Observation of Vinen turbulence during far-from-equilibrium Bose-Einstein condensation

Researchers observed and characterized decaying Vinen turbulence in a homogeneous 3D atomic Bose gas, demonstrating that the decay of vortex line density follows ultraquantum predictions and exhibits incompressible hydrodynamic behavior consistent with strongly interacting superfluid helium, regardless of the gas's weak interatomic interactions.

The Gist

Imagine a giant, invisible bowl filled with a special kind of gas so cold that the atoms inside stop acting like individual particles and start moving as a single, synchronized team. This is a Bose-Einstein Condensate, a state of matter often called a "superfluid." In this state, the gas flows without any friction, like a perfect ice skater who never slows down.

However, if you disturb this calm team, things get chaotic. The paper describes what happens when scientists take this super-cold gas, shake it up to create a mess, and then watch it try to calm itself down again.

Here is the story of their discovery, broken down into simple concepts:

1. The "Hair Tangle" Problem

When the gas is disturbed, it doesn't just swirl like water in a bathtub. Instead, it gets tangled up with invisible, microscopic "whirlpools" called vortices.

Think of these vortices like strands of hair. If you take a handful of hair and shake it, the strands get knotted and tangled in a messy, random ball. In this experiment, the gas became a 3D ball of tangled "hair" (vortex lines). The scientists wanted to know: How does this messy tangle untangle itself?

2. The Magic Camera Trick

The problem is that these vortex lines are incredibly thin—about 1,000 times thinner than a human hair. If you just took a photo of the whole cloud of gas, the vortices would be too small to see, like trying to spot a single thread in a giant ball of yarn from across the room.

To solve this, the scientists used a clever "magic trick" with light:

• Magnification: They used a special lens made of light waves to blow up the gas cloud, making it about 3.5 times bigger.
• The Slice: Instead of looking at the whole giant ball, they took a picture of just a very thin slice of it (like slicing a loaf of bread and looking at just one slice).

When they did this, the invisible vortex lines appeared as dark, high-contrast lines running through the slice, looking like cracks in a frozen lake or dark threads in a piece of fabric.

3. The "Untangling" Dance

The team watched these dark lines over time. At first, the gas was a chaotic mess of tiny ripples. But as time passed (over about 80 milliseconds), the ripples organized themselves into clear, distinct lines.

Then, the real magic happened: The lines started to disappear.
The scientists measured how much "length" of these vortex lines existed in the gas. They found that the lines were decaying (disappearing) at a very specific, predictable speed. It wasn't a slow, random fade; it followed a strict mathematical rule known as Vinen turbulence.

4. The Surprising Connection

Here is the most fascinating part of the discovery.

• The Gas: The scientists were using a gas made of Potassium atoms. It is very light, very cold, and the atoms barely bump into each other (it's "weakly interacting").
• The Comparison: They compared their results to superfluid helium, a liquid helium that has been cooled to near absolute zero. Helium is heavy, dense, and the atoms bump into each other constantly (it's "strongly interacting").

Despite these two fluids being completely different on a microscopic level (like comparing a swarm of bees to a crowd of people in a mosh pit), they untangled in exactly the same way.

The rate at which the vortex lines disappeared was identical in both the light gas and the heavy helium. This suggests that once the chaos gets big enough, the specific details of the atoms don't matter. The "rules of the game" for how turbulence dies out are universal.

The Bottom Line

The paper shows that when a quantum fluid is thrown into chaos, it creates a tangled mess of invisible whirlpools. By using a special "zoom-in" camera technique, the scientists proved that this mess untangles itself following a universal law.

It's as if you took a bowl of spaghetti (the gas) and a bowl of wet sand (the helium), shook them both up, and found that they both settled down into a neat pile at the exact same speed, regardless of whether they were made of pasta or sand. This confirms a theory that has been around for decades: that the way quantum turbulence decays is a fundamental rule of nature, not just a quirk of a specific material.

Technical Summary

1. Problem Statement

The relaxation of quantum fluids from far-from-equilibrium states is theoretically linked to the emergence of long-range order and the decay of a turbulent, isotropic tangle of quantized vortex lines. This regime is known as Vinen turbulence (or "ultraquantum" turbulence), characterized by a random tangle of vortices without large-scale classical flow.

While Vinen turbulence has been studied extensively in superfluid helium (where it was linked to the decay of vortex line density $L$ following the law $dL/dt = -BL^2$), direct observation and quantitative measurement of this phenomenon in homogeneous 3D atomic Bose gases have remained a challenge. Previous studies in atomic gases relied on indirect measurements (e.g., momentum distributions) that inferred vortex presence but did not directly image the vortex tangle. The specific challenges include:

• Directly visualizing randomly oriented vortex lines in a 3D volume.
• Quantifying the vortex line-length density ($L$) and its decay dynamics.
• Determining if the decay constant $B$ depends on interaction strength in weakly interacting, highly compressible gases compared to the incompressible superfluid helium.

2. Methodology

The authors employed a novel experimental protocol using a homogeneous 3D atomic gas of Potassium-39 ($^{39}$K) trapped in an optical box potential.

• System Preparation:

  • The gas was prepared in a far-from-equilibrium state by starting with a Bose-Einstein Condensate (BEC), perturbing it to create an incoherent state, and then initiating relaxation by tuning the scattering length $a$ via a Feshbach resonance at $t=0$.
  • The gas density was $n \approx 2.2 \, \mu\text{m}^{-3}$, with a healing length $\xi \sim 1 \, \mu\text{m}$.
  • Interactions were tuned to three different scattering lengths ($a = 300, 430, 1400 \, a_0$) to test universality.

• Imaging Technique (Matter-Wave Magnification & Slicing):

  • To resolve the small vortex cores ($\sim 1 \, \mu\text{m}$), the team used a matter-wave lens (pulsed harmonic potential) to magnify the cloud by a factor of $M \approx 3.5$ in the $x-y$ plane.
  • Crucially, they imaged only a thin central slice ($d \approx 15 \, \mu\text{m}$) of the magnified cloud. This technique prevents the "washing out" of vortex signatures that occurs when integrating over the entire 3D volume.
  • Vortex lines crossing this slice appear as high-contrast dark lines (density dips) in the 2D absorption images.

• Quantitative Analysis:

  • Vortex Line-Length Density ($L$): Defined as the total length of vortex lines per unit volume.
  • Using the geometric relation for an isotropic random tangle, $L = 4N_i/S$, where $N_i$ is the number of vortex intersections with the imaged slice area $S$.
  • $N_i$ was determined by counting distinct dark clusters in the images (after thresholding and grouping pixels).
  • The decay of $L$ was analyzed against the theoretical prediction for Vinen turbulence:
    $$\frac{dL}{dt} = -BL^2 - \frac{L}{\tau}$$
    where the first term represents vortex-vortex interactions (turbulence decay) and the second term represents one-body decay (annihilation at boundaries).

3. Key Contributions

1. Direct Visualization: First direct observation of randomly oriented vortex lines in a homogeneous 3D atomic gas during far-from-equilibrium condensation.
2. Quantitative Decay Law: Direct measurement of the vortex line-length density $L(t)$ and verification of the nonlinear decay law characteristic of Vinen turbulence.
3. Universality of the Decay Constant: Demonstration that the decay prefactor $B$ is independent of the interaction strength (scattering length $a$) in the weakly interacting regime.
4. Incompressible Hydrodynamics in Compressible Gases: Evidence that despite the gas being highly compressible, its large-scale turbulence dynamics mimic those of an incompressible hydrodynamic fluid (like superfluid helium).

4. Key Results

• Vortex Emergence: At early times ($t < 80$ ms), only small-scale density fluctuations were observed. Well-defined, separated vortex lines appeared at $t \gtrsim 80$ ms, coinciding with the onset of universal coarsening in the coherence length $\ell$ (where $\ell \propto L^{-1/2}$).
• Decay Dynamics:
  • The measured $L(t)$ followed the predicted decay curve.
  • At early times, the decay was faster than exponential, driven by vortex-vortex interactions ($L^2$ term).
  • At very long times (low $L$), the decay became exponential ($L/\tau$), attributed to individual vortices annihilating at the trap walls.
• The Decay Constant ($B$):
  • The fitted value was $B \approx 1.0(2) \hbar/m$.
  • This value is consistent across different interaction strengths ($a$).
  • This result matches the value observed in superfluid $^4$He ($B \approx \hbar/m_{He}$), despite the microscopic differences (compressibility, interaction strength) and the fact that the ratio of vortex separation to core size in the atomic gas is $\sim 10^5$ times smaller than in helium.
• Universality: When data from different scattering lengths were scaled by the interaction-dependent time offset $t^*$, they collapsed onto a single universal curve defined by $B = 1.0 \hbar/m$.

5. Significance

• Validation of Theoretical Models: The results provide strong experimental validation for theoretical models linking far-from-equilibrium Bose-Einstein condensation to the decay of a random vortex tangle (Vinen turbulence).
• Bridge Between Systems: The finding that $B \approx \hbar/m$ in a weakly interacting, compressible atomic gas, matching the value in strongly interacting, incompressible superfluid helium, suggests a deep universality in quantum turbulence dynamics that transcends specific microscopic details.
• Methodological Advancement: The combination of matter-wave magnification and slice imaging establishes a powerful new tool for studying 3D quantum turbulence, allowing for direct tomography of vortex structures that was previously impossible in 3D atomic gases.
• Future Directions: The work opens avenues for studying the transition from wave-dominated to vortex-dominated turbulence, exploring dynamics near criticality, and performing full 3D tomography of vortex tangles by imaging multiple slices.

In summary, this paper successfully bridges the gap between theoretical predictions of quantum turbulence in atomic gases and experimental observation, confirming that the decay of a random vortex tangle follows the universal Vinen law regardless of the specific interaction strength of the gas.