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Topological Pumping Through a Localized Bulk in a Photonic Hofstadter System

This paper demonstrates a photonic realization of the Harper-Hofstadter and Aubry-André models in a multilayer Bragg stack, revealing a crossover from adiabatic Thouless pumping to a Landau-Zener-mediated topological pump where chiral edge states persist despite strong quasiperiodic disorder-induced bulk localization.

Original authors: Kyle Linn, Megan Goh, Sachin Vaidya, Christina Jörg, Mikael C. Rechtsman

Published 2026-03-25

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CC BY 4.0

Original authors: Kyle Linn, Megan Goh, Sachin Vaidya, Christina Jörg, Mikael C. Rechtsman

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

The Big Idea: A Light Highway with a Twist

Imagine you are trying to drive a car from one side of a city to the other. Usually, you need a clear road. But what if the road is full of potholes, construction zones, and random obstacles? In the world of physics, this is called disorder, and it usually stops things from moving.

However, there is a special kind of "magic highway" in physics called a Topological System. On this highway, the traffic (light or electrons) is so protected by the laws of physics that it can ignore potholes and keep flowing, even if the road is messy. This is the "Quantum Hall Effect."

This paper is about building a photonic (light-based) version of this magic highway. The researchers wanted to see what happens when they make the road extremely bumpy and chaotic. Do the cars stop? Or do they find a new way to keep moving?

The Setup: A Synthetic Dimension

To build this highway, the researchers didn't use a giant 2D map. Instead, they used a 1D stack of glass and silicon layers (like a very thin, multi-layered cake).

The Trick: They treated the thickness of these layers as a "knob" they could turn. By changing the thickness in a specific, repeating-but-never-exactly-the-same pattern (called quasiperiodic), they created a "fake" second dimension.
The Analogy: Imagine a long hallway (the 1D stack). Usually, you can only walk forward or backward. But if every floor tile in the hallway is painted with a different shade of blue, and the shades follow a complex pattern, you can imagine "walking" through the colors as if you were walking sideways. The researchers used this "color dimension" to simulate a 2D magnetic field for light.

The Discovery: The "Butterfly" and the "Edge"

When they mapped out how light moves through this stack, they saw a famous pattern called the Hofstadter Butterfly.

The Butterfly: Imagine a fractal butterfly made of energy bands. The "wings" are the places where light can travel freely (extended states). The "gaps" between the wings are places where light is blocked.
The Edge States: Inside the gaps, there are special "lanes" where light can travel along the very edge of the stack without getting stuck. These are the Chiral Edge States. They are like a one-way street that is immune to traffic jams.

The Experiment: Turning Up the Chaos

The researchers asked: What happens if we make the "bumps" in our road really, really big?

Low Chaos (Weak Modulation): When the layer thicknesses vary slightly, the "bulk" (the middle of the stack) is like a wide-open field. Light flows through the middle and loops around to the edges. This is called Thouless Pumping. It's like a conveyor belt moving smoothly.
High Chaos (Strong Modulation): When they increased the variation in layer thickness, something wild happened. The "bulk" (the middle of the stack) suddenly became frozen. The light couldn't move through the middle anymore; it got stuck in place. This is called Localization.

The Surprise: Even though the middle of the stack was frozen solid, the edge lanes kept working! The light still traveled from one side to the other, even though the "highway" in the middle was blocked.

How Does It Work? The "Landau-Zener" Handoff

If the middle is frozen, how does the light get from Edge A to Edge B?

The Old Way (Adiabatic): In the low-chaos version, the light smoothly glides through the bulk, like a surfer riding a wave.
The New Way (Landau-Zener): In the high-chaos version, the light can't glide. Instead, it has to perform a series of quantum jumps.

The Metaphor: Imagine you are trying to cross a frozen lake (the localized bulk) to get to the other side. You can't walk on the ice because it's too slippery and broken.

Instead, you see a series of small, isolated ice floes (localized states) floating close to each other.
You have to jump from one floe to the next.
If you jump at the exact right moment with the right energy, you make it across. If you miss, you fall in.
The researchers found that the light is doing exactly this: it is tunneling (jumping) between these frozen pockets of light. It's a "handoff" mechanism where the light hops from one localized spot to another until it reaches the other edge.

Why Does This Matter?

Robustness: It proves that topological protection is incredibly strong. Even when the system is so disordered that the "middle" stops working, the "edge" keeps going.
New Physics: It shows a crossover from a smooth, flowing transport (like a river) to a hopping, jumping transport (like a frog on lily pads).
Applications: This could lead to better optical devices that are immune to manufacturing defects. If you build a laser or a sensor using this principle, it will keep working even if the materials aren't perfect, because the light knows how to "hop" through the mess.

Summary

The researchers built a special stack of glass layers that acts like a 2D magnetic world for light. They cranked up the disorder until the middle of the stack froze solid. Surprisingly, the light didn't stop; it just changed its strategy. Instead of flowing smoothly, it started hopping between frozen spots to get to the other side. This proves that topological highways are so tough, they can survive even when the road is completely broken, as long as you know how to jump.

1. Problem Statement

The Quantum Hall Effect (QHE) and topological transport are typically studied in two-dimensional electron gases or solid-state systems. While the robustness of chiral edge states against weak perturbations (like fabrication defects) is well-understood, the interplay between strong perturbations (specifically strong disorder or quasiperiodicity) and topological transport remains less explored.

The Gap: In solid-state systems, it is difficult to simultaneously control strong quasiperiodic modulation and quantum Hall physics. Strong random disorder usually leads to Anderson localization, suppressing transport, though it can sometimes create topological Anderson insulators.
The Question: How does topological pumping behave when the bulk states of a topological system become fully localized due to strong quasiperiodic modulation? Does the transport mechanism change from adiabatic pumping to a different regime?

2. Methodology

The authors utilize a one-dimensional (1D) multilayer photonic crystal (PhC) (a Bragg stack) as a highly tunable platform to emulate higher-dimensional physics.

Physical Platform: A stack of alternating Silicon (Si) and Silicon Dioxide (SiO $_2$ ) layers.
Synthetic Dimension: The system maps a 1D physical structure to a 2D parameter space.
- The real dimension is the physical stacking direction ( $x$ ).
- The synthetic dimension is encoded in the layer thickness modulation parameter $\phi$ .
Mathematical Model: The layer thicknesses $t_n$ $t_{n}$ are modulated according to the Aubry-André-Harper (AAH) equation:
$t_n = t_0 [1 + A \cos(2\pi\beta n + \phi)]$
- $A$ : Modulation amplitude (strength of quasiperiodicity).
- $\beta$ : Spatial frequency (analogous to magnetic flux per plaquette).
- $\phi$ : Phase offset (acts as the synthetic momentum/transverse momentum).
Simulation & Experiment:
- Numerical: Used MIT Photonic Bands (MPB) software to solve the 1D Maxwell eigenvalue equation and calculate eigenmodes, Inverse Participation Ratios (IPR), and transmission spectra.
- Fabrication: Created Si/SiO $_2$ stacks using Plasma-Enhanced Chemical Vapor Deposition (PECVD) on borosilicate glass.
- Measurement: Measured transmission spectra using a UV-Vis-NIR spectrophotometer.

3. Key Contributions

Realization of the Photonic Hofstadter Model: Successfully mapped a 1D photonic crystal with modulated geometry to the 2D Harper-Hofstadter model, reproducing the fractal "butterfly" spectrum and chiral edge states.
Discovery of a Localization Transition: Demonstrated that increasing the quasiperiodic modulation amplitude ( $A$ ) induces a wavelength-selective localization transition. Specific bulk Chern bands become fully localized in real space, while chiral edge states persist.
Mechanism Shift (Thouless $\to$ Landau-Zener): Identified a fundamental crossover in the transport mechanism:
- Weak Modulation: Transport occurs via adiabatic Thouless pumping, where edge states smoothly connect to extended bulk states.
- Strong Modulation: With a localized bulk, transport proceeds via Landau-Zener transitions between localized states across avoided crossings in the synthetic dimension.

4. Key Results

A. The Photonic Hofstadter Butterfly

By sweeping $\beta$ across different samples, the authors reconstructed the Hofstadter butterfly spectrum.
By fixing $\beta$ and sweeping $\phi$ , they observed chiral edge states traversing the bandgaps.
Validation: Experimental transmission spectra matched Transfer-Matrix simulations, confirming the existence of edge states and the fractal nature of the spectrum.

B. Localization Transition

At low modulation ( $A=0.2$ ), bulk bands are extended (delocalized), and edge states connect smoothly to them.
At high modulation ( $A=0.5$ , specifically using the inverse golden ratio $\beta \approx 0.618$ ), specific bulk bands near $\lambda/\langle a \rangle \sim 3.5$ become fully localized (high IPR).
Crucially, the chiral edge states persist and continue to wind across the gap despite the bulk being localized.

C. Transport Mechanism Analysis

Center-of-Mass Analysis: By tracking the real-space center-of-mass of the energy density, the authors showed that in the localized regime, an eigenstate does not flow smoothly through the bulk. Instead, it encounters avoided crossings (mini-gaps) between localized states.
Landau-Zener Tunneling: The transport across the system is mediated by non-adiabatic Landau-Zener transitions between these nearly degenerate localized states. The synthetic dimension ( $\phi$ ) provides the connectivity required for this "handoff" mechanism.
Experimental Confirmation: Transmission measurements on samples of varying sizes ( $N=5, 8, 13, 21$ ) showed that while extended bulk bands maintain transmission, the localized bulk bands and edge states exhibit sharp attenuation as system size increases, confirming the localization length is finite and short.

5. Significance

New Transport Paradigm: This work demonstrates that topological transport can survive even when the bulk is fully localized, challenging the conventional view that extended bulk states are necessary for adiabatic pumping.
Controlled Physics: It provides a compact, tunable photonic platform to study the complex interplay between topology, quasiperiodicity, and localization, which is difficult to achieve in solid-state systems.
Mechanism Insight: The identification of Landau-Zener-mediated topological pumping offers a new theoretical framework for understanding transport in disordered or quasiperiodic topological systems.
Applications: These findings could inform the design of robust photonic devices, such as topological lasers and optical isolators, that remain functional even in the presence of strong structural disorder or intentional quasiperiodic modulation.

In summary, the paper establishes a crossover from adiabatic Thouless pumping to a Landau-Zener-mediated topological pump in a photonic system where the bulk is localized, proving that topological protection can persist through a sequence of non-adiabatic transitions between localized states.