Universal Family-Vicsek scaling in quantum gases far from equilibrium

This paper experimentally demonstrates that the universal Family-Vicsek scaling laws, originally established for classical surface growth, also govern the non-equilibrium dynamics of quantum fluctuations in a one-dimensional Bose gas, thereby unifying the understanding of universality across classical and quantum systems.

The Gist

Imagine you are watching a crowd of people trying to cross a narrow bridge. Sometimes, they move in a chaotic, jostling mess; other times, they move in a perfectly synchronized, fluid line. For decades, physicists have had a special set of rules to predict how these crowds behave, but those rules were mostly written for classical things—like water flowing, sand piling up, or bacteria growing.

This paper is a groundbreaking discovery because it shows that these same rules also apply to the quantum world—the weird, tiny world of atoms and subatomic particles where things behave like waves and particles at the same time.

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

1. The "Rough Surface" Analogy

To understand the experiment, imagine a long, flat table covered in a layer of sand.

• The Goal: You want to see how "rough" the surface of the sand gets over time as the grains start to move.
• The Measurement: If the sand is perfectly flat, the roughness is zero. If the sand piles up into hills and valleys, the roughness increases.
• The Discovery: In the 1980s, scientists found that for many different systems (from burning paper to growing bacteria), this roughness follows a universal pattern called Family-Vicsek (FV) scaling. It's like a "law of nature" that says: No matter how big your table is or how fast the sand moves, if you measure the roughness correctly, all the data points will collapse into a single, perfect curve.

2. The Quantum Experiment: A Line of Atoms

The researchers at KAIST (in Korea) and their collaborators wanted to see if this "law of roughness" works for quantum gases.

• The Setup: They trapped a line of Lithium atoms in an "optical lattice." Think of this as a grid made entirely of laser light. The atoms are like marbles sitting in the cups of an egg carton, but the cups are made of light.
• The Initial State: They arranged the atoms in a perfect checkerboard pattern (an atom, a gap, an atom, a gap). This is like a perfectly smooth, flat surface.
• The "Quench": They suddenly turned up the "tunneling" power, allowing the atoms to jump between the laser cups. This is like shaking the table. The atoms start to move, creating a chaotic dance.

3. The Two Types of Chaos

The team ran the experiment in two different ways to see how the "roughness" (the fluctuation of atoms) evolved.

Scenario A: The "Super-Speed" Run (Ballistic)

First, they let the atoms move in a clean, quiet environment.

• What happened: The atoms moved incredibly fast and coherently, like a synchronized swim team. They didn't bump into each other randomly; they moved as a wave.
• The Result: The roughness grew quickly and followed the "Ballistic" pattern. The math showed that the atoms were moving at a constant, maximum speed allowed by quantum mechanics.
• The Analogy: Imagine a perfectly organized marching band. Even though they are moving, they stay in a tight, predictable formation. The "roughness" of their formation grows in a very specific, predictable way.

Scenario B: The "Chaotic" Run (Diffusive)

Next, they introduced "temporal disorder." This means they randomly flicked the laser lights on and off at different spots, creating random bumps in the road for the atoms.

• What happened: The perfect synchronization broke. The atoms started bumping into these random bumps, getting confused, and moving more like a drunk person stumbling through a crowd.
• The Result: The roughness still followed a universal pattern, but it was a different pattern. It slowed down and became "diffusive" (like smoke spreading in a room).
• The Analogy: Now imagine that same marching band, but someone is randomly throwing confetti and tripping them. They stop moving in a line and start wandering aimlessly. The "roughness" of their formation grows, but much slower and in a different way.

4. Why This Matters

The most exciting part of this paper is that both scenarios worked.

• Before this: We knew these scaling laws worked for classical things (like sand). We thought they might work for quantum things, but we hadn't proven it experimentally.
• Now: The researchers proved that quantum systems obey the same universal rules as classical systems.
  • They showed that you can take a quantum system, mess it up with random noise, and it will still settle into a predictable, universal pattern.
  • They measured the "exponents" (the secret numbers that define the pattern) and found they matched the theoretical predictions perfectly.

The Big Picture Takeaway

Think of the universe as having a "universal instruction manual" for how things grow and change when they are out of balance.

• For a long time, we thought this manual only applied to the big, slow world we can see (classical physics).
• This paper proves that the manual applies to the tiny, fast, weird world of atoms (quantum physics) too.

Whether it's a growing crystal, a spreading fire, or a cloud of atoms dancing in a laser trap, nature seems to use the same underlying code to describe how chaos turns into order (or how order turns into chaos). This discovery bridges the gap between the two worlds, giving us a unified framework to understand how the universe behaves when things are in a state of flux.

Technical Summary

Here is a detailed technical summary of the paper "Universal Family–Vicsek scaling in quantum gases far from equilibrium."

1. Problem Statement

Universal scaling laws, particularly the Family–Vicsek (FV) scaling, are well-established frameworks for describing the stochastic growth of surfaces in classical systems (e.g., fluid interfaces, combustion). These laws characterize surface roughness ($w$) via universal exponents: roughness exponent ($\alpha$), growth exponent ($\beta$), and dynamic exponent ($z$).

While recent theoretical studies suggested that FV scaling might extend to quantum many-body systems far from equilibrium (driven by intrinsic quantum fluctuations rather than external noise), experimental validation has been lacking. Specifically, there was no experimental demonstration of:

1. The full spatiotemporal scaling collapse (all exponents) in a quantum system.
2. The ability to tune the universality class of a quantum system from ballistic to diffusive regimes via controlled disorder.
3. The robustness of these scaling laws against integrability-breaking perturbations in a closed quantum system.

2. Methodology

The authors utilized a one-dimensional (1D) Bose-Hubbard quantum simulator using ultracold $^7$Li atoms in an optical lattice.

• Experimental Setup:

  • Quantum Gas Microscope: High-resolution imaging (single-site resolution) allowed for the measurement of atom occupation numbers.
  • Programmable Potential: A Digital Micromirror Device (DMD) was used to create arbitrary site-addressable potentials. This enabled the preparation of specific initial states and the introduction of dynamic disorder.
  • System Parameters: The system operated in the strongly interacting regime ($U/J \approx 39$) with low filling, effectively mapping the Bose-Hubbard model to a spin-1/2 XX model (hardcore bosons).

• Protocol:

  1. Initial State: A Charge Density Wave (CDW) state ($\uparrow, \downarrow, \uparrow, \downarrow, \dots$) was prepared. This corresponds to a "flat" surface with zero initial roughness.
  2. Quench: The lattice depth was suddenly lowered to initiate dynamics, allowing atoms to tunnel.
  3. Surface Height Definition: A quantum "surface height" operator $\hat{h}_j$ was defined as the cumulative sum of local density fluctuations (mapped to spin operators): $\hat{h}_j = \sum_{i=1}^j (\hat{n}_i - \nu)$, where $\nu=1/2$.
  4. Roughness Measurement: The surface roughness $w(t, L)$ was calculated as the standard deviation of the surface height at the chain center ($L/2$).
  5. Two Regimes:
     • Ballistic Regime: Dynamics in a clean lattice (static potential).
     • Diffusive Regime: Dynamics under a temporal random potential. Local repulsive barriers were switched on and off at random times (Poisson process) to break integrability and induce dephasing.

• Analysis:

  • Finite-Size Scaling: Experiments were performed on various system sizes ($L = 8$ to $24$).
  • Data Collapse: The authors tested the FV scaling ansatz $w(L, t) = L^\alpha f(t/L^z)$ to extract exponents $\alpha, \beta, z$.
  • Numerical Simulation: Theoretical calculations were performed using the Schrödinger equation for the XX model, incorporating experimental imperfections (mixed initial states) to validate the results.

3. Key Contributions

• First Experimental Observation of FV Scaling in Quantum Gases: The study provides the first experimental confirmation that the full Family–Vicsek scaling law holds in a closed quantum many-body system, capturing the entire relaxation process from fluctuation growth to saturation.
• Control of Universality Classes: The authors demonstrated the ability to switch the system's universality class by introducing temporal disorder:
  • Clean System: Exhibits Ballistic transport (Integrable XX model).
  • Disordered System: Exhibits Diffusive transport (Edwards-Wilkinson class) due to the destruction of integrability and momentum-space dephasing.
• Robustness of Scaling: The study confirms that FV scaling persists even under strong temporal disorder, provided the system remains in the diffusive regime.
• Correlation Dynamics: The work links surface roughness scaling to the spreading of density-density correlations, showing a transition from ballistic (Lieb-Robinson velocity) to diffusive spreading under disorder.

4. Key Results

A. Ballistic Universality Class (Clean System)

• Observation: In the absence of disorder, the surface roughness grows and saturates.
• Extracted Exponents:
  • $\alpha \approx 0.49(1)$
  • $\beta \approx 0.41(1)$
  • $z \approx 1.00(4)$
• Interpretation: These exponents align with the ballistic universality class (specifically the KPZ-like or free-fermion behavior expected for the integrable XX model). The dynamic exponent $z=1$ confirms ballistic propagation.
• Scaling Collapse: Data for different system sizes collapsed onto a single universal curve when rescaled by $L^\alpha$ and $L^z$.

B. Edwards-Wilkinson Universality Class (Disordered System)

• Observation: When temporal random potentials were applied, the system lost integrability.
• Extracted Exponents:
  • $\alpha \approx 0.49(1)$ (unchanged)
  • $\beta \approx 0.23(2)$
  • $z \approx 1.9(2)$
• Interpretation: The exponents shifted to match the Edwards-Wilkinson (EW) diffusive class (theoretical values: $\alpha=0.5, \beta=0.25, z=2$). The increase in $z$ from 1 to 2 indicates a transition from ballistic to diffusive transport.
• Correlation Spread: Density correlations spread diffusively ($\sim \sqrt{t}$) with a diffusion constant $D \approx 0.9 J a_{lat}^2 / \hbar$, contrasting with the linear ($\sim t$) ballistic spread in the clean case.

C. Numerical Validation

• Simulations confirmed that the choice of the height operator definition (cumulative sum vs. charge transfer) does not affect the universal scaling exponents in the long-time limit.
• Simulations verified that the scaling is robust against different initial states and the specific nature of the random kicks.

5. Significance

• Unification of Classical and Quantum Universality: The results establish a unified framework for nonequilibrium universality, proving that the same scaling laws governing classical surface growth also describe the relaxation of quantum fluctuations in many-body systems.
• Quantum Simulation of Non-Equilibrium Physics: The experiment demonstrates the power of quantum simulators to explore complex non-equilibrium phenomena (like universality class transitions) that are difficult to access in classical materials or purely theoretical models.
• Insight into Integrability Breaking: The study provides a clear mechanism for how temporal disorder destroys integrability and drives a quantum system toward stochastic diffusive behavior, bridging the gap between coherent quantum dynamics and classical hydrodynamics.
• Future Directions: The work opens avenues for investigating other universality classes (e.g., KPZ in quantum systems, anomalous scaling in quasi-periodic systems) and applying finite-size scaling to study non-Hermitian phase transitions and many-body localization.

In summary, this paper successfully bridges the gap between theoretical predictions and experimental reality, demonstrating that Family–Vicsek scaling is a universal feature of far-from-equilibrium quantum gases, with tunable dynamics ranging from ballistic to diffusive regimes.