⚛️ quantum physics

Non-Abelian interference of topological edge states

This paper demonstrates the implementation of non-Abelian quantum interference and entanglement generation in coupled Su-Schrieffer-Heeger chains, where dual symmetries protect tunable topological particle transfer and the creation of spatially entangled NOON states dictated by permutation sequences.

Original authors: Shi Hu, Meiqing Hu, Zhoutao Lei

Published 2026-02-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shi Hu, Meiqing Hu, Zhoutao Lei

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 have a set of magical, one-way highways for tiny particles (like electrons or photons). In the world of quantum physics, these are called topological edge states. They are special because they are incredibly tough; they can't be knocked off course by bumps or noise, and they only travel in one direction.

This paper is about taking these tough highways and adding a new layer of magic: Non-Abelian Interference.

To understand what that means, let's use a simple analogy: The Train Switching Game.

The Setup: The Train Tracks

Imagine you have three parallel train tracks (let's call them Track A, Track B, and Track C). At the very ends of these tracks, there are special "edge stations" where particles like to hang out.

Normally, if you send a train from Track A to Track B, it just goes there. It's a simple swap. But in this paper, the authors built a system where the tracks are connected by a complex, time-shifting mechanism (like a giant, rotating switchyard).

The Magic Trick: The "Order Matters" Rule

The core discovery here is about Non-Abelian behavior. In everyday language, this means the order in which you do things changes the final result.

Think of it like putting on your shoes and socks:

Abelian (Normal): If you put on socks then shoes, you get dressed. If you put on shoes then socks (and somehow manage), you might get a weird result, but usually, math works like addition: $2 + 3$ is the same as $3 + 2$ .
Non-Abelian (This Paper): Imagine a magical wardrobe where the order of dressing changes who you become.
- If you put on the Red Hat then the Blue Scarf, you turn into a Wizard.
- If you put on the Blue Scarf then the Red Hat, you turn into a Pirate.

In this paper, the "Red Hat" and "Blue Scarf" are the permutation sequences (the order in which the tracks swap connections).

If the tracks swap in Sequence 1, a particle starting on Track A ends up on Track C.
If the tracks swap in Sequence 2 (even if the same tracks are involved), that same particle ends up on Track B!

The destination isn't just about where you go; it's about the history of the journey.

The Experiment: The Quantum Dance

The researchers simulated this using a system of coupled chains (like the train tracks). Here is what they achieved:

Tunable Transport: They showed they could send a single particle from one end of a track to the other, but they could control exactly how much of the particle went to the left end versus the right end of the destination track. It's like a faucet you can turn to get 10% water or 90% water, just by adjusting the timing of the "dance."
The Hong-Ou-Mandel (HOM) Effect: This is a famous quantum trick where two identical particles meet and "bunch" together. Usually, they just stick together.
- In this paper, they did something cooler. They sent two particles in.
- Because of the "Non-Abelian" magic (the order of the track swaps), the two particles didn't just stick together; they became entangled.
- They created a NOON state. Imagine two particles that are so linked that they are either both at the left end of the track OR both at the right end, but never one at each. It's like a coin flip where the coin is in two places at once until you look.

Why is this a Big Deal?

Robustness: Because these states are "topological," they are protected by the laws of physics (symmetries). Even if the system gets a little noisy or imperfect, the particles don't get lost. They stay on their magical highways.
New Computing Power: Current quantum computers are very fragile. If you make a mistake in the order of operations, the whole calculation fails. This paper suggests a way to build quantum logic where the order of operations is a feature, not a bug. It allows for a new kind of "braiding" of information, similar to how you can braid hair.
The "Time-Dependent" Shield: The authors used a special trick where the rules of the system change over time (like a rotating switch). This dynamic change protects the particles, ensuring they follow the complex braiding path without getting confused.

The Bottom Line

The authors have built a theoretical "quantum playground" where they can braid the paths of particles like shoelaces. By changing the order in which they braid the laces, they can send particles to different destinations and create highly entangled states.

This opens the door to quantum computers that are not only faster but also much harder to break, using the "braiding" of particles as a way to store and process information safely. It's like upgrading from a simple straight road to a complex, magical interchange where the route you take defines your destination.

1. Problem Statement

Topological boundary states are renowned for their robustness against backscattering and unidirectional propagation, making them ideal candidates for quantum state transfer and quantum information processing. While the topological protection of quantum entanglement and the observation of Hong-Ou-Mandel (HOM) interference in topological systems have been explored, the specific role of non-Abelian topology in mediating quantum interference and entanglement remains largely unexplored.

Existing research on non-Abelian braiding often relies on quasi-static processes in non-Hermitian systems, which limits practical application. The authors aim to address this gap by demonstrating a platform where non-Abelian quantum interference and entanglement generation are protected by time-dependent symmetries, enabling operation beyond the quasi-static limit.

2. Methodology

The authors propose a periodically modulated one-dimensional multi-chain system based on coupled Su-Schrieffer-Heeger (SSH) chains. The core mechanism relies on the interplay between two symmetries:

Time-independent Inversion Symmetry ( $\hat{P}$ ): Protects the parity of edge states, ensuring components with different parities evolve independently.
Time-dependent Interchain Symmetry ( $\hat{S}(t)$ ): A dynamic symmetry that induces the swapping (braiding) of edge state branches over time.

The system is modeled using a time-dependent Hamiltonian where intrachain hopping ( $v$ ) and interchain coupling ( $J$ ) and on-site energy offsets ( $\Delta$ ) are modulated sinusoidally with frequency $\omega$ .

Double-Coupled System (Sec. II): Serves as a foundational model to demonstrate Abelian braiding and tunable particle transfer between two chains (A and B).
Triple-Coupled System (Sec. III): The primary platform for demonstrating non-Abelian effects. This system involves three chains (A, B, C) with specific time-dependent couplings designed to create a cyclic permutation of three pairs of edge states.

3. Key Contributions

A. Symmetry-Protected Non-Abelian Permutation

The authors demonstrate that the time-dependent interchain symmetry causes the eigenvalues of the symmetry operator to swap cyclically over one period $T$ .

In the triple-chain system, this results in a twisted energy spectrum where three branches of edge states undergo cyclic mutual transformation.
Crucially, the permutation is non-Abelian: the final state depends on the order of operations.
- Sequence $M_1 M_2$ (braiding operator $\tau_1^{-1}\tau_2^{-1}$ ) results in a cyclic shift $A \to B \to C \to A$ .
- Sequence $M_2 M_1$ (braiding operator $\tau_2^{-1}\tau_1^{-1}$ ) results in a different shift $A \to C \to B \to A$ .
This operation-sequence dependence is the hallmark of non-Abelian statistics.

B. Tunable Topological Transport

Using the adiabatic evolution of edge states protected by the dual symmetries, the authors achieve:

Single-Particle Transfer: A particle injected at one end of a chain can be transferred to the two ends of a different target chain.
Tunability: The ratio of the particle appearing at the two ends of the target chain is continuously tunable (from 0 to 1) by controlling the accumulated dynamical phase ( $\theta_d$ ) during the evolution.
Destination Control: The specific target chain is determined solely by the permutation sequence (the order of braiding), independent of the dynamical phase.

C. Non-Abelian Hong-Ou-Mandel (HOM) Interference

The system is extended to two identical bosons injected into the ends of a single chain.

By tuning the dynamical phase to a 50:50 ratio ( $\theta_d = 2.5\pi$ ), the system suppresses the antibunching term, generating a spatially entangled NOON state ( $|1,1\rangle + |L,L\rangle$ ).
Non-Abelian Nature of Interference: The spatial location of the generated NOON state is dictated by the permutation sequence.
- Sequence $M_1 M_2$ generates the NOON state on Chain C.
- Sequence $M_2 M_1$ generates the NOON state on Chain A.
This demonstrates that the interference outcome (entanglement location) is non-commutative with respect to the control parameters (braiding order).

4. Results

Numerical Simulations: The authors performed numerical simulations for systems with $L=14$ $L = 14$ lattice sites.
- Fidelity: The fidelity between the final evolved state and the target state (tunable transfer or NOON state) reached 0.998 for single-particle transfer and 0.995 for the two-particle NOON state.
- Adiabaticity: The study confirmed that while inversion symmetry provides strict parity conservation, the time-dependent symmetry requires a sufficiently long (but finite) evolution time $T$ to ensure the system remains within the desired adiabatic branch.
- Visualization: Density distribution plots clearly showed the particle moving from Chain B to Chain C (for $M_1 M_2$ ) or Chain A (for $M_2 M_1$ ), and the formation of correlated bunching at the ends of the target chains for the two-particle case.

5. Significance

New Pathway for Non-Abelian Topology: This work establishes a novel pathway to explore non-Abelian topology in quantum science, distinct from traditional anyon-based approaches, by utilizing time-dependent symmetry protection.
Quantum Control: It offers a robust method for tunable quantum state transfer and entanglement generation where the destination and the nature of the entanglement can be controlled by the sequence of external modulation (braiding order) rather than just static parameters.
Experimental Feasibility: The authors propose that this protocol is experimentally feasible using photonic waveguide arrays. By slowly varying the spacing between waveguides along the propagation axis, the required adiabatic modulation of the SSH model can be realized, building upon existing experimental demonstrations of topological pumping and HOM interference in photonics.
Foundational Impact: The work bridges the gap between non-Abelian topology and practical quantum information processing tasks (like quantum gates and metrology), highlighting the potential of time-dependent symmetries to stabilize quantum effects against perturbations.