phases of all four elements of the time-evolution operator $T_{t_n}=e^{-iH' t_n}$ at five evolution times $t_n=\{0.05\pi, 0.10\pi, 0.15\pi, 0.20\pi, 0.25\pi\}$. Each subpanel shows the experimentally measured data together with the corresponding theoretical curves for the four complex matrix elements. Panels (b–e) (1-4 in (a)) correspond to points on the Fermi arc at $(p,q) = (0.1\pi,0.99\pi)$, $(0.5\pi,0.5\pi)$, $(1.0\pi,0.5\pi)$, and $(1.85\pi,0.3\pi)$, respectively. Panels (f–g) (5 and 6 in (a)) correspond to off-arc points at $(p,q) = (1.4\pi,0.9\pi)$ and $(0.75\pi,0.8\pi)$, which exhibit a finite imaginary eigenvalue gap. These measurements validate the accurate implementation of the non-unitary dynamics and form the basis for the Hamiltonian reconstruction shown in Fig. \ref{['f4']}.
• Figure 4: Theoretical vs. experimentally reconstructed Hamiltonian elements.(a–f) Each subpanel compares the real and imaginary parts of the four Hamiltonian matrix elements for six representative parameter points. The upper panel in each subfigure shows the theoretically calculated components, while the lower panel shows the corresponding experimentally reconstructed values obtained from fitting the measured time-evolution operators. Panels (a–d) correspond to points located on the Fermi arc, whereas panels (e–f) correspond to points in gapped regions away from the arc.