⚛️ quantum physics

Reciprocal Floquet thermalization in one-dimensional Rydberg atom array

This paper proposes and demonstrates a square-wave-modulated Floquet protocol in one-dimensional Rydberg atom arrays that reveals a disorder-free reciprocal thermalization mechanism, where specific combinations of laser detuning and interactions trigger rapid equilibration while suppressing thermalization between these resonant conditions.

Original authors: Yunhui He, Yuechun Jiao, Jianming Zhao, Weibin Li

Published 2026-03-18

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CC BY 4.0

Original authors: Yunhui He, Yuechun Jiao, Jianming Zhao, Weibin Li

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 long line of tiny, super-excited marbles (Rydberg atoms) sitting on a table. Normally, if you shake the table rhythmically (periodic driving), these marbles would eventually get so hot and chaotic that they forget where they started. They would mix up completely, reaching a state of "thermal equilibrium"—like a cup of coffee that has cooled down to room temperature, where you can no longer tell which part was the hot milk and which was the cold coffee.

In the world of quantum physics, this "forgetting" is called thermalization, and it's usually a problem. It destroys the delicate quantum information scientists want to study.

This paper introduces a clever new trick to control this chaos. The researchers found a way to make the marbles either mix up completely (thermalize) or stay frozen in their original pattern (localize), simply by tuning the rhythm of the shaking and the strength of the "glue" between the marbles.

Here is the breakdown of their discovery using everyday analogies:

1. The Setup: The "Square-Wave" Shaker

Usually, scientists shake these quantum systems with a smooth, sine-wave motion (like a gentle wave). But here, the researchers used a square-wave motion.

The Analogy: Imagine a light switch. Instead of dimming the light slowly up and down, you flip it instantly ON (bright) for a split second, then instantly OFF (dark) for the same amount of time, and repeat.
The Effect: This creates a very sharp, "bang-bang" rhythm. The atoms are either being strongly pushed by a laser or left alone to interact with their neighbors.

2. The "Reciprocal" Secret: The Perfect Timing

The magic happens when the timing of the laser (the "shaking") perfectly matches the natural "buzz" of the atoms interacting with each other.

The Analogy: Think of a playground swing. If you push the swing at exactly the right moment in its arc, it goes higher and higher (resonance). If you push at the wrong time, it stops.
The Discovery: The researchers found a specific condition where the laser's timing and the atoms' interaction strength are "reciprocally" matched (like a perfect lock and key).
- When they match: The system goes into a frenzy. The atoms absorb energy rapidly, forget their past, and mix up completely. This is Thermalization.
- When they don't match (even slightly): The system refuses to mix. The atoms stay in their original pattern, effectively "freezing" in time. This is Localization.

3. The "Narrow Peaks" of Chaos

One of the most surprising findings is how precise this needs to be.

The Analogy: Imagine a piano. If you press a key, you get a note. But in this system, chaos only happens if you press the key exactly right. If you move your finger just a tiny bit to the left or right, the music stops, and the system becomes quiet and orderly again.
The Result: The researchers saw "narrow peaks" of chaos. If you tune the system to the exact peak, it's a chaotic mess. If you tune it just slightly away from the peak, it becomes a calm, predictable system. This allows scientists to switch between "chaos mode" and "order mode" with extreme precision.

4. Why This Matters: The "Reset Button"

In quantum computing, you often want to keep information safe (localization) or test how a system behaves when it's fully mixed (thermalization).

The Problem: Usually, getting a system to thermalize quickly without using messy, random disorder (like throwing sand on the track) is hard.
The Solution: This method works in a clean, perfect system. It's like having a "reset button" for quantum matter. You can force the atoms to forget everything and start fresh, or force them to remember their history forever, just by adjusting the laser's timing.

5. The "Rydberg" Advantage

Why use Rydberg atoms? They are like giant, fluffy atoms that interact strongly with each other from far away.

The Analogy: Imagine normal atoms are like people in a crowded room who only talk to the person standing right next to them. Rydberg atoms are like people with super-hearing; they can hear and react to people across the entire room.
The Benefit: Because they interact so strongly and specifically (via a force called Van der Waals), they create these sharp, distinct "peaks" of chaos that other systems (like superconducting circuits) don't show.

Summary

The paper demonstrates a new way to steer quantum systems. By using a specific "on-off" laser rhythm and tuning it to a precise mathematical relationship with the atoms' interactions, scientists can:

Trigger rapid thermalization: Make the system heat up and mix instantly (useful for studying how things break down).
Suppress thermalization: Keep the system frozen and ordered (useful for protecting quantum information).

It's like finding a master key that can instantly turn a chaotic, boiling pot of soup into a perfectly arranged set of frozen cubes, and back again, just by tapping the spoon at the exact right rhythm. This opens the door to better quantum simulations and more robust quantum computers.

1. Problem Statement

Periodically driven (Floquet) quantum systems offer a powerful method for engineering exotic quantum phases and effective Hamiltonians. However, a major limitation is Floquet heating: isolated quantum systems driven periodically tend to absorb energy continuously, leading to rapid thermalization to an infinite-temperature state. This thermalization washes out quantum correlations and suppresses non-trivial dynamics, limiting the utility of Floquet engineering.

While techniques like Many-Body Localization (MBL) or dynamical localization can suppress heating, they often rely on disorder or specific high-frequency limits. The authors address the challenge of controlling and steering thermalization dynamics in a clean (disorder-free) Rydberg atom array using a specific driving protocol, aiming to identify regimes where thermalization is either triggered or suppressed.

2. Methodology

The authors propose and analyze a square-wave modulated Floquet engineering protocol applied to a 1D array of $N$ Rydberg atoms with open boundary conditions.

System Hamiltonian: The system consists of atoms in a ground state $|g\rangle$ $∣ g ⟩$ and a Rydberg state $|s\rangle$ $∣ s ⟩$ . The Hamiltonian $\hat{H}(t)$ $\hat{H} (t)$ alternates between two phases within a period $T = 2\tau$ $T = 2 τ$ :
1. Driving Phase ( $0 < t < \tau$ ): A strong Rabi frequency $\Omega_0$ couples $|g\rangle$ and $|s\rangle$ (term $\propto \hat{\sigma}^x$ ).
2. Interaction Phase ( $\tau < t < 2\tau$ ): The Rabi frequency is zero (or negligible $\Omega_l \ll \Omega_0$ ), and the system evolves under detuning $\Delta$ and van der Waals (vdW) interactions $V_{jk} \propto 1/|j-k|^6$ .
Floquet Operator: The stroboscopic evolution is governed by the Floquet operator $\hat{U}_F = e^{-i\hat{H}_2\tau}e^{-i\hat{H}_1\tau}$ .
Key Parameters: The study focuses on the interplay between the nearest-neighbor (NN) interaction strength $V_0$ , the Rabi frequency $\Omega_0$ , and the detuning $\Delta$ . They define a shifted detuning $\Delta_0 = \Delta + V_0$ .
Numerical Techniques:
- Exact Diagonalization (ED): Used for small system sizes ( $N \approx 12-14$ ) to compute eigenphases, level spacing statistics, and entanglement entropy.
- Time-Dependent Variational Principle (TDVP): Used for larger systems and to verify thermalization scaling.
- Master Equation: Simulated to account for finite Rydberg lifetimes (decay).

3. Key Contributions

A. Identification of "Reciprocal Floquet Thermalization"

The authors discover a mechanism where thermalization is triggered only when the system parameters satisfy a specific reciprocal condition:
$(\Delta + V_0)T \approx 2K\pi$
where $K$ is an integer. When this condition is met, the system exhibits rapid thermalization. Conversely, between these resonance peaks, the system remains in an integrable (localized) phase where thermalization is suppressed.

B. Mechanism of Quantum Chaos vs. Integrability

Chaos Peaks: At the reciprocal condition, the unperturbed energies of the many-body basis states become approximately integers. The periodic drive couples these states, generating a Gaussian envelope of Floquet eigenphases. The level spacing ratio $\langle r \rangle$ follows the Circular Orthogonal Ensemble (COE) statistics ( $\approx 0.527$ ), indicating Many-Body Quantum Chaos (MBQC).
Integrable Valleys: Away from the peaks, the eigenphases form a "banded" structure. The drive couples only near-resonant sub-bands, preventing global energy redistribution. The level spacing ratio drops to Poissonian statistics ( $\approx 0.386$ ), indicating integrability.
Role of vdW Interaction: The authors emphasize that the specific $1/r^6$ decay of the vdW interaction is crucial. Other interaction types (e.g., dipolar $1/r^3$ or Coulomb) typically produce broad chaotic phases rather than the sharp, narrow "reciprocal" peaks observed here.

C. Experimental Feasibility

The study demonstrates that this phenomenon is observable within realistic experimental constraints:

Disorder-Free: No random disorder is required; the effect arises from the deterministic interplay of laser parameters and interactions.
Timescales: Thermalization occurs within the Rydberg lifetime (e.g., $\sim 125 \mu s$ for $^{87}\text{Rb}$ in the $|60S\rangle$ state).
Initial States: The protocol works with experimentally accessible product states (e.g., all atoms in $|g\rangle$ or alternating superpositions), rather than requiring difficult-to-prepare mid-spectrum states.

4. Key Results

Level Statistics: The average level spacing ratio $\langle r \rangle$ shows sharp peaks at $\Delta_0 T = 2K\pi$ (COE statistics) separated by valleys of low $\langle r \rangle$ (Poisson statistics).
Stroboscopic Dynamics:
- Thermalization Phase: Starting from a product state (e.g., $|gg\dots g\rangle$ ), the atomic population $\langle \hat{S}^z \rangle$ rapidly saturates to zero, and the energy saturates to $\epsilon N/2$ . The half-chain entanglement entropy grows to the Page value (volume law), confirming thermalization.
- Integrable Phase: The population and energy remain close to their initial values. Entanglement entropy remains low (area law or sub-volume law).
Edge Modes: In the integrable phase, strong persistent fluctuations are observed in edge modes (two-time correlations), whereas these decay rapidly in the chaotic phase.
Robustness to Decay: Simulations including Rydberg decay show that the distinction between thermalizing and localized phases remains clear even with finite lifetimes, as the decay rate is small compared to the driving frequency.

5. Significance and Impact

New Control Paradigm: This work establishes a robust framework for steering thermalization in clean quantum systems without disorder. By tuning the laser detuning and interaction strength, one can switch the system between chaotic (thermalizing) and integrable (localized) regimes.
Quantum Simulation: The ability to engineer effective Hamiltonians and control the onset of chaos makes Rydberg atom arrays a superior platform for studying non-equilibrium dynamics, prethermalization, and the transition from localization to thermalization.
Fundamental Physics: The discovery of "reciprocal" thermalization highlights the importance of the interaction potential exponent ( $\alpha=6$ for vdW) in determining non-equilibrium phases. It suggests that specific interaction ranges can induce unique spectral structures (narrow chaos peaks) not seen in other platforms like superconducting circuits or trapped ions.
Applications: The findings have potential applications in digital quantum simulation, where controlling thermalization is essential for preserving quantum information, and in the design of Floquet time crystals or other exotic phases where heating must be managed.

In summary, the paper provides a theoretically sound and experimentally viable protocol to manipulate the thermalization dynamics of Rydberg atom arrays, revealing a novel "reciprocal" mechanism where thermalization is resonantly triggered by the interplay of interaction and driving frequency.