Realization of fractional Fermi seas

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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 a crowded dance floor. In the world of quantum physics, there are two main types of dancers: Bosons and Fermions.

Bosons are the social butterflies. They love to huddle together in the exact same spot, forming a giant, synchronized blob (this is called a Bose-Einstein Condensate).
Fermions are the introverts with a strict "personal space" rule. Thanks to the Pauli Exclusion Principle, no two fermions can ever occupy the same spot. They stack up like books on a shelf, filling every available seat from the bottom up until they hit a "Fermi energy" line. This stack is called a Fermi Sea.

For decades, physicists thought these were the only two options. But what if there was a middle ground? What if you could have a "Fermi Sea" where the dancers are allowed to share seats, but only partially? This is the concept of a Fractional Fermi Sea (FFS).

This paper describes how a team of scientists in Innsbruck successfully built this exotic "fractional" dance floor using ultracold atoms. Here is how they did it, explained simply:

1. The Setup: A One-Dimensional Dance Line

The scientists took a cloud of Cesium atoms (which are naturally Bosons) and trapped them in a grid of thousands of tiny, vertical tubes made of laser light. Think of it like turning a 3D dance floor into thousands of 1D hallways. In these narrow hallways, the atoms are forced to interact in a very specific way.

2. The Magic Trick: The "Holonomy Cycle"

The core of the experiment is a special routine they call an interaction cycle. Imagine the atoms are connected by invisible springs.

Step 1: They start with the springs pushing the atoms apart (strong repulsion). The atoms are forced to act like Fermions (the introverts) because they can't get close to each other.
Step 2: They slowly change the springs to pull the atoms together (attraction). This is dangerous because usually, attractive atoms crash into each other and collapse.
Step 3: They cross a "zero point" where the springs do nothing, and then pull them together even harder.
Step 4: They bring it back to the start.

In a normal world, this cycle would destroy the atoms. But because these atoms are in a 1D line and follow special quantum rules, the system is "integrable." This is a fancy way of saying the system has so many hidden conservation laws that it acts like a perfect, frictionless machine that doesn't lose its shape easily.

3. The Result: Fractional Occupancy

When they ran this cycle, something magical happened. The atoms didn't just return to their original state. Instead, they settled into a new, excited state where the "dance floor" was filled differently.

In a normal Fermi Sea, every seat up to a certain line is 100% full. In this new Fractional Fermi Sea, the seats are only half-full (or a quarter-full, depending on the cycle).

Analogy: Imagine a parking garage. A normal Fermi Sea is a garage where the bottom 10 floors are completely packed with cars, and the top floors are empty. A Fractional Fermi Sea is a garage where the bottom 20 floors are only half-parked. It's a "fractional" occupancy of the available space.

4. The Proof: The "Friedel Oscillation"

How did they know they succeeded? They looked at the momentum distribution (how fast and in what direction the atoms were moving).

In a normal gas, the atoms move in a smooth, predictable way. But in this fractional state, the atoms started "wiggling" in a specific pattern called Friedel Oscillations.

Analogy: If you drop a stone in a calm pond, you get smooth ripples. If you drop a stone in a pond with a hidden, structured grid underneath, the ripples bounce off the grid and create a complex, repeating interference pattern.
The scientists saw this "interference pattern" in the atoms' movement. It was the "smoking gun" that proved the atoms had rearranged themselves into this exotic fractional structure.

Why Does This Matter?

This isn't just a cool party trick.

New Physics: It proves that we can create "super-fermionic" states where particles share space in fractional ways, bridging the gap between bosons and fermions.
Stability: They managed to keep these excited, high-energy states stable for over 5 seconds. Usually, excited quantum states fall apart instantly.
Future Tech: Understanding these states could help us build better quantum computers or sensors. If we can control how particles share space, we might be able to engineer materials with totally new properties, like super-efficient electricity conductors or ultra-sensitive detectors.

In summary: The team took a group of "social" atoms, forced them into a narrow line, and performed a delicate dance of attraction and repulsion. The result was a stable, high-energy state where the atoms occupied space in a fractional, "half-full" pattern, revealing a new chapter in the story of how matter behaves.

1. Problem and Motivation

Fundamental Physics: The Pauli exclusion principle dictates that fermions fill energy states up to a Fermi energy, forming a "Fermi sea," while bosons can condense into a single state. However, in one-dimensional (1D) systems, interactions and statistics are intrinsically linked, allowing bosons to acquire fermionic traits (dynamical fermionization).
Theoretical Gap: Haldane's Generalized Exclusion Statistics (GES) extends the Pauli principle, suggesting that adding a particle reduces the available state space by a factor $\alpha$ . While $\alpha=0$ corresponds to bosons and $\alpha=1$ to fermions, the theory predicts "super-fermionic" states where $\alpha > 1$ . In such states, particles occupy a fractional number of momentum states, leading to Fractional Fermi Seas (FFS) characterized by uniform but fractional momentum occupancies ( $n(k) = 1/\alpha$ ).
Challenge: While FFS have been predicted theoretically as excited many-body states in the Lieb-Liniger (LL) model, their experimental realization has been elusive due to the difficulty of stabilizing highly excited, non-thermal states in the presence of attractive interactions, which typically lead to collapse or thermalization.

2. Methodology

The researchers utilized ultracold Cesium (Cs) atoms confined in an array of approximately 7,000 vertical 1D tubes created by a 2D optical lattice.

System Control:
- Interaction Tuning: The 1D interaction strength ( $g_{1D}$ ) was dynamically tuned using Feshbach resonances combined with a confinement-induced resonance (CIR). This allowed the system to traverse from repulsive ( $g_{1D} > 0$ ) to attractive ( $g_{1D} < 0$ ) regimes, passing through the non-interacting point ( $g_{1D}=0$ ).
- Interaction Cycles (Holonomy Cycles): The core protocol involved "holonomy cycles" where $g_{1D}$ was ramped from $0 \to +\infty$ (Tonks-Girardeau phase) $\to -\infty$ (super-Tonks-Girardeau phase) $\to 0$ .
- State Parameterization: The resulting states are parameterized by a charge parameter $\ell$ (where $\alpha = \ell$ ). The experiment targeted $\ell = 2$ and $\ell = 4$ , corresponding to fractional occupancies of $1/2$ and $1/4$ .
Stabilization Strategy:
- To prevent collapse in the attractive regime, the team optimized the loading protocol. They loaded the 1D tubes at a higher initial scattering length ( $a_{3D} \approx 500 a_0$ ) compared to previous experiments, resulting in lower atomic density per tube. This significantly enhanced the stability of the super-Tonks-Girardeau (sTG) state, achieving lifetimes $>5$ seconds.
Measurement & Theory:
- Momentum Distribution: Measured via Time-of-Flight (ToF) expansion.
- Correlation Functions: The first-order correlation function $G^{(1)}(x)$ was derived via Fourier transform of the momentum distribution.
- Modeling: The experiment was modeled using Generalized Hydrodynamics (GHD), a framework tailored for integrable systems (like the LL model) that describes non-equilibrium dynamics using Generalized Gibbs Ensembles (GGE).

3. Key Contributions

First Experimental Realization of FFS: The team successfully prepared and stabilized excited 1D Bose gas states exhibiting fractional Fermi sea characteristics for $\alpha = 2$ and $\alpha = 4$ .
Observation of Friedel Oscillations (FO): They identified the "smoking-gun" signature of FFS: pronounced Friedel oscillations in the first-order correlation function $G^{(1)}(x)$ . Unlike ground-state bosons where FOs are subleading, these non-equilibrium states exhibited strong, clear oscillations.
Demonstration of Irreversibility: The study revealed a strong asymmetry in the interaction cycles. While forward ramps (creating FFS) were relatively stable, reverse ramps (returning to the ground state) resulted in significant atom loss (up to 45%). This was attributed to the excitation of Bethe strings (bound states) when crossing the non-interacting point from the repulsive side, a phenomenon predicted by GHD but difficult to observe directly.
Validation of GHD: The experimental data showed excellent agreement with GHD simulations, confirming that the system evolves into specific non-thermal macrostates described by GGEs rather than thermalizing.

4. Key Results

Momentum Distributions:
- $\ell=0$ (Ground State): Narrow momentum distribution (FWHM $\approx 1.72 \, \mu m^{-1}$ ).
- $\ell=2$ : Broadened and flattened distribution with a "box-like" profile ( $n(k) \approx 1/2$ ) and FWHM $\approx 5.19 \, \mu m^{-1}$ .
- $\ell=4$ : Further broadening with $n(k) \approx 1/4$ and FWHM $\approx 6.69 \, \mu m^{-1}$ .
- Note: While the trap potential smoothed the sharp edges of the ideal "box" distribution, the overall broadening matched theoretical predictions.
Correlation Functions ( $G^{(1)}(x)$ ):
- For $\ell=0$ , $G^{(1)}(x)$ decayed smoothly and remained positive.
- For $\ell=2$ and $\ell=4$ , $G^{(1)}(x)$ developed oscillatory behavior, dipping below zero.
- The first minima occurred at $\approx 1.6 \, \mu m$ ( $\ell=2$ ) and $\approx 1.2 \, \mu m$ ( $\ell=4$ ), consistent with the theoretical prediction of Friedel oscillations arising from the fractional Fermi momentum.
Stability and Irreversibility:
- The optimized loading protocol allowed for multiple cycles with minimal atom loss in the forward direction.
- Reverse cycles triggered bound-state formation (Bethe strings), leading to rapid atom loss, confirming the non-reversible nature of accessing these excited states via slow ramps through the non-interacting point.

5. Significance

New Quantum States: This work provides the first experimental evidence of "super-fermionic" statistics and fractional Fermi seas, expanding our understanding of quantum statistics beyond the boson-fermion dichotomy.
Non-Equilibrium Thermodynamics: It demonstrates a controllable method to engineer and stabilize highly excited, non-thermal states in integrable systems, offering a platform to study quantum thermodynamics and Generalized Gibbs Ensembles.
Diagnostic Tools: The observation of Friedel oscillations serves as a direct diagnostic for the underlying many-body structure and fractional occupancy.
Future Applications: The ability to engineer these states opens pathways for:
- Investigating the dynamics of impurities in FFS (e.g., emergent superfluidity, Bloch oscillations).
- Developing new protocols for quantum information and sensing based on geometric control of many-body states.
- Exploring the transition between integrable and non-integrable dynamics in the presence of exotic statistics.

In summary, the paper successfully bridges advanced theoretical concepts of generalized exclusion statistics with experimental reality, using ultracold atoms to create and characterize exotic fractional Fermi seas, while providing deep insights into the stability and irreversibility of non-equilibrium quantum matter.