Silicon photonic circuit maps exceptional topology on a Klein bottle

A programmable silicon photonic circuit has experimentally emulated a two-band non-Hermitian system whose momentum-like parameter space forms a Klein bottle, a nonorientable surface. The researchers used differential loss to reproduce non-unitary dynamics without adding optical gain to the chip, then reconstructed the complex Hamiltonian from measured amplitudes and phases. Their measurements identified a single Fermi arc and provided evidence for same-charge exceptional points, or an exceptional-point monopole, that cannot annihilate locally. The study reports the first photonic realization of exceptional topology on a nonorientable manifold.

The experiment implements a Hamiltonian with two coupled modes and a detuning that varies across parameters p and q. Its non-symmorphic symmetry identifies H(p, q) with H(−p, q + π), reducing the fundamental domain to a rectangle with an orientation-reversing boundary. That geometry turns the parameter rectangle into a Klein bottle rather than an orientable surface. The nonorientability modifies the global rules governing degeneracies: the usual fermion-doubling constraint becomes a relation between boundary braids, and a phase with a nonzero invariant can remain gapless while hosting exceptional points whose topological charges cannot be removed locally. For the two-band model used here, the relevant braid-group elements reduce to integer winding numbers.

An exceptional point is a non-Hermitian degeneracy in which eigenvalues and their corresponding eigenstates coalesce, reducing the dimensionality of the eigenspace. This differs from a diabolic point in a Hermitian system, where eigenvalues can coincide while the associated eigenstates remain orthogonal. The non-diagonalizable character of exceptional points allows eigenvalues to exchange when a degeneracy is encircled in parameter space. Their winding properties therefore encode how the spectrum changes under continuous parameter variation. In this experiment, that information was read from the orientation of a Fermi arc, defined as the curve in parameter space where the imaginary part of the eigenvalue gap closes.

Non-Hermitian photonic systems commonly obtain their behavior from optical gain and loss, but the researchers used a different strategy for a linear, CMOS-compatible silicon circuit. They first shifted the Hamiltonian so that all eigenvalues lay in the lower half of the complex plane, then embedded the desired non-unitary evolution in a larger unitary operation through auxiliary, or ancilla, modes. The unitary operation coherently redirects energy into those modes, which are excluded from the nominal output, while reproducing the effective differential-loss dynamics in the measured system. The circuit therefore simulated the target Hamiltonian without incorporating a nonlinear gain material or a dedicated optical amplifier.

The experimental platform was an 8-by-8 silicon photonic integrated circuit containing 28 thermally tunable Mach–Zehnder interferometers and 56 heating electrodes. Each interferometer combined multimode interference couplers with internal and external phase shifters, allowing independent control of beam splitting and relative phase. A 1550-nanometer pulsed diode laser supplied coherent light, and a polarization controller selected the transverse electric mode in the waveguides. One optical switch selected among the principal input modes, while a second rapidly scanned the eight output channels so the full intensity distribution could be recorded with a power meter. Calibrated electrical currents controlled the phase shifters, allowing each target transformation to be synthesized on the mesh and reconfigured for different parameter values.

Initial tests covered coupling strengths from 1 to 12 and compared measured intensities in the first two modes with numerical simulations of oscillation and decay. Across five selected combinations of the coupling c and the phase parameters p and q, the study reported weighted correlation coefficients above 0.97. These measurements checked the implementation over a broad range of couplings before the topological analysis focused on c = 1. The demonstrated quantity was a controlled set of optical transformations rather than a direct physical image of the mathematical surface: the Klein-bottle structure arose from the boundary relations imposed on the programmed two-mode Hamiltonians.

To recover the effective non-Hermitian Hamiltonian, the team measured all four complex elements of the two-mode evolution operator at five times: 0.05π, 0.10π, 0.15π, 0.20π and 0.25π. The investigation sampled 11 points along the predicted Fermi arc and eight points in gapped regions. At points on the arc, the two diagonal evolution-matrix elements showed minimal amplitude separation, accompanied by symmetric phase bifurcation. At off-arc points, their amplitudes diverged markedly and the phase bifurcation was asymmetric. Fitting the measured matrix elements across the five times produced an effective generator whose real and imaginary components could be compared with theoretical calculations. The overall reported agreement between experiment and theory for the illustrated reconstruction reached 98.83%, while small nonzero real diagonal terms in one comparison were attributed to residual experimental noise.

The phase diagram itself was computed numerically, with a small perturbation applied to sharpen the visibility of the Fermi arc, and the experiment supplied targeted measurements around that predicted structure. The calculation showed one Fermi arc crossing only the orientation-reversing boundary associated with the p-axis. The arc retained a consistent orientation on either side of that boundary, while the q-axis boundary carried a trivial braid. In the authors’ topological analysis, this meant that the p-direction boundary braid was nontrivial and that the total exceptional-point charge was plus or minus two. The nonzero flow had to be sourced by exceptional points at which a Fermi arc ends or, in a more finely tuned configuration, by a monopole at which several arcs emerge. The measured orientation therefore supported the presence of two same-charge exceptional points, or an exceptional-point monopole, that could not annihilate locally on the Klein bottle.

The result shows that reconfigurable linear photonics can emulate complex non-Hermitian Hamiltonians and probe their exceptional topology without material gain. Each coordinate in the mapped parameter space is generated through calibrated internal and external phase settings, allowing the circuit to be reset for different target evolution operators. The authors distinguish this approach from related demonstrations in stabilized fiber rings, metasurfaces, optical cavities, synthetic dimensions, dual-ring fibers and real-space lattices, where non-Hermitian landscapes are produced by tuning physical degrees of freedom. The present experiment instead embeds a specifically nonorientable model directly in a programmable integrated circuit. The demonstrated result is a classical photonic emulation rather than a quantum implementation, although the authors note that coherently redirecting energy into ancilla modes could support future quantum state-engineering work, including coherent absorption measurement.

Subject of Research: Experimental photonic emulation and mapping of non-Hermitian exceptional topology on a Klein bottle using a programmable silicon circuit.

Article Title: Non-Hermitian exceptional topology on a Klein bottle photonic circuit

Article References: Xu, Z.-S., König, J. L. K., Cataldo, A., Yadgirkar, R., Krishna, G., Deenadayalan, V., Zwiller, V., Preble, S., Bergholtz, E. J., Gao, J., & Elshaari, A. W. (2026). Non-Hermitian exceptional topology on a Klein bottle photonic circuit. Light: Science & Applications, 15(1), Article 390. https://doi.org/10.1038/s41377-026-02516-x

Image Credits: AI Generated

DOI: 10.1038/s41377-026-02516-x

Keywords: non-Hermitian physics, exceptional points, Klein bottle, photonic integrated circuit, Fermi arc, silicon photonics, topological physics, Mach–Zehnder interferometers, quantum photonics, unitary dilation, Non-Hermitian, exceptional