Selective braiding of different anyons in the even-denominator fractional quantum Hall effect

This paper demonstrates a gate-tunable Fabry-Pérot interferometer that enables the selective control and measurement of distinct statistical phases ($\pi$ and $\pi/2$) for different anyon types in even-denominator fractional quantum Hall states, thereby resolving individual anyon tunneling events and addressing a key challenge in observing non-Abelian braiding.

The Gist

Imagine a world where the rules of how things move and interact are completely different from our everyday experience. In this world, particles don't just bump into each other; they "dance" around one another, and the way they dance changes the very fabric of reality. This is the world of Quantum Hall Effects, a strange state of matter that occurs when electrons are trapped in a thin sheet and subjected to extreme cold and powerful magnetic fields.

In this paper, scientists from the Weizmann Institute of Science have performed a high-stakes experiment to watch these particles dance, specifically focusing on a mysterious type of particle called an anyon.

Here is the story of their discovery, explained simply:

1. The Stage: A Quantum Dance Floor

Think of the electrons in this experiment as a crowded dance floor. Usually, if two people swap places, nothing special happens. But in this quantum dance floor, the electrons are "fractionalized." They act like tiny, ghostly fragments of an electron (called quasiparticles).

When two of these ghosts swap places, they don't just return to normal. They leave a "memory" of their dance.

• Abelian Anyons: If they swap, the universe gets a little "twist" (a phase shift). It's like turning a page in a book; the story is the same, but you've moved to the next page.
• Non-Abelian Anyons: This is the magic. If they swap, the universe doesn't just twist; it changes into a completely different story. The book doesn't just turn a page; the plot changes entirely. This is the holy grail for building future quantum computers.

2. The Problem: A Noisy Crowd

For years, scientists have tried to watch these dances to see if the "Non-Abelian" ones exist. They use a device called a Fabry-Pérot Interferometer. Imagine a racetrack where the particles run in a loop. To see the dance, scientists send a stream of particles around the loop and watch how they interfere with each other, like ripples in a pond.

The problem? The loop isn't empty. There are "ghosts" (localized anyons) hiding inside the loop, waiting to be danced around.

• If you don't know how many ghosts are hiding, or what kind of ghosts they are, the dance gets messy. The ripples get scrambled, and you can't tell if you saw a simple twist or a plot change.
• Previous experiments were like trying to hear a whisper in a hurricane. The "ghosts" kept popping in and out randomly, creating noise.

3. The Solution: The "Gatekeeper" Antidot

The team's breakthrough was building a Gatekeeper.
They built a tiny, isolated island (an antidot) right in the middle of the racetrack. Think of this island as a VIP lounge with a bouncer.

• By adjusting the voltage on this "bouncer," they could control exactly how many ghosts were allowed to sit in the VIP lounge.
• They could say, "Okay, today we have exactly two ghosts," or "Today, we have one ghost."
• This allowed them to selectively choose which type of ghost the main dancers would circle around.

4. The Discovery: Two Different Dances

By tuning this gatekeeper, the scientists observed two distinct types of "twists" in the data:

• The $\pi$ Twist (The Simple Swap): When the main particle circled a specific type of ghost (an e/2 particle), the signal jumped by a full half-turn ($\pi$). This confirmed they were dancing with a standard "Abelian" ghost.
• The $\pi/2$ Twist (The Plot Change): When they tuned the gatekeeper to trap a different, rarer ghost (an e/4 particle), the signal jumped by only a quarter-turn ($\pi/2$).

Why is the $\pi/2$ twist so exciting?
In the world of quantum mechanics, a $\pi/2$ jump is the signature of a Non-Abelian particle. It suggests that the universe did change its story. The scientists successfully isolated a situation where they could watch a particle dance around a "plot-changing" ghost without the noise of other ghosts ruining the view.

5. The "Telegraph" Effect: Watching the Ghosts Blink

The paper also describes a fascinating side effect. Sometimes, the number of ghosts in the VIP lounge would change spontaneously.

• Imagine a light switch that flips on and off all by itself.
• The scientists watched their data "telegraph" (flip back and forth) as a single ghost tunneled into or out of the loop.
• They could literally see the interference pattern shift in real-time, proving they were tracking individual quantum particles, not just a blurry average.

The Big Picture: Why This Matters

This paper is a major step toward Topological Quantum Computing.

• Current computers are like fragile glass; if you drop them (or if there's a little noise), they break.
• Quantum computers based on these "Non-Abelian" anyons would be like a knot. You can shake the knot, but the knot itself (the information) stays safe because it's protected by the laws of topology.

The scientists didn't just see the knot; they built a tool to control the knot. They proved they can select which type of particle is dancing and watch the dance happen cleanly.

In summary:
The team built a quantum racetrack with a VIP lounge in the middle. By controlling who sits in the lounge, they could force the racing particles to dance around specific partners. They successfully identified two different dance moves, one of which is the legendary "Non-Abelian" move that could power the super-computers of the future. They turned a chaotic, noisy quantum world into a controlled, observable stage.

Technical Summary

Here is a detailed technical summary of the paper "Selective braiding of different anyons in the even-denominator fractional quantum Hall effect" by Kim et al.

1. Problem Statement

Fractional Quantum Hall (FQH) states host emergent quasiparticles known as anyons, which possess fractional exchange statistics. While Abelian anyons acquire a phase upon exchange, non-Abelian anyons can transform the many-body wavefunction into an orthogonal state, a property essential for topological quantum computing.

• The Challenge: Observing non-Abelian braiding requires precise control over two distinct types of anyons simultaneously:
  1. Interfering anyons: Those forming the current beam in the interferometer.
  2. Localized anyons: Those trapped within the interference loop (the "bulk").
• Previous Limitations: In previous experiments (particularly in GaAs and graphene), fluctuations in the number of localized anyons caused random phase shifts, obscuring the specific statistical phase associated with braiding. Furthermore, distinguishing between different anyon types (e.g., $e/2$ vs. $e/4$) within the loop was difficult, preventing the isolation of specific braiding events.
• Specific Context: The even-denominator filling factor $\nu = 1/2$ (or $-1/2$ in hole-doped systems) is predicted to host non-Abelian anyons ($e/4$) alongside Abelian ones ($e/2$), making it a prime candidate for observing non-Abelian statistics.

2. Methodology

The authors utilized a gate-tunable Fabry-Pérot Interferometer (FPI) fabricated on a high-mobility bilayer graphene heterostructure encapsulated in hexagonal boron nitride (hBN) and graphite.

• Device Architecture:
  • Interferometer: Two Quantum Point Contacts (QPCs) define an interference loop.
  • Embedded Antidot: A key innovation is a gate-defined antidot (a small island of area $0.1 , \mu m^2$) located inside the interference loop. This antidot is controlled by a dedicated gate voltage ($V_{ADG}$).
  • Control Mechanism: By tuning $V_{ADG}$, the authors can locally adjust the filling factor ($\nu_{AD}$) of the antidot island, effectively controlling the number and type of localized anyons trapped within the loop.
• Experimental Conditions:
  • Measurements performed at ultra-low temperatures ($T \approx 10$ mK) and high magnetic fields (up to 12 T).
  • Measurement Technique: The diagonal resistance ($R_D$) was measured while sweeping the plunger gate voltage ($V_{PG}$) to modulate the interference phase.
  • Time-Resolved Analysis: Instead of single sweeps, the authors performed repeated sweeps over long durations (seconds to minutes) to construct histograms of $R_D$. This allowed them to resolve discrete jumps in the interference pattern caused by individual anyon tunneling events.

3. Key Contributions

1. Selective Control of Localized Anyons: The embedded antidot allows for the selective trapping of specific anyon types ($e/2$ or $e/4$) by tuning the local filling factor, a capability previously lacking in FPI experiments.
2. Resolution of Distinct Braiding Phases: The experiment successfully resolved two distinct statistical phases:
   • $\theta_{braid} = \pi$ (associated with $e/2$ anyons).
   • $\theta_{braid} = \pi/2$ (associated with $e/4$ anyons).
3. Direct Observation of Single Anyon Tunneling: By analyzing time-dependent data, the authors directly observed individual tunneling events of localized anyons into and out of the interference loop, distinguishing between different occupancy states.
4. Validation of Topological Order: The results provide strong evidence for the topological order of the $\nu = 1/2$ state, supporting the Moore-Read Pfaffian (or anti-Pfaffian) theory which predicts the coexistence of $e/2$ and $e/4$ quasiparticles.

4. Key Results

A. Interference at $\nu = -1/2$ (Even-Denominator)

• Antidot at $\nu_{AD} \approx -1/2$: When the antidot filling matched the bulk, the interference pattern exhibited phase jumps of $\pi$. This corresponds to an interfering $e/2$ quasiparticle encircling a localized $e/2$ anyon.
• Antidot at $\nu_{AD} \approx 0$: When the antidot was tuned to a different filling (near zero), the phase jumps changed to $\pi/2$. This indicates the interfering $e/2$ quasiparticle is now encircling a localized $e/4$ anyon.
• Dynamics: The transition between these states occurred via discrete tunneling events. The authors observed "telegraph noise" where the system switched between two interference patterns (ground state and excited state) with a lifetime of approximately 11 seconds, allowing for the statistical analysis of the phase shifts.
• Parity Dominance: The observation of rigid $\pi/2$ shifts implies that one specific parity of the $e/2$ anyon (likely the even-parity fermion) dominates the interference, as the cancellation of opposite parities did not occur.

B. Interference at $\nu = -1/3$ (Odd-Denominator Control)

• As a control, experiments were performed at $\nu = -1/3$.
• The system exhibited phase jumps of **$2\pi/3$**, consistent with the braiding of interfering$e/3$anyons around localized$e/3$ anyons.
• This confirmed the methodology's ability to resolve fractional statistics distinct from the even-denominator case.

C. Comparison with Integer States

• In integer quantum Hall states ($\nu = -1$), the antidot tuning produced a "sawtooth" pattern in the phase due to the addition/removal of single electrons, resetting the phase by $2\pi$. This contrasted with the fractional states where phase shifts were fractional ($\pi, \pi/2, 2\pi/3$) and did not reset the phase, confirming the fractional nature of the excitations.

5. Significance

• Step Toward Non-Abelian Braiding: This work addresses one of the two major hurdles in observing non-Abelian statistics: the selective control of localized anyons. By proving that specific anyon types can be trapped and their braiding phases measured, the authors have established a critical pathway toward demonstrating non-Abelian statistics (which would require braiding $e/4$ with $e/4$).
• Material Platform Validation: The results validate bilayer graphene as a superior platform for studying FQH states, offering high mobility and tunability that allows for the resolution of subtle topological effects often obscured in GaAs systems.
• Theoretical Confirmation: The observation of $\pi/2$ phase jumps provides experimental support for the existence of non-Abelian $e/4$ quasiparticles in the $\nu = 1/2$ state, a long-standing prediction in condensed matter physics.
• Future Outlook: While the current experiment controls the localized anyons, the authors note that the interfering anyons remain $e/2$. The next critical step is to tune the device to make $e/4$ the interfering particle, which would complete the requirements for a full non-Abelian braiding experiment.

In summary, this paper demonstrates a breakthrough in the manipulation of anyonic statistics, using a gate-defined antidot to selectively trap and identify different quasiparticle types, thereby resolving distinct braiding phases ($\pi$ and $\pi/2$) in an even-denominator fractional quantum Hall state.