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Experimental observation of energy-band Riemann surface

This paper presents the first experimental observation of an energy-band Riemann surface in a non-Hermitian photonic system using a synthetic frequency dimension, thereby revealing key topological features such as complex-energy winding, the generalized Brillouin zone, and branch points to establish a unified framework for studying non-Hermitian topological physics.

Original authors: Dali Cheng, Heming Wang, Janet Zhong, Eran Lustig, Charles Roques-Carmes, Shanhui Fan

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dali Cheng, Heming Wang, Janet Zhong, Eran Lustig, Charles Roques-Carmes, Shanhui Fan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a map of a city. Usually, maps are flat: you have streets (locations) and buildings (energy levels). If you walk down a street, you know exactly how much energy it takes to get from point A to point B. This is how most physics works in a "perfect" world where nothing is lost or gained.

But in the real world, things aren't perfect. Systems lose energy (like friction) or gain energy (like a battery). In physics, we call this non-Hermiticity. When you add this "messiness" to the map, things get weird. The map stops being flat and starts twisting into a 3D shape, like a spiral staircase or a twisted ribbon. Physicists call this shape a Riemann surface.

For a long time, scientists had a very detailed mathematical map of this twisted ribbon, but they had never actually seen it. They could calculate it on a computer, but they couldn't touch it or measure it in the real world.

This paper is the first time someone has built a machine to actually "see" and measure this twisted ribbon.

Here is how they did it, using some creative analogies:

1. The Playground: A Ring of Light

Instead of using a real city, the researchers used a ring of light (a photonic ring resonator). Imagine a race track where light particles (photons) run around in circles.

The Tracks: They treated different colors (frequencies) of light as different "lanes" on the track.
The Modulators: They added special gadgets that wiggle the light as it runs, creating connections between the lanes. This turns the ring of light into a giant, invisible "lattice" or grid, similar to a chessboard made of light.

2. The Magic Trick: The "Imaginary" Gauge

The big problem was that this twisted ribbon exists in a world of "complex numbers" (a mix of real numbers and "imaginary" numbers). You can't just measure an "imaginary" distance with a ruler.

To solve this, the researchers used a clever mathematical trick called an Imaginary Gauge Transformation.

The Analogy: Imagine you are trying to map a mountain range that exists in a parallel dimension. You can't climb it directly. But, if you change your perspective slightly—like putting on special 3D glasses that shift your view—you can see parts of that mountain range that were previously hidden.
In the Lab: By tweaking the wiggling gadgets (modulators) in a very specific way, they effectively "shifted" the light's perspective. This allowed them to probe the system at different "imaginary" angles, revealing different slices of the twisted ribbon.

3. Reconstructing the Map

By taking many measurements at different "angles" (different settings of their magic wiggles), they were able to stack these slices on top of each other.

The Result: They built a complete, 3D hologram of the Energy-Band Riemann Surface. It's like taking a 2D map of a city and realizing the streets actually loop up and down in a spiral, and then finally building a physical model of that spiral.

4. What Did They Find on the Map?

Once they had the full map, they could spot some very strange and important features:

The Branch Points (The Knots): On a twisted ribbon, there are spots where the ribbon splits or knots together. In physics, these are called Branch Points. The researchers found these knots. They are like "traffic jams" for energy where the rules of physics get very strange.
The Open-Boundary Spectrum (The Edge Effect): Usually, if you cut a piece of paper, the edge is just a line. But on this twisted ribbon, the edge behaves totally differently than the middle. The researchers showed that the "edge" of the system has its own unique energy signature, which is a key feature of non-Hermitian physics (often called the "Skin Effect," where particles pile up at the edges).
The Winding (The Spiral): They watched how the energy "wound" around as they moved along the ribbon. Sometimes it loops once, sometimes twice. This "winding number" tells you if the system is topologically stable or if it's about to change state.

Why Does This Matter?

Think of this discovery like the first time humans saw the curvature of the Earth. Before that, people thought the world was flat, and their maps were wrong for long-distance travel.

Better Lasers and Sensors: Understanding these twisted ribbons helps engineers design lasers that are more efficient and sensors that are incredibly sensitive (able to detect a single virus, for example).
New Physics: It proves that the "weird" math of complex numbers isn't just a theoretical game; it describes real, physical shapes that light and matter actually form.
Unified View: It gives scientists a single, unified map to understand many different strange phenomena that were previously thought to be unrelated.

In short: The researchers took a mathematical concept that looked like a twisted, multi-layered ribbon, built a machine to "see" it in real life, and proved that this twisted shape is the secret code behind how energy behaves in systems that gain or lose energy.

1. Problem Statement

Non-Hermitian physics, which describes systems exchanging energy with their environment (e.g., via gain and loss in photonics), has revealed novel topological phenomena such as the non-Hermitian skin effect, exceptional points, and unique symmetry classes. The theoretical foundation of these phenomena is non-Hermitian energy band theory, where both energy ( $E$ ) and wavevectors ( $k$ ) can take complex values.

In this framework, the band structure is not a simple curve but a Riemann surface—a multi-sheeted complex manifold connecting complex energies to complex wavevectors. Key physical observables, including the spectrum under open boundary conditions (OBC), the generalized Brillouin zone (GBZ), energy band windings, and branch points, are intrinsically linked to the topology of this surface. Despite decades of theoretical interest dating back to Kohn (1959) and recent theoretical advances, no experimental study had successfully mapped the entire energy-band Riemann surface in any physical system. The primary challenge lies in the inability to probe systems at arbitrary complex wavevectors, as standard experiments are typically limited to real wavevectors (Periodic Boundary Conditions, PBC).

2. Methodology

The authors developed an experimental platform using photonic synthetic frequency dimensions to reconstruct the full energy-band Riemann surface.

Experimental Platform: A ring resonator supporting equally spaced frequency modes. The frequency modes act as lattice sites, and periodic phase and amplitude modulations create couplings between these sites, effectively forming a 1D tight-binding lattice model in the frequency domain.
Imaginary Gauge Transformation: To probe complex wavevectors ( $k = k_{real} + i\sigma$ $k = k_{r e a l} + iσ$ ), the team implemented a tunable imaginary gauge transformation. Mathematically, this corresponds to substituting $\Omega_R t \to \Omega_R t + i\sigma$ $Ω_{R} t \to Ω_{R} t + iσ$ in the modulation signals, where $\sigma$ $σ$ is a real parameter.
- In the physical setup, this was achieved by modifying the time-dependent transmittance of phase ( $\tau_{PM}$ ) and amplitude ( $\tau_{AM}$ ) modulators using hyperbolic functions ( $\cosh \sigma, \sinh \sigma$ ).
- This transformation effectively shifts the imaginary part of the wavevector ( $\text{Im}(k)$ ) without altering the physical boundary conditions of the ring, allowing the system to be probed at arbitrary $\sigma$ .
Measurement Protocol:
1. The resonator is excited by a continuous-wave laser with a frequency detuning $\delta\omega$ .
2. The transmission intensity $\xi(\delta\omega, t)$ is measured as a function of detuning and modulation time $t$ .
3. In this synthetic dimension, time $t$ maps to the wavevector $k$ .
4. The resonance center location yields $\text{Re}(E)$ , and the linewidth yields $\text{Im}(E)$ .
5. By varying $\sigma$ , the team measured the intersection of the Riemann surface with different iso- $\text{Im}(k)$ planes. Stacking these slices reconstructs the 3D surface.

3. Key Contributions

First Experimental Reconstruction: This work presents the first experimental observation and reconstruction of a complete energy-band Riemann surface in a non-Hermitian system.
Unified Framework: It demonstrates that diverse non-Hermitian phenomena (OBC spectrum, GBZ, windings, branch points) are unified manifestations of the topology of the energy-band Riemann surface.
Bulk Measurement of Boundary Phenomena: The authors demonstrated a method to extract the Open Boundary Condition (OBC) spectrum and the Generalized Brillouin Zone (GBZ) from bulk measurements (PBC spectra with imaginary gauge), bypassing the need to physically construct open chains or interfaces.

4. Results

The study focused on a specific non-Hermitian lattice model ( $H_1$ ) with symmetric nearest-neighbor coupling ( $g$ ) and anti-symmetric next-nearest-neighbor coupling ( $\pm \kappa$ ).

Surface Reconstruction: The team successfully mapped the complex energy $E$ as a function of complex wavevector $k$ . The data showed excellent agreement with theoretical predictions, visualizing the multi-sheeted structure of the Riemann surface.
Complex-Energy Windings: By analyzing the trajectories of complex energies as $\text{Re}(k)$ varies from $0$ to $2\pi$ at different $\sigma$ values, the authors observed variations in point-gap topology and winding numbers.
OBC Spectrum and GBZ Identification:
- The authors identified that the OBC spectrum and GBZ correspond to specific branch cuts on the Riemann surface (specifically the cut between the $r$ -th and $(r+1)$ -th Riemann sheets).
- They established a criterion based on winding numbers: A self-intersection point in the complex-energy winding belongs to the OBC spectrum if and only if it is surrounded by four regions with winding numbers $0, +1, 0, -1$ .
- Using this criterion on the measured data, they extracted the OBC spectrum and the GBZ (in the complex- $z$ plane, where $z=e^{ik}$ ), confirming they match theoretical expectations.
Branch Points and Ramification Points:
- The study located branch points (where $\partial E / \partial k = 0$ ), which act as Van Hove singularities.
- They experimentally verified the topological behavior near these points: a trajectory in the complex- $k$ plane encircling a ramification point results in the energy winding twice around the corresponding branch point in the complex- $E$ plane, whereas a non-encircling trajectory results in a simple loop.

5. Significance

Fundamental Physics: This work bridges the gap between abstract mathematical concepts (Riemann surfaces) and physical reality in non-Hermitian systems, providing a direct experimental validation of the theoretical framework.
Methodological Advancement: The technique of using imaginary gauge transformations in synthetic dimensions offers a powerful, versatile tool for probing complex band structures without the limitations of physical boundary engineering.
Future Applications: The unified framework allows for the prediction and measurement of non-Hermitian phenomena (like the skin effect or exceptional points) through bulk measurements. The methodology is generalizable to multi-dimensional and multi-band systems, as well as other degrees of freedom (polarization, orbital angular momentum), and could be extended to continuous photonic systems like photonic crystals.

In conclusion, this paper provides a definitive experimental demonstration that the complex topology of energy-band Riemann surfaces is the underlying mechanism governing the unique spectral and transport properties of non-Hermitian systems.