Energy-Resolved Eigenmode Spectroscopy of 1-D and 2-D Non-Hermitian Skin Effects

This paper reports the first energy- and band-resolved spectroscopy of non-Hermitian skin modes in both 1D and 2D frequency lattices realized via an electro-optically modulated ring resonator, directly revealing boundary-localized states and their energy-dependent spatial displacement while establishing a versatile platform for Hamiltonian engineering.

The Gist

Imagine you have a giant, invisible drum made of light. Usually, when you hit a drum, the sound waves bounce around evenly, filling the whole space. But in this experiment, the researchers built a special kind of "drum" where the rules of physics are slightly broken. They created a world where light doesn't just bounce; it gets sucked to one side, piling up like water against a wall.

This phenomenon is called the Non-Hermitian Skin Effect. In simple terms, it's a situation where almost all the energy in a system gets trapped at the very edges, leaving the middle empty.

Here is how the researchers did it and what they found, explained through everyday analogies:

1. The "Synthetic" Ladder

Usually, to study how particles move in a grid (a lattice), you need a physical grid of atoms or wires. But this team used a clever trick. They used a ring of fiber optic cable (like a loop of glass tubing).

Inside this loop, light travels in specific "colors" (frequencies). Instead of moving left or right in space, the light hops from one color to the next. The researchers treated these different colors as if they were rungs on a ladder. This is their "synthetic dimension." It's like playing a piano where the keys aren't arranged in a line, but the sound jumps between them to create a new kind of map.

2. Building the Walls (The Boundaries)

To see the "skin effect," you need a ladder with an end. If the ladder goes on forever, the light just keeps hopping.

• The Trick: They used a second, smaller ring of fiber to act as a "mirror." Every time the light tried to hop to a specific rung on the ladder, this mirror blocked it.
• The Result: They created a finite ladder with clear walls on both sides. This is crucial because the "skin effect" only happens when the light hits a wall and can't go further.

3. The One-Way Slide (Non-Reciprocity)

In a normal hallway, if you walk forward, you can walk backward just as easily. In this experiment, the researchers used electronic modulators to make the hallway one-way.

• Imagine a hallway with a gentle slope. If you walk forward, you glide easily. If you try to walk backward, you have to fight against a strong wind.
• In their light-ladder, the light could hop forward easily but struggled to hop backward. This imbalance is what causes the "skin effect."

4. The Great Pile-Up (The Skin Effect)

Because the light can slide forward easily but gets stuck trying to go back, it doesn't stay in the middle of the ladder.

• The Analogy: Imagine a crowd of people in a hallway where everyone is trying to move forward, but the doors at the back are locked. Everyone eventually piles up against the front door.
• The Discovery: The researchers found that the light energy didn't stay in the middle of their synthetic ladder. Instead, it collapsed and piled up exponentially against one of the boundaries (the "walls" they built). This is the Non-Hermitian Skin Effect.

5. Taking a "Snapshot" of the Light (Spectroscopy)

The hardest part of this research was not just seeing the pile-up, but taking a picture of exactly how the light looked at every single step of the process.

• The Problem: Usually, scientists can only guess what the light is doing inside the system.
• The Solution: They used a high-speed camera technique (heterodyne measurement) to take a "snapshot" of the light at every single rung of the ladder, for every possible energy level.
• The Result: They created a detailed map showing that the light wasn't just randomly stuck at the edge; it formed specific patterns depending on its energy. Some energy levels piled up right at the wall, while others were slightly further back. They called this "Eigenmode Spectroscopy"—essentially, a direct X-ray of the light's behavior.

6. From a Ladder to a Grid (2D)

So far, they had a 1D ladder. But they wanted to see what happens in 2D (a grid).

• The Challenge: In previous experiments, trying to make a 2D grid out of light often resulted in a twisted tube shape (like a Möbius strip), which isn't a true flat grid.
• The Breakthrough: Because they built such strong "walls" (boundaries) in their system, they could connect multiple ladders together without twisting them. They created a true, flat 2D grid of light.
• The Observation: In this 2D grid, they could control the light to flow in specific diagonal directions (like southeast or southwest). They showed that they could trap the light along the edges of this 2D grid, creating "edge states" in two dimensions.

Summary

In short, the researchers built a special playground for light using fiber optics. They created a world where light prefers to move in one direction, causing it to crash and pile up against the walls. They didn't just guess this was happening; they took a high-resolution "movie" of the light to prove exactly how it behaved. Finally, they expanded this from a single line into a flat grid, showing they can control where the light goes with incredible precision.

This work proves that we can now directly "see" and map these strange, edge-trapping behaviors of light, which is a big step toward building better sensors and simulators in the future.

Technical Summary

Technical Summary: Energy-Resolved Eigenmode Spectroscopy of 1-D and 2-D Non-Hermitian Skin Effects

Problem Statement
Non-Hermitian (NH) lattices exhibit the non-Hermitian skin effect (NHSE), where a macroscopic fraction of bulk eigenstates collapse into exponentially localized edge modes. While this phenomenon underpins anomalous bulk-boundary correspondence and enhanced sensing capabilities, direct access to the eigenmodes of NH lattices has been limited. Previous characterizations in frequency-domain photonic platforms have predominantly relied on reciprocal-space ($k$-space) measurements, offering little information about lattice-space occupations. Furthermore, existing frequency synthetic dimension experiments have been largely "band-agnostic," lacking the ability to resolve eigenmodes across the full energy spectrum. A critical challenge has been the realization of stable, strong boundaries in synthetic dimensions to prevent the "twisted" boundary conditions often found in previous implementations, which effectively reduce 2D systems to folded 1D tubes.

Methodology
The authors utilize a frequency synthetic dimension platform based on an electro-optically modulated fiber ring resonator. The system treats resonant frequency modes as synthetic lattice sites, with inter-site coupling engineered via intra-cavity phase and amplitude modulation.

• Boundary Realization: To create finite lattices with strong boundaries, the authors couple the primary ring to an auxiliary ring that is 7 times shorter. This hybridization introduces a mode splitting of $\sim$8 MHz at every 7th resonance of the main ring. This splitting acts as a "frequency mirror," distorting local resonance spacing and suppressing further coupling, thereby terminating the synthetic lattice and creating well-defined open boundary conditions (OBC).
• Hamiltonian Engineering: The platform implements the Hatano-Nelson (HN) model, a foundational 1D NH lattice with nonreciprocal couplings ($C \pm \Delta e^{\pm i\phi}$). Nonreciprocity is controlled electronically by tuning the relative strength and phase of the modulation.
• Spectroscopy Technique: The authors employ a "band- and energy-resolved eigenmode spectroscopy" approach.
  • Band Structure: Laser detuning maps to quasi-energy ($E$), while time-resolved transmission maps to quasi-momentum ($k$). This allows direct measurement of the band structure.
  • Site-Resolved Readout: A cascaded optical-electrical heterodyne scheme enables low-noise, one-shot measurements of site occupations across nearly two full finite lattices ($\sim$12 sites). By sweeping the input laser across the entire lattice, the authors reconstruct the spatial profile of each mode at specific eigenenergies.
• 2D Construction: To extend beyond 1D, the authors incorporate long-range couplings between distinct finite lattices (separated by frequency mirrors). This architecture breaks 1D translational symmetry explicitly, enabling the construction of "genuine" 2D rectangular lattices rather than the twisted tube geometries common in previous frequency implementations.

Key Results

• 1D Eigenmode Spectroscopy: The authors successfully demonstrate the NHSE in a 6-site and 7-site finite lattice. They observe that under OBC, the complex energy spectrum collapses onto the real axis, and eigenmodes become exponentially localized at the boundaries. Crucially, the spectroscopy reveals that eigenmodes closer to extremal eigenvalues exhibit a slight inward shift of their peak amplitude away from the edge, a trend confirmed by tight-binding simulations.
• Energy-Dependent Localization: The study provides the first direct, energy-resolved characterization of NH eigenmodes. The data shows that the spatial profile of the modes varies significantly with detuning (eigenenergy), confirming that the skin effect is not uniform across the spectrum but depends on the specific eigenenergy.
• 2D Directional Transport: In the constructed 2D frequency lattices, the authors observe tunable directional transport. By adjusting coupling asymmetries along the synthetic axes, they demonstrate accumulation of excitations toward specific corners (e.g., southeast or southwest).
• Edge Localization in 2D: When exciting a site at the boundary of the 2D lattice, the population localizes along the edge in one dimension while remaining unbounded in the other, effectively reducing the system to an unbounded 1D lattice along the $Y$-direction. This confirms the presence of edge localization in two synthetic dimensions.

Significance and Claims
The paper claims to provide one of the first experimental demonstrations of the NHSE in synthetic frequency dimensions with strong boundaries. Its primary contributions are:

1. Direct Eigenmode Spectroscopy: Establishing a method for direct, band- and energy-resolved access to NH eigenmodes, moving beyond indirect inferences or purely reciprocal-space measurements.
2. Genuine 2D Lattices: Demonstrating that strong frequency boundaries allow for the construction of irreducible 2D lattices, circumventing the twisted boundary conditions that have limited previous frequency-domain experiments to folded 1D systems.
3. Platform Versatility: Validating frequency-domain synthetic photonic platforms as a flexible tool for Hamiltonian engineering, capable of simulating complex non-Hermitian physics and tunable directional transport.

The authors position these results as a step toward more versatile quantum simulators and non-Hermitian sensors, noting that the platform's ability to engineer strong boundaries and reconfigurable coupling geometries makes it suitable for exploring higher-dimensional non-Hermitian phenomena.