Riemann Rarefaction Waves in a Strongly Interacting Fermi Gas

This study demonstrates that the expansion dynamics of a strongly interacting Fermi gas in a shock tube geometry closely follow Riemann's inviscid Euler solution across a wide temperature range, with self-similarity persisting even on the BCS side despite significant increases in viscosity.

The Gist

Imagine you have a crowded room of people (the gas) packed tightly inside a long, narrow hallway (the "shock tube"). Suddenly, one of the walls at the end of the hallway vanishes, and everyone rushes out into an empty, infinite space (a vacuum).

This paper is about watching exactly how that crowd spreads out, but with a very special kind of "people": ultracold atoms that are interacting so strongly they act like a single, perfect fluid.

Here is the story of what the scientists found, broken down into simple concepts:

1. The Perfect Fluid and the "Magic" Point

Usually, when things flow, they get messy. Honey flows slowly and sticks to itself (viscosity); water splashes and swirls. But these scientists were studying a specific state of matter called unitarity.

Think of unitarity as a "Goldilocks" zone for these atoms. It's a special setting where the atoms interact with each other just right—neither too weakly nor too strongly. At this point, the gas becomes a "perfect fluid." It has almost no internal friction (viscosity) and doesn't care about its size or shape (scale invariance). It's like a crowd of people who can move past each other without ever bumping or slowing down.

2. The "Riemann" Recipe

When the wall drops and the gas rushes out, the scientists wanted to know: What does the crowd look like as it spreads?

They turned to a 19th-century math recipe called the Riemann solution. This recipe predicts how a fluid should spread if it has zero friction. The recipe says the spreading should be self-similar.

The Analogy: Imagine taking a photo of the crowd spreading out at 1 second, then another at 2 seconds, then 3 seconds. If you stretch the 1-second photo to be twice as wide, and the 2-second photo to be four times as wide, they would all look exactly the same. The shape of the crowd doesn't change; it just gets bigger. This is what "self-similar" means.

3. The Experiment: A "Shock Tube"

The scientists built a tiny, invisible box using laser beams to hold their gas. It was shaped like a cylinder.

• The Setup: They held the gas in place, then suddenly turned off one laser "door."
• The Result: The gas rushed out. They took pictures of the density (how crowded it was) at different times.

What they found at the "Magic" Point (Unitarity):
The results were perfect. The gas spread out exactly as the 19th-century math recipe predicted. No matter how hot the gas was or how long they waited, if you adjusted the picture for the speed of the spread, every single photo collapsed into one single, perfect curve. The gas was behaving like a frictionless, ideal fluid.

4. Pushing the Limits: What if the fluid isn't perfect?

The scientists then asked: What happens if we change the rules? They moved the gas away from that "perfect" point.

• On one side (BEC): The atoms clumped together like molecules.
• On the other side (BCS): The atoms barely talked to each other.

In these "imperfect" states, the fluid has friction (viscosity). In the real world, friction usually messes up perfect patterns. It should make the spreading look different at different times, breaking the "self-similar" rule.

The Surprise:
Even when they added a lot of friction (making the gas 20 times more "sticky" than before), the gas still looked almost exactly like the perfect, frictionless recipe!

Why?
The scientists explain this with a "time" analogy. Friction needs time to mess things up.

• Imagine a drop of ink in a glass of water. At first, the ink is a sharp dot. Over time, it spreads and blurs.
• In this experiment, the gas was spreading out so fast and so far that the "blurring" effect of friction hadn't had enough time to ruin the pattern yet.
• It's like running a race: If you run fast enough, you can stay ahead of the wind for a long time. The gas was expanding so quickly that it stayed "self-similar" for a long time, even though it wasn't perfectly frictionless.

5. The Bottom Line

This paper shows that:

1. Perfect Fluids exist: At a specific setting, ultracold atoms act like a frictionless fluid that follows simple, elegant math rules perfectly.
2. Robustness: Even when the fluid gets "messy" and develops friction, it still looks like the perfect math model for a surprisingly long time.
3. A New Playground: This experiment gives scientists a clean, controllable way to study how fluids behave when they are pushed to their limits, acting like a test tube for the complex physics of how things flow.

In short, they watched a crowd of atoms rush out of a box and found that, whether the crowd was perfectly coordinated or a bit clumsy, they all spread out in a beautiful, predictable pattern that matches a math formula from 150 years ago.

Technical Summary

Technical Summary: Riemann Rarefaction Waves in a Strongly Interacting Fermi Gas

Problem Statement
Understanding transport in strongly interacting Fermi systems is a central challenge in many-body physics, relevant to contexts ranging from high-temperature superconductors to neutron stars and quark-gluon plasmas. In these systems, collision rates near the Fermi energy invalidate the weakly interacting quasi-particle picture, necessitating a hydrodynamic description governed by conservation laws. While the values of transport coefficients (e.g., viscosity, thermal conductivity) are difficult to predict, strongly interacting Fermi gases of ultracold atoms offer a highly controllable platform to test transport theories. Previous studies of nonlinear hydrodynamics in these gases were often complicated by inhomogeneous trapping potentials, which created spatially varying entropy distributions and ill-defined transport coefficients in low-density regions. This work addresses the need to study nonlinear hydrodynamics in a homogeneous setting to isolate the fundamental dynamics of expansion into a vacuum.

Methodology
The authors utilize a "shock tube" geometry to create a homogeneous, strongly interacting Fermi gas. The system consists of a balanced spin mixture of the $|1\rangle$ and $|3\rangle$ hyperfine states of $^6$Li, prepared at the Feshbach resonance near 690 G (unitarity). The gas is confined in a cylindrical box (length $L=90$ $\mu$m, diameter 130 $\mu$m) formed by blue-detuned laser beams and green light sheets.

The experimental protocol involves:

1. Initial State: Preparing a degenerate, homogeneous gas with a typical density $n_{3D} \sim 1$ $\mu$m$^{-3}$ and Fermi energy $E_F \sim h \cdot 10$ kHz.
2. Expansion: At $t=0$, one end cap of the box is suddenly removed, allowing the gas to expand into a vacuum along the cylinder's symmetry axis ($x$).
3. Imaging: After variable expansion times, polarization rotation imaging yields the radially integrated density profile $n(x,t)$.
4. Parameter Variation: The study probes the gas at unitarity across various temperatures ($T/T_F = 0.15$ to $0.32$) and across the BEC-BCS crossover by adiabatically tuning the interaction parameter $(k_F a)^{-1}$ from unitarity to both the BEC and BCS sides.

Key Contributions and Theoretical Framework
The primary theoretical contribution is the application of Riemann's solution to the Euler equations for a 1D expansion problem. In the inviscid limit (Euler hydrodynamics), the expansion of a homogeneous gas into a vacuum constitutes a Riemann problem with a step-function initial condition.

• Self-Similarity: Because the initial pressure and density of the unitary gas do not define a characteristic length scale (only a velocity scale $\sqrt{P/\rho}$), the dynamics are predicted to be self-similar, depending solely on the ratio $\xi = x/t$.
• Rarefaction Wave Solution: For a unitary gas with a polytropic equation of state $P \propto \rho^\gamma$ (where $\gamma = 5/3$), the authors derive an analytical density profile:
  $$\tilde{\rho} = \left( \frac{3 - \tilde{\xi}}{4} \right)^3$$
  where $\tilde{\xi} = x / (c_0 t)$, $c_0$ is the initial speed of sound, and $\tilde{\rho}$ is the normalized mass density. This solution predicts a specific density value at the origin ($x=0$) of $(3/4)^3$ at all times.

Results

1. Unitary Gas at Unitarity:

   • Experimental data for the expansion of the unitary Fermi gas shows that density profiles at different times and temperatures collapse onto a single curve when plotted against the reduced coordinate $\tilde{\xi} = x/(c_0 t)$.
   • The experimental data exhibits excellent, fit-free agreement with the derived Riemann solution (Eq. 5). The measured density at $x=0$ matches the theoretical prediction of $(3/4)^3$.
   • Small deviations, such as the rounding of the sharp cusp at $\tilde{\xi} = -1$, are attributed to finite viscosity and the finite thickness of the box wall at $t=0$.

2. Variation of Interactions (BEC-BCS Crossover):

   • Self-Similarity Robustness: Even when interactions are tuned away from unitarity into the BCS regime (where sound diffusivity increases twenty-fold), the data largely retains self-similarity. The RMS deviations from self-similarity remain nearly constant as interactions move into the BCS regime.
   • Deviations from Riemann Solution: While self-similarity (collapsing onto a single curve) is preserved, the shape of the collapsed profiles in the deep BCS limit deviates from the specific Riemann solution derived for the unitary gas. This is attributed to the increasing role of viscosity and thermal conductivity, which introduce a new length scale ($D/c_0$, where $D$ is sound diffusivity).
   • Time Scale Argument: The authors argue that self-similarity persists because the hydrodynamic length scale $c_0 t$ eventually becomes much larger than the viscous length scale $D/c_0$ (i.e., the Reynolds number grows with time), rendering viscous effects negligible for the observed time scales ($t \ge 1.0$ ms).

Significance
The paper demonstrates that strongly interacting Fermi gases serve as a highly controllable testbed for nonlinear hydrodynamics. The key findings are:

• Validation of Euler Hydrodynamics: The unitary Fermi gas behaves as a nearly inviscid fluid where rarefaction waves follow the self-similar Riemann solution of the Euler equations across a wide range of temperatures.
• Robustness of Self-Similarity: Self-similarity is remarkably robust even when the system is moved away from the scale-invariant unitary point and into regimes with significantly higher dissipation (BCS side). This suggests that 1D Navier-Stokes rarefaction flows can approach Euler self-similar solutions at long times, even in the presence of increased viscosity.
• Transition to Navier-Stokes: The observed deviations from the specific Riemann profile in the BCS regime highlight the transition to full Navier-Stokes dynamics where dissipative effects (viscosity and thermal conductivity) become relevant, distinguishing the flow from the ideal inviscid case.

The work establishes a foundation for studying entropy production and nonlinear flows in a controlled environment, with potential extensions to controlled 1D shock waves.