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

The research centers on a class of scattering phenomena that occur when light interacts with open systems capable of both gain and loss. In conventional time-independent settings, a laser threshold marks the point at which a resonant mode’s amplification exactly balances its leakage out of the cavity, producing self-oscillation at a finite frequency. Coherent perfect absorption, sometimes described as a time-reversed laser or an anti-laser, instead occurs when incoming waves from two or more channels are tuned in amplitude and phase so that all of the incident energy is absorbed or trapped, with nothing reflected or transmitted. Until recently, these two conditions were treated largely as separate design goals.

The new analysis shows that in Floquet scattering systems, where a modulation frequency Ω periodically reshapes the system, these phenomena can be understood through the quasi-energy spectrum and the analytic structure of the scattering matrix in the complex frequency plane. Because time modulation is not energy-conserving, the scattered field can exchange integer multiples of the modulation quantum ħΩ with the drive, giving rise to sidebands at frequencies shifted by nΩ. The scattering problem therefore lives not on a single frequency axis but on a ladder of frequency channels, and lasing and absorption thresholds become conditions connecting entire columns of this generalized scattering matrix.

A central concept in the work is the exceptional point, the degeneracy at which two or more eigenmodes of an open system coalesce in both their eigenvalues and their eigenvectors. Exceptional points are the hallmark of non-Hermitian physics, the mathematics of systems with gain, loss, and exchange of energy with the environment. At an exceptional point, the usual rules of spectral perturbation break down: eigenvalue shifts scale with the square root of a small perturbation rather than linearly, and the response of the system becomes hypersensitive. The new study maps out how exceptional points of the quasi-energy spectrum govern the onset of lasing and the onset of coherent perfect absorption as modulation parameters are varied.

Technically, the authors formulate the problem using the Floquet scattering matrix, an infinite-dimensional matrix whose columns correspond to scattering channels labeled by their sideband index. Lasing thresholds correspond to zeros of the determinant of this matrix at real frequencies, meaning the system emits spontaneously without any input. Coherent perfect absorption corresponds instead to poles of the scattering matrix at real frequencies, meaning the incoming waves excite a mode with no outgoing component. In time-periodic systems, the time-reversal relationship between the two effects becomes subtle, because time modulation itself breaks time-translation symmetry and therefore mixes forward and reversed propagation in a way that static systems never do.

One of the striking conclusions is that periodically driven systems can host simultaneous lasing and coherent perfect absorption at different sideband frequencies within the same device. A structure can be lasing at one frequency channel while perfectly absorbing coherent radiation at another, a possibility with no analogue in time-independent optics. The researchers show that the modulation frequency and amplitude act as tuning knobs that sweep the system’s singularities through the real-frequency axis, switching these regimes on and off in a controllable fashion. This transforms what was previously a matter of carefully engineered material gain and loss into a matter of dynamical design.

The framework also clarifies the role of exceptional points in shaping the emission and absorption spectra. Near an exceptional point, the density of quasi-energy states is strongly modified, and the coupling between sidebands can either enhance or suppress the net gain of a mode. The authors demonstrate that exceptional points in the Floquet spectrum signal the merging of lasing and absorption conditions and can be used to coalesce multiple lasing modes into a single-mode state, a result of direct relevance to building stable single-frequency microlasers. Conversely, exceptional points associated with the absorption branch determine how sharply the anti-laser condition can be satisfied and how sensitive it is to phase errors in the incoming beams.

From an experimental standpoint, the findings arrive at a moment when time-modulated photonic platforms are maturing rapidly. Electro-optically modulated resonators, temporally modulated metamaterials, and acousto-optically driven waveguides all provide realistic settings in which the modulation frequency can reach a sizable fraction of the optical carrier detuning between channels. The theory indicates that such platforms should be able to realize Floquet lasers and Floquet anti-lasers without requiring exotic gain media, simply by programming the temporal profile of the modulation. The sideband structure that once appeared as an unwanted parasitic effect is instead promoted to the central resource that enables simultaneous emission and absorption.

The broader significance of the work lies in its unification of several threads of modern optics research. Non-Hermitian physics has shown over the past decade that loss, usually seen as a nuisance, can be a design tool; topological photonics has shown that periodic driving can create band structures with no static counterpart; and coherent perfect absorption has established that absorption can be engineered with the same precision as emission. By treating all three within the common language of Floquet scattering, the study provides a conceptual map for devices in which light is steered, amplified, trapped, and switched by time alone. Potential applications range from on-chip single-mode lasers and sensitive modulators to unidirectional absorbers, wireless power delivery schemes, and fundamental tests of time-reversal symmetry in driven open systems.

The authors emphasize that their results are general and apply to any linear scattering system with periodic time modulation, whether implemented in photonics, acoustics, microwaves, or even matter waves. As experimental groups begin to test the predictions, the exceptional points that once lived only in the mathematics of non-Hermitian operators may soon be observed at work inside operating laser cavities, telling light precisely when to shine and when to vanish.

Subject of Research: Exceptional points, lasing, and coherent perfect absorption in Floquet scattering systems

Article Title: Exceptional points, lasing, and coherent perfect absorption in Floquet scattering systems

Article References: Globosits, D., Garg, P., Hüpfl, J., Canós Valero, A., Weiss, T., Rockstuhl, C., & Rotter, S. (2026). Exceptional points, lasing, and coherent perfect absorption in Floquet scattering systems. Light: Science & Applications, 15(1), Article 375. https://doi.org/10.1038/s41377-026-02380-9

Image Credits: AI Generated

DOI: 10.1038/s41377-026-02380-9

Keywords: exceptional points, Floquet scattering, lasing, coherent perfect absorption, non-Hermitian physics, time-modulated photonics, quasi-energy, anti-laser, sidebands, photonic resonators, open systems, Exceptional