Characterising transport in a quantum gas by measuring Drude weights

This study experimentally validates hydrodynamic predictions of nearly dissipationless transport in a one-dimensional ultracold bosonic gas by measuring Drude weights through two distinct current-inducing protocols, confirming that integrability governs large-scale dynamics via ballistically propagating quasi-particles.

The Gist

Imagine you have a long, narrow hallway filled with thousands of tiny, invisible balls (atoms) that are bouncing around. In the real world, if you push these balls, they usually bump into walls, each other, and lose energy, eventually slowing down like a car driving through thick mud. This is how most materials conduct electricity or heat: with friction and resistance.

But in this specific experiment, the scientists created a special "hallway" where the balls behave like ghosts. They don't bump into each other in a way that slows them down. Instead, they zip through the hallway at high speed without losing any energy. This is called ballistic transport.

The paper is about measuring exactly how well these ghost-balls move. To do this, the researchers used a concept called the Drude weight.

The "Stiffness" of the Flow

Think of the Drude weight as a measure of "flow stiffness."

• If a material is like a sponge (an insulator), it soaks up the push. The balls don't move much, and the "stiffness" is zero.
• If a material is like a super-highway (a metal or superconductor), the balls zoom through effortlessly. The "stiffness" is high.

The scientists wanted to measure this "stiffness" in a gas of atoms that was cooled down to almost absolute zero (colder than outer space) and squeezed into a one-dimensional line.

The Two Experiments: Pushing and Mixing

To measure this stiffness, the team used two different tricks, like two different ways to test how fast water flows in a pipe:

1. The Tilted Floor (Constant Force):
   Imagine the hallway of atoms is sitting on a flat floor. The researchers suddenly tilted the floor slightly, creating a gentle slope. Gravity (or in this case, a magnetic force) pulled the atoms down the slope. They measured how fast the atoms accelerated. Because the atoms were so "ghostly" (due to a property called integrability), they didn't slow down from friction; they just kept speeding up linearly. The rate of this speed-up told them the Drude weight.

2. The Dam Break (Bipartition):
   Imagine the hallway was split in the middle. On the left side, the atoms were packed tightly together. On the right side, they were spread out loosely. The researchers suddenly removed the wall in the middle. The atoms from the crowded side rushed into the empty side, creating two waves moving outward. By watching how these waves spread, they could calculate the "stiffness" of the flow.

The Secret Sauce: Physics-Informed Neural Networks

Here is the tricky part: The researchers couldn't see the "speed" of the atoms directly; they could only see where the atoms were (their density). It's like trying to guess how fast a river is flowing just by looking at a photo of the water's surface, without seeing the current underneath.

To solve this, they used a special computer program called a Physics-Informed Neural Network (PINN). Think of this AI as a super-smart detective.

• The detective knows the "rules of the game" (the laws of physics, like conservation of mass and energy).
• The detective looks at the blurry photos of the atoms.
• The detective uses the rules to fill in the missing pieces, calculating exactly how fast the atoms and energy were moving, even though they couldn't see it directly.

The Big Discovery

The results were a perfect match for a new theory called Generalized Hydrodynamics (GHD).

• The Theory: GHD predicted that even though the atoms were warm (relatively speaking) and interacting with each other, they would move without any friction.
• The Reality: The experiments confirmed this. The Drude weight was high, meaning the transport was almost entirely "dissipationless" (no energy lost to heat).

Why This Matters (According to the Paper)

The paper claims that this experiment proves that these "ghost-like" atoms follow the rules of Generalized Hydrodynamics perfectly. It shows that in these specific, one-dimensional quantum systems, the Drude weight is the key number that describes how the system moves on a large scale.

The authors also note that their method (using the AI detective to find currents from density) isn't just for this specific gas. It could be used to study other complex quantum materials where it's hard to see what's happening inside.

In short: The scientists built a friction-free highway for atoms, measured how "stiff" the flow was using two different methods, and used a smart AI to prove that the atoms moved exactly as a new, complex theory predicted—zooming forever without slowing down.

Technical Summary

Problem and Context
Transport properties are fundamental to classifying materials as insulators, metals, or superconductors. A key parameter in this classification is the Drude weight, which quantifies the ballistic transport of charge carriers. In solid-state systems, a finite Drude weight in the thermodynamic limit distinguishes metals from insulators. While generally expected to vanish at finite temperatures for interacting systems, notable exceptions exist, such as graphene and integrable models, where conserved quantities prevent the complete decay of currents.

Recent theoretical advances, specifically Generalized Hydrodynamics (GHD), have provided a framework for understanding transport in integrable systems. GHD posits that conserved charges are carried by interacting quasi-particles with infinite lifetimes that propagate ballistically, even at finite temperatures. Consequently, Drude weights should fully characterize the large-scale dynamics of these systems. However, despite theoretical progress and experimental observations of hydrodynamic evolution, direct measurements of transport coefficients like the Drude weight have not yet been performed.

Methodology
The authors experimentally measure the Drude weights of a one-dimensional (1D) ultracold gas of interacting bosonic atoms ($^{87}\text{Rb}$), a system that is integrable and described by the Lieb-Liniger model. The experiment is conducted on an Atom Chip, where atoms are confined to a single dimension by a tight transverse magnetic trap. The longitudinal dynamics are manipulated using a Digital Micromirror Device (DMD) to create custom optical dipole potentials.

To extract the Drude weights, the authors employ two distinct experimental protocols to induce currents in response to external perturbations:

1. Constant Force Protocol: A homogeneous gas is prepared in a box trap. At $t=0$, the potential is tilted to create a constant gradient (equivalent to a constant force). The resulting acceleration of the atomic current is measured by tracking the imbalance of atoms between the left and right halves of the system.
2. Bipartition Protocol: Two subsystems prepared in different equilibrium states (with a chemical potential imbalance $\delta\mu$) are joined at a point contact. The subsequent flow of charges between the subsystems is observed.

To determine the local atomic and energy currents from the measured atomic density profiles, the authors utilize Physics-Informed Neural Networks (PINNs). These networks are trained to satisfy the continuity equations for mass and energy, allowing for the accurate inference of current fields from sparse density data. The experimental results are compared against theoretical predictions derived from GHD and the Kubo formalism.

Key Results
The study successfully extracts Drude weights for both atomic and energy currents.

• Scaling Behavior: The Drude weight characterizing the atomic current response to a density perturbation ($D_{nn}$) is found to be proportional to the mean atomic density divided by the mass ($\bar{n}/m$). This simple scaling holds despite the system being at finite temperature and dominated by interactions.
• Energy Current: The Drude weight quantifying the ballistic energy current induced by coupling to atomic density ($D_{n\epsilon}$) also shows agreement with theoretical predictions.
• Dissipationless Transport: The results demonstrate that the transport is almost fully dissipationless. The measured Drude weights align closely with GHD predictions for thermal states at $T=50$ nK.
• Independence from Parameters: Analysis of residuals reveals that the measured responses are independent of temperature (within the quasi-condensate regime) and perturbation amplitude, confirming the linear response regime and the dominance of ballistic transport over diffusive processes on experimental timescales.
• Discrepancies: The experimentally extracted Drude weights are slightly lower than theoretical predictions. The authors attribute this to imperfections in the optical dipole potential, which introduce density variations and disorder not accounted for in the ideal GHD simulations.

Significance and Claims
The paper claims to provide the first direct experimental measurement of Drude weights in an ultracold quantum gas. The findings offer experimental validation of Generalized Hydrodynamics predictions, confirming that Drude weights almost fully characterize the large-scale transport of integrable systems, even in the presence of finite temperature and interactions.

The authors emphasize that the methodologies employed—specifically the use of controlled perturbations and PINNs for current extraction—are generalizable to other condensed matter systems. This work establishes a foundation for further studies on the transport properties of strongly correlated quantum matter, including non-thermal equilibrium states and systems where integrability is weakly broken. The simplicity of the 1D Bose gas serves as an ideal platform for verifying these transport coefficients, bridging the gap between theoretical hydrodynamic models and experimental observation.