⚛️ quantum physics

Non-Abelian Thouless pumping based on the global adiabatic criterion in Rydberg synthetic lattices

This paper proposes and numerically validates a quantum implementation of non-Abelian Thouless pumping in Rydberg synthetic lattices using a global adiabatic criterion to optimize pulse timing, demonstrating superior population transfer and non-commuting geometric operations compared to existing methods.

Original authors: Jin-Kang Guo, Jin-Lei Wu, Chuan-Cun Shu

Published 2026-07-09

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CC BY 4.0

Original authors: Jin-Kang Guo, Jin-Lei Wu, Chuan-Cun Shu

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 very special, invisible train track made of energy levels inside an atom. This isn't a normal track; it's a "synthetic" one, built by tuning the atom with microwave signals, much like a conductor tuning an orchestra. The scientists in this paper used this setup to perform a complex quantum dance called Thouless pumping.

Here is the story of what they did, explained simply:

1. The Stage: A Quantum Train Station

Think of the atom's energy levels as a small train station with three platforms (called "cells"). On this station, there are special "waiting rooms" where passengers (quantum states) can sit without paying any energy cost. These are called zero-energy states.

The scientists built a station with 12 specific tracks (energy levels).
Six of these tracks are the "waiting rooms" (the zero-energy states) where the magic happens.
The other six tracks are "bright states." If a passenger accidentally steps onto these, they get kicked off the train (they leak away and are lost).

2. The Goal: The Non-Abelian Dance

Usually, if you move a train around a loop and come back to the start, you end up exactly where you began. But in this special quantum world, the rules are different.

The Twist: If you move the train in one order (Loop A then Loop B), the passengers end up in one specific seat. If you do the loops in the reverse order (Loop B then Loop A), the passengers end up in a different seat.
This is called Non-Abelian behavior. It's like a magic trick where the order of your moves changes the final result. The scientists wanted to prove they could do this reliably on their atomic train station.

3. The Problem: The "Too Fast" vs. "Too Slow" Dilemma

To make this quantum dance work, the scientists have to wiggle the microwave signals (the train tracks) in a specific cycle.

If they go too slow: The atom gets tired. The passengers start leaking out of the waiting rooms into the "bright states" and disappear due to natural decay.
If they go too fast: The passengers get confused and jump to the wrong tracks because the movement wasn't smooth enough.

They needed a way to find the perfect speed for the dance so the passengers stayed safe and ended up in the right seats.

4. The Solution: The "Global Adiabatic Criterion" (GAC)

The scientists invented a new rulebook called the Global Adiabatic Criterion (GAC).

The Analogy: Imagine you are driving a car on a bumpy road. You want to drive fast enough to get there quickly, but slow enough not to spill your coffee.
Instead of just guessing, the GAC acts like a smart dashboard. It looks at the "bumps" (leakage risks) along the entire route. It calculates the average bumpiness and how much the bumps fluctuate (change suddenly).
By adjusting the timing of the microwave pulses (like shifting when you hit the gas or brake), they found a schedule that minimized the "spills" (leakage) without needing to add extra, complicated machinery to the car.

5. The Results: A Perfect Performance

When they tested this new timing schedule:

The Order Matters: They ran the dance twice. Once with Order A-B, and once with Order B-A. The passengers ended up in different seats, exactly as the math predicted. This proved the "Non-Abelian" magic was real.
Better than the Old Way: They compared their GAC timing against two other common timing methods used in previous studies. Their method kept more passengers in the correct seats and lost fewer to the "bright states."
Sturdy: Even when they added "noise" (like static on the radio or slight errors in the microwave signals), their method held up better than the others.

Summary

In short, the scientists built a tiny, artificial quantum train station inside an atom. They figured out the perfect rhythm to move the trains so that the order of their movements changed the final destination (a quantum magic trick). They did this by using a new "smart timing" rule (GAC) that kept the passengers safe from leaking away, proving that this method is more robust and efficient than previous techniques. This brings us one step closer to using these atomic systems for powerful quantum computers.

Technical Summary: Non-Abelian Thouless Pumping in Rydberg Synthetic Lattices

Problem Statement
Thouless pumping typically converts cyclic parameter modulation into geometric transport. While Abelian pumping relies on scalar geometric phases (Berry curvature), non-Abelian pumping occurs within degenerate subspaces, where adiabatic operations are represented by matrix-valued holonomies (Wilczek-Zee). A critical challenge in implementing non-Abelian pumping in quantum systems is the finite-time constraint: adiabatic evolution requires slow modulation to avoid leakage to non-degenerate "bright" states, yet slow evolution exacerbates population loss due to radiative decay and imperfect state preparation in Rydberg atoms. Existing schemes often lack a robust method to optimize the relative timing of control pulses to balance adiabaticity against decoherence without introducing complex auxiliary couplings.

Methodology
The authors propose a quantum implementation of non-Abelian Thouless pumping using a synthetic Lieb lattice encoded in twelve selected microwave-coupled Rydberg levels of $^{39}\text{K}$ .

System Construction: The lattice consists of three unit cells, each containing four states ( $|A_i\rangle, |B_i\rangle, |C_i\rangle, |D_i\rangle$ ), forming a finite three-cell structure. The microwave couplings create a tight-binding Hamiltonian with a bipartite structure that supports six degenerate zero-energy states. These states constitute the working subspace, while the remaining "bright" states act as leakage channels.
Protocol Design: Two elementary pumping cycles ( $P_1$ and $P_2$ ) are defined by Gaussian microwave-coupling envelopes. $P_2$ is derived from $P_1$ by reversing the temporal order of the $\Omega_C$ and $\Omega_D$ pulses. The composition of these cycles in opposite orders ( $\hat{U}_2\hat{U}_1$ vs. $\hat{U}_1\hat{U}_2$ ) is designed to probe non-commutativity.
Global Adiabatic Criterion (GAC): To address finite-time constraints, the authors introduce a GAC. Instead of adding counter-diabatic terms, this method optimizes the relative timing (delay parameter $\alpha$ ) of the Gaussian pulses within a fixed pulse family. The GAC evaluates the local nonadiabatic factor $Q(t)$ over a representative $\Lambda$ -type transfer step. It selects the timing that minimizes both the mean value ( $\bar{Q}$ ) and the temporal fluctuation ( $\sigma_Q^2$ ) of this factor, theoretically suppressing leakage to bright states.
Simulation: The dynamics are simulated using the Lindblad master equation, incorporating state-dependent Rydberg decay (at 10 K), systematic errors (Rabi amplitude, detuning gradients), and static disorder.

Key Contributions

Implementation Scheme: The paper establishes a concrete protocol for non-Abelian Thouless pumping in a finite synthetic Rydberg lattice, utilizing the six zero-energy states of a three-cell Lieb geometry as the computational subspace.
Timing Optimization via GAC: The authors develop and apply the Global Adiabatic Criterion to select the optimal relative timing of Gaussian pulses. This approach avoids the complexity of shortcut-to-adiabaticity protocols (which require auxiliary couplings) by strictly optimizing the timing of existing control fields.
Demonstration of Non-Commutativity: The study demonstrates that reversing the temporal order of the pumping cycles results in distinct projected population maps in the zero-energy subspace, confirming the non-commutative nature of the matrix-valued adiabatic operations.
Robustness Analysis: The GAC-selected schedule is shown to outperform two literature-adapted Gaussian pulse schedules in terms of target-state population and stability against decay and perturbations.

Results

Order-Dependent Dynamics: Simulations confirm that the composed cycles $\hat{U}_2\hat{U}_1$ and $\hat{U}_1\hat{U}_2$ produce different final states for the same initial conditions (e.g., starting from $|C_3\rangle$ ). The population transfer maps in the six-dimensional computational basis differ significantly, validating the non-Abelian character of the transport.
Performance of GAC: Within the same Gaussian pulse family, the timing selected by the GAC ( $\alpha \approx 0.18$ ) yields a higher target-state population ( $P_{tar}$ ) compared to reference schedules across various total evolution times. The GAC schedule exhibits smoother population curves with reduced finite-time oscillations.
Robustness: Under state-dependent Rydberg decay, global Rabi amplitude errors, deterministic detuning gradients, and static on-site/coupling disorder, the GAC-selected schedule maintains higher fidelity and shows slower degradation compared to the reference schedules. The spread of results under disorder is narrower for the GAC schedule.

Significance
The paper claims that this work represents a major milestone in finite-time geometric control. By enabling non-Abelian Thouless pumping with optimized timing via the GAC, the study paves the way for transformative applications in holonomic quantum computing using Rydberg synthetic lattices. The results support the viability of Rydberg synthetic dimensions as a controllable setting for coherent operations in degenerate working subspaces, specifically addressing the trade-off between adiabaticity and decoherence without altering the fundamental pumping geometry or pulse family.