🔬 mesoscale physics

Expansion of a free Fermi gas released from an isotropic trapping potential

This paper utilizes semi-classical techniques to derive explicit expressions for the universal late-time spatial and momentum profiles of a large number of non-interacting fermions expanding ballistically from an isotropic trap, revealing a generalized edge exponent of $d/a$ that extends the Wigner semi-circle and Thomas-Fermi laws to arbitrary power-law potentials and dimensions.

Original authors: J M Luck, P L Krapivsky

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J M Luck, P L Krapivsky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 where the dancers are tiny, invisible particles called fermions. These aren't just any dancers; they follow a strict rule known as the "no-sharing" policy. In the quantum world, this means no two fermions can ever occupy the exact same spot or move in the exact same way at the same time. It's like a cosmic game of musical chairs where, as soon as the music stops, everyone must find a unique seat, and if the chairs are full, new dancers have to stand on the very tips of their toes, vibrating with high energy.

Now, picture these dancers trapped inside a bowl-shaped cage. This cage is a "trapping potential," a force field that keeps them confined. In many experiments with ultracold atoms, scientists use these bowls to study how matter behaves when it's super cold and super crowded. But what happens if you suddenly yank the cage away? The dancers are suddenly free to run. If they don't bump into each other (which they don't, in this specific scenario), they will fly outward in straight lines, like confetti thrown from a cannon. This is called "ballistic expansion." Scientists care about this because watching how the cloud of particles spreads out tells them everything about how the particles were moving and arranged before the cage was removed. It's like looking at the pattern of raindrops on a windshield to figure out how fast the car was going and what the road looked like before the storm started.

This paper takes that idea and asks a big question: If we have a huge crowd of these non-interacting fermions trapped in a bowl of any shape (not just a perfect curve, but maybe a steep cliff or a flat-bottomed box), what does the expanding cloud look like after a long time? The authors, J. M. Luck and P. L. Krapivsky, use a mix of math tricks and "semi-classical" thinking (which is like treating the particles as tiny billiard balls for the big picture) to predict the final shape of this flying cloud.

Here is what they found. When the trap is released, the cloud doesn't just puff out randomly; it expands in a very specific, predictable way. The particles fly outward at speeds they had while trapped, and because the fastest particles were at the edge of the trap, the whole cloud stretches out into a giant, growing ball (or a long line in one dimension). The most exciting discovery is about the edge of this cloud.

If the trap was a perfect bowl (a harmonic potential), the edge of the cloud looks like a smooth, rounded hill, a shape known as the "Wigner semi-circle law." But the authors show that if you change the shape of the trap—making it steeper, flatter, or shaped like a box—the edge of the cloud changes too. It doesn't stay a smooth hill; instead, it develops a specific "power-law" shape. Think of it like this: if the trap is a gentle slope, the cloud's edge is a soft curve. If the trap is a steep cliff, the cloud's edge becomes a sharper, more pointed drop-off. The paper provides a single, universal formula that predicts exactly how sharp or soft that edge will be, based on how steep the original trap was and how many dimensions the particles are moving in.

They also looked at a very specific case: a "square-well" trap, which is like a box with perfectly flat bottom and vertical walls. In this case, the math predicts that the particles inside will have a uniform momentum distribution across the bulk of the cloud, meaning most particles have speeds spread evenly up to a maximum limit. When the box opens, the cloud spreads out evenly until it hits a sharp edge. However, the authors note that right at this very edge, there's a tiny, microscopic "fuzziness." In a perfect box, this drop-off is so sharp that a few particles manage to sneak just a little bit outside the main group, and the number of these "escapees" grows slowly (logarithmically) as you add more particles. In contrast, for the smooth bowl trap, the fuzziness is a smooth, gentle curve, and the number of escapees is a tiny, constant number that doesn't grow with the crowd size.

The paper doesn't just guess these shapes; they derive them using rigorous math for any number of dimensions (1D, 2D, 3D, or more) and any trap shape. They confirm that for a huge number of particles, the cloud's shape is a generalized version of the famous Wigner semi-circle law, but with a "tail" that changes its sharpness depending on the trap. They also point out that in higher dimensions (like 2D or 3D), the exact number of particles needed to fill up the "shells" of energy levels can be a bit tricky and irregular, leading to small wiggles in the predictions, but the overall big picture remains clear.

In short, this work gives us a universal rulebook for how a crowd of quantum particles behaves when set free. Whether they were trapped in a soft bowl or a hard box, the authors have shown us exactly how the cloud will look as it flies apart, turning a complex quantum problem into a beautiful, predictable pattern of expansion.

Technical Summary: Expansion of a Free Fermi Gas Released from an Isotropic Trapping Potential

Problem Statement
This work investigates the non-equilibrium dynamics of a system of $N$ non-interacting, spinless (or spin-degenerate) fermions initially prepared in the many-body ground state of an isotropic trapping potential $V(\mathbf{x})$ . The study focuses on the regime following the instantaneous release of the potential at $t=0$ . Specifically, the authors analyze the ballistic expansion of the fermionic cloud at late times, aiming to derive the full spatial and momentum profiles of the gas in the limit of large fermion number $N$ . The primary goal is to determine how the initial trapping potential shape (characterized by a power-law exponent $a$ ) and the spatial dimension $d$ influence the asymptotic density profile and the edge behavior of the expanding cloud.

Methodology
The authors employ a combination of exact analytical solutions for specific cases and semi-classical (WKB) approximations for the general case of large $N$ .

Exact Solutions (Harmonic and Square-Well Potentials):
- For a harmonic potential ( $V \propto x^2$ ), the authors utilize the exact time-dependence of the many-body wavefunction, which is governed by the Ermakov-Pinney equation. They demonstrate that the rescaled positions of the fermions follow the eigenvalue distribution of a Gaussian Unitary Ensemble (GUE) random matrix.
- For a one-dimensional square-well potential, exact wavefunctions (involving Bessel functions in 2D and trigonometric functions in 1D) are used to compute the momentum distribution $f_N(p)$ for finite $N$ , allowing for an analysis of edge behaviors and finite-size corrections.
Semi-Classical Approximation (General Potentials):
- For arbitrary isotropic power-law potentials $V(r) \propto r^a$ in any dimension $d$ , the authors apply the stationary-phase approximation to the time-evolved wavefunction. This establishes that at late times, the position $\mathbf{x}$ of a fermion is related to its initial momentum $\mathbf{p}$ by ballistic scaling: $\mathbf{x} \approx \mathbf{p}t$ .
- The momentum distribution $f_N(p)$ is derived by summing the squared modulus of the Fourier transforms of the occupied single-particle eigenstates. In the large $N$ limit, this sum is converted into an integral using the semi-classical quantization rule and the Thomas-Fermi approximation for the density of states.
- The microcanonical momentum distribution at a fixed energy $E$ is calculated based on the classical phase-space volume, which is then integrated over the occupied energy states up to the Fermi energy $E_F$ .

Key Contributions and Results

Generalized Density Profiles:
The paper derives explicit expressions for the late-time spatial density profile $\rho_N(r; t)$ and the momentum distribution $f_N(p)$ for arbitrary power-law potentials $V(r) \propto r^a$ in $d$ dimensions.
- Momentum Distribution: For $p < p_F$ , the distribution scales as:
  $f_N(p) \sim \frac{N}{p_F^d} \left( 1 - \frac{p^2}{p_F^2} \right)^{d/a}$
- Spatial Profile: Consequently, the late-time spatial profile of the cloud (for $r < p_F t$ ) is:
  $\rho_N(r; t) \sim \frac{N}{(p_F t)^d} \left( 1 - \frac{r^2}{(p_F t)^2} \right)^{d/a}$
  Here, $p_F$ is the Fermi momentum, which scales with $N$ as $p_F \sim N^{a/((a+2)d)}$ .
Universal Edge Exponent:
A central finding is the identification of a universal edge exponent $d/a$ . The density profiles exhibit power-law singularities at the moving edges ( $r = p_F t$ ) rather than the square-root singularities characteristic of the Wigner semi-circle law (which corresponds to the specific case $d=1, a=2$ ) or the Thomas-Fermi distribution ( $d=3, a=2$ ). The results generalize these known laws to all power-law potentials and dimensions.
Edge Behavior and Scaling Functions:
- Harmonic Trap ( $a=2$ ): The edge of the Wigner semi-circle law is shown to be broadened into a smooth crossover with a relative width scaling as $N^{-2/3}$ . The profile near the edge is described by a scaling function involving the Airy function derivative squared, $A(\xi) = \text{Ai}'(\xi)^2 - \xi \text{Ai}(\xi)^2$ . The mean number of fermions outside the Wigner interval is a small, finite constant ( $\approx 0.061$ ).
- Square-Well Potential ( $a \to \infty$ ): In the limit of an infinite well, the momentum distribution becomes uniform inside the Fermi ball. However, the drop to zero at the edge is described by a different scaling function $F(\xi)$ involving digamma and trigamma functions. Unlike the harmonic case, the sharp potential leads to a long algebraic tail in the distribution ( $F(\xi) \sim 1/\xi$ ), causing the number of fermions outside the Fermi interval to grow logarithmically with $N$ ( $N_{out} \sim \ln N$ ).
Higher-Dimensional Effects:
In dimensions $d \ge 2$ , the presence of orbital degeneracy is addressed. The authors show that for specific fermion numbers corresponding to complete shells of degenerate states, the system possesses a unique isotropic ground state. For an infinite circular well in 2D, the paper analyzes the oscillatory component of the integrated density of states, demonstrating that the Fermi momentum exhibits irregular oscillations even when restricted to complete shell numbers, a feature linked to the periodic orbits of the classical billiard.

Significance and Claims
The authors claim that these outcomes are novel, providing a comprehensive analytical framework for the expansion of free fermions that generalizes the Wigner semi-circle law and the Thomas-Fermi distribution. The work highlights that the late-time ballistic expansion maps the initial momentum distribution directly to the spatial profile, with the edge behavior universally determined by the ratio of the spatial dimension to the potential exponent ( $d/a$ ).

The paper emphasizes the utility of free fermionic systems in providing detailed analytical access to non-equilibrium dynamics, which is often unavailable in interacting systems. It establishes a clear connection between the geometry of the initial confinement (via the exponent $a$ ) and the universality class of the expanding cloud's edge. The findings regarding the logarithmic growth of particles outside the Fermi edge in square-well potentials versus the constant number in harmonic traps illustrate how the sharpness of the initial potential dictates the nature of edge fluctuations. The work serves as a follow-up to previous studies on return probabilities, extending the analysis to full density profiles and higher dimensions.

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