Observation of end-to-end pumping in a quasiperiodic Fibonacci-type photonic chain

This paper theoretically and experimentally demonstrates that a finite quasiperiodic Fibonacci-type photonic chain can achieve robust, end-to-end topological pumping of light between non-neighboring elements, even in the presence of structural defects.

The Gist

Imagine you have a long row of 11 light pipes (waveguides) sitting side-by-side. In a normal setup, if you shine a flashlight into the first pipe on the far left, the light would naturally spread out, leaking into the neighboring pipes as it travels down the line. By the time it reaches the end, the light would be scattered all over the place, and very little would make it to the last pipe on the far right.

This paper describes a clever trick to force that light to travel from the very first pipe on the left to the very last pipe on the right, without getting lost in the middle. They call this "end-to-end pumping."

Here is how they did it, using simple analogies:

1. The "Fibonacci" Pattern

Instead of spacing the pipes evenly, the researchers arranged them in a specific, repeating pattern based on the Fibonacci sequence (a famous number pattern found in nature, like in sunflower seeds).

• Some pipes are close together (strong connection).
• Some pipes are far apart (weak connection).
• This creates a "quasiperiodic" chain—a pattern that repeats but never exactly the same way twice.

2. The "Special Guest" (The Pumping State)

In this specific arrangement, there is a special "mode" or state where light likes to hide.

• At the start: If you set the pipes up one way, this "special guest" light hides exclusively in the leftmost pipe.
• At the end: If you change the setup slightly, that same "special guest" light hides exclusively in the rightmost pipe.

The researchers found a way to gently move the light from the left hiding spot to the right hiding spot without it ever getting stuck in the middle.

3. The "Bending" Trick (The Pump)

Usually, to move light from one side to the other in these systems, you would need to constantly tweak the distance between every single pipe along the entire line. That is like trying to walk a tightrope while constantly adjusting the tension of every rope in the room—it's incredibly difficult and prone to mistakes.

The breakthrough in this paper:
They discovered that they only needed to bend two specific pipes (the second one and the tenth one) to make the whole system work.

• Imagine the row of pipes is a train. Instead of changing the engine of every car, they just gently curved the second and tenth cars.
• As the light travels down the line, this gentle curve acts like a conveyor belt, smoothly guiding the light from the left end to the right end.

4. The "Robustness" (The Bumpy Road)

One of the biggest worries with these delicate light systems is that if you make a small mistake—like a pipe being slightly too close or too far (a "defect")—the whole system breaks.

The researchers tested this by intentionally messing up the spacing of the pipes in the middle of the line.

• The Result: The light still made it from left to right! In fact, in some cases, the "mistakes" actually made the light transfer better because they created a stronger "gap" that kept the light from leaking out.
• The Analogy: It's like driving a car on a bumpy road. Usually, bumps make the ride worse. But in this specific system, the bumps actually helped keep the car in its lane, ensuring it reached the destination safely.

Summary

The team built a special row of light pipes arranged in a Fibonacci pattern. They proved that by bending just two pipes, they could act as a "pump" to move light from one end of the row to the other. They showed that this method is:

1. Simple: You don't need to control the whole system, just two points.
2. Robust: It works even if the pipes are slightly damaged or spaced incorrectly.

This is a "proof of principle" showing that we can move information (light) across a network efficiently and reliably, even if the network isn't perfect.

Technical Summary

Technical Summary: Observation of End-to-End Pumping in a Quasiperiodic Fibonacci-Type Photonic Chain

Problem Statement
Topological pumps offer a mechanism for robust state transfer between non-neighboring elements in a network, a capability valuable for quantum communication and state transfer. While end-to-end pumping has been demonstrated in various platforms, including the Su–Schrieffer–Heeger (SSH) model and Fibonacci quasicrystals, existing schemes face a significant practical limitation: they typically require continuous, exhaustive tuning of bond strengths along the entire length of the system. This requirement increases experimental complexity and introduces accumulated imperfections, hindering scalability and robustness. The authors identify a need for a pumping scheme that achieves efficient end-to-end transfer with minimal structural modifications and high resilience to disorder.

Methodology
The study employs a combined theoretical and experimental approach using a finite quasiperiodic Fibonacci-type photonic chain realized in an array of coupled optical waveguides.

• Theoretical Framework: The system is modeled using a tight-binding Hamiltonian under the paraxial approximation, where the propagation of light mimics the Schrödinger equation. The chain consists of $N=11$ waveguides with nearest-neighbor separations following a Fibonacci sequence, defined by two characteristic distances, $d_A$ (weak coupling, $J_A$) and $d_B$ (strong coupling, $J_B$). The system's structure is tuned via a "phason angle" ($\phi$), which alters the sequence of bond strengths.
• Pumping Protocol: Unlike previous schemes requiring global tuning, the authors propose a protocol where only two specific waveguides (the second and tenth) are bent along the propagation direction ($z$). This bending modulates only four hopping parameters (two at each end) to interpolate between two configurations: Configuration 1 (phason angle $\phi_1$) and Configuration 2 (phason angle $\phi_2$). The transfer protocol follows a cosine-like trajectory for the inter-waveguide distances.
• Experimental Implementation: Waveguide arrays were fabricated in transparent glass using femtosecond laser direct writing. The coupling strengths were calibrated to extract the bare hopping amplitude ($J \approx 63.5 \text{ mm}^{-1}$) and decay length ($\xi \approx 2.2 \text{ \mu m}$). Light from a 780 nm laser was injected into a single waveguide at one end ($z=0$), and the output intensity profile was recorded at the opposite end ($z=L=40 \text{ mm}$) using a CCD camera.
• Robustness Testing: The system's stability was tested by introducing controlled structural defects, where specific inter-waveguide distances were modified to a third value ($d_D = 8 \text{ \mu m}$), replacing either $d_A$ or $d_B$ bonds.

Key Contributions and Results

1. Minimalist Pumping Scheme: The authors demonstrate that end-to-end pumping can be achieved in a quasiperiodic Fibonacci chain by modifying only two waveguides at the system's boundaries. This contrasts sharply with SSH-based or previous quasicrystal proposals that require tuning the entire chain.
2. Defect-State Mechanism: The pumping mechanism relies on a localized "defect state" rather than a winding state. Numerical simulations and experiments confirm that this state is localized at the left end in Configuration 1 and at the right end in Configuration 2. As light propagates through the array under the bending protocol, it adiabatically transfers from one end to the other.
3. Experimental Verification: The study provides direct experimental evidence of end-to-end pumping. Light injected into the leftmost waveguide emerges predominantly at the rightmost waveguide, confirming the theoretical prediction of adiabatic transport across the bulk.
4. Enhanced Robustness: A critical finding is that the pumping process is not only resilient to structural defects but can be enhanced by them. Introducing specific defects (e.g., replacing $d_A$ bonds with $d_D$) was shown to increase the excitation gap between the pumping state and the bulk states. This larger gap improves adiabaticity, leading to more pronounced pumping efficiency in the presence of defects compared to the defect-free case.
5. Protocol Flexibility: The study verifies that the pumping effect persists under different bending profiles, including linear changes in inter-waveguide distances, not just the cosine profile used in the primary demonstration.

Significance
The paper establishes quasiperiodic Fibonacci-type photonic lattices as a robust and experimentally viable platform for disorder-resilient state transfer. The primary significance lies in the dramatic simplification of the experimental setup; by requiring only local structural changes (bending two waveguides) rather than global tuning, the scheme reduces experimental complexity and potential sources of error. Furthermore, the demonstration that structural defects can enhance, rather than degrade, the pumping performance challenges the conventional view of disorder in topological systems. This work provides a proof-of-principle for efficient, minimal-tuning topological pumps, offering a pathway toward more scalable and fault-tolerant photonic networks for state transfer.