Energy Dynamics of a Nonequilibrium Unitary Fermi Gas

By periodically modulating the trapping potential of a spherical unitary Fermi gas to excite a dissipationless breathing mode via SO(2,1) symmetry, researchers precisely measured the system's nonequilibrium energy evolution, revealing that trapping and internal energies oscillate out of phase and are governed by the dynamic virial theorem rather than equilibrium predictions.

The Gist

Imagine a group of tiny, invisible dancers (atoms) trapped inside a perfectly round, invisible ballroom. These dancers are special: they interact with each other so strongly that they act like a single, unified fluid. In physics, this is called a "unitary Fermi gas."

Usually, if you try to shake this ballroom to get the dancers moving, the friction between them would quickly turn that energy into heat, and they would settle down into a calm, sleepy state. This is what happens in most systems: you add energy, and it dissipates (spreads out and disappears) until things go back to normal.

The Magic Trick: A Perfectly Elastic Bounce
The researchers in this paper discovered a way to shake this ballroom without any friction. Because of a special mathematical symmetry in the system (called SO(2,1) symmetry), the "friction" that usually stops the motion vanishes.

Think of it like pushing a child on a swing. In a normal playground, air resistance and chain friction eventually stop the swing. But in this experiment, the swing is in a vacuum with no friction. If you push it at just the right rhythm, it keeps swinging higher and higher, or in this case, the whole cloud of atoms expands and contracts (breathes) forever without losing energy.

The Experiment: Shaking the Trap
The scientists used a laser "trap" to hold these atoms. They then rhythmically squeezed and relaxed this trap (like squeezing a stress ball) to pump energy into the system.

• The Result: Instead of the atoms getting hot and settling down, the entire cloud started "breathing"—expanding and contracting in a perfect, rhythmic dance.
• The Measurement: Because this breathing motion lasts for a very long time without fading, the scientists could use it as a perfect ruler to measure exactly how much energy they had pumped into the system. It's like being able to measure the speed of a car by watching how high a perfectly bouncy ball on its roof jumps, knowing the ball never stops bouncing.

What They Found

1. Energy Swapping: As they kept shaking the trap, they noticed two types of energy: the energy of the "walls" holding the atoms (trapping potential) and the energy of the atoms moving inside (internal energy). These two energies were like a seesaw. When the wall energy went up, the internal energy went down, and vice versa. They were perfectly out of sync, oscillating like two people on a seesaw.
2. The "Too Hard" Shake: When they shook the trap too violently (large amplitude), the perfect rhythm broke. Why? Because the laser trap isn't a perfect, smooth bowl; it gets a little bumpy at the edges (anharmonicity). When the atoms got too big, they hit these bumps, and the energy injection became less efficient. It's like trying to push a swing when the chains are getting tangled; the motion becomes messy and less efficient.
3. The Rules of the Game: The scientists compared their results to a set of rules called the "dynamic virial theorem." For normal, calm systems, there is a rule about how energy balances. But for this shaking, non-equilibrium system, the old rules didn't apply. Instead, a new, time-dependent rule predicted exactly what they saw. The experiment matched the new rule perfectly.

Why It Matters
This work is like learning how to keep a pot of soup boiling without ever turning off the stove or letting the heat escape. By understanding how to inject energy into a system without it leaking away, the scientists have created a long-lasting, non-equilibrium state. This gives them a clear window to watch how energy moves and rearranges itself in a quantum world, something that is usually too fast or too messy to see.

In short, they found a way to make a quantum gas "breathe" forever, allowing them to measure exactly how energy flows in a system that never settles down.

Technical Summary

Technical Summary: Energy Dynamics of a Nonequilibrium Unitary Fermi Gas

Problem Statement
The investigation of nonequilibrium dynamics in strongly interacting quantum systems is a fundamental challenge in quantum many-body physics. While ultracold atoms provide a versatile platform for studying such dynamics, a significant obstacle remains: in most scenarios, energy injection is accompanied by dissipation, which drives the system back to equilibrium rapidly, obscuring the transient energy evolution. Furthermore, while the static virial theorem accurately describes equilibrium systems (where total energy equals twice the trapping potential energy), the energy dynamics of nonequilibrium systems are governed by the time-dependent dynamic virial theorem. However, experimental observation of these energy dynamics in a long-lived nonequilibrium state has been lacking. Specifically, there is a need to precisely measure how energy is injected, redistributed between internal and trapping potential components, and evolves over time without the confounding effects of dissipation.

Methodology
The authors utilize a spherically trapped unitary Fermi gas of $^{6}$Li atoms as a model system. The experiment exploits the SO(2,1) dynamical symmetry inherent to unitary Fermi gases in isotropic harmonic traps, which suppresses bulk viscosity and allows for undamped breathing modes.

The experimental protocol involves:

1. System Preparation: A degenerate Fermi gas ($N \approx 9.6 \times 10^4$, $T/T_F = 0.24$) is prepared at the Feshbach resonance ($B = 832.2$ G) in a spherical trap with frequency $\omega_0 = 2\pi \times 720$ Hz.
2. Energy Injection: The trapping potential is modulated isotropically at a frequency of $2\omega_0$ for a duration $t_1$. The modulation follows $\omega^2(t) = \omega_0^2 [1 + \beta \sin(2\omega_0 t)]$, where $\beta$ is the amplitude. This drives the system away from equilibrium without inducing dissipation due to the SO(2,1) symmetry.
3. Measurement via Breathing Mode: After modulation ($t > t_1$), the trap frequency is held constant at $\omega_1$. The system exhibits a long-lived, undamped breathing oscillation. The mean square cloud size $\langle r^2 \rangle(t)$ is measured after a 1 ms time-of-flight (TOF) expansion.
4. Energy Extraction: The center position of the fitted breathing oscillation is directly related to the total energy of the nonequilibrium system via the relation $\langle r^2 \rangle_1 = E/m\omega_1^2$ (corrected for TOF expansion).
5. Component Analysis: The total energy is decomposed into trapping potential energy ($E_{ho}$) and internal energy ($E_{int}$). $E_{int}$ is extracted by measuring the expansion velocity of the cloud after switching off the trap, while $E_{ho}$ is deduced from the in-situ cloud size.
6. Theoretical Framework: The dynamics are analyzed using the dynamic virial theorem, $E(t) = 2E_{ho}(t) + \frac{1}{4}\frac{d^2I(t)}{dt^2}$, and scaling relations derived from hydrodynamic theory. For large modulation amplitudes, the effects of trap anharmonicity are incorporated into the theoretical model.

Key Contributions and Results

• Precision Measurement of Nonequilibrium Energy: The authors successfully demonstrate a method to precisely measure the energy evolution of a nonequilibrium system by utilizing the long-lived breathing mode excited by the SO(2,1) symmetry. This avoids the rapid equilibration seen in dissipative systems.
• Validation of the Dynamic Virial Theorem: The measured energy evolution $E(t)$ during the modulation process agrees quantitatively with the predictions of the dynamic virial theorem. This contrasts with equilibrium systems where the static virial theorem applies, confirming that nonequilibrium energy is governed by time-dependent collective motions.
• Energy Redistribution and Phase Opposition: The study reveals that the trapping potential energy ($E_{ho}$) and internal energy ($E_{int}$) both increase with modulation time but oscillate nearly $180^\circ$ out of phase. This indicates a continuous conversion between potential and internal energy during the modulation, a behavior derived analytically from linearized scaling equations.
• Impact of Trap Anharmonicity: At large modulation amplitudes ($\beta = 0.1$), the energy injection efficiency is significantly reduced compared to harmonic trap predictions. The authors attribute this to trap anharmonicity, which breaks the SO(2,1) symmetry, introduces damping, and detunes the modulation from the resonant frequency. The experimental data aligns with theoretical calculations that include fourth-order anharmonic corrections to the trapping potential.
• Measurement Accuracy: The study highlights that measuring energy via the long-lived breathing oscillation yields significantly higher accuracy and lower fluctuation compared to summing separately measured $E_{ho}$ and $E_{int}$ components, which are subject to larger experimental noise.

Significance
The paper claims to provide a novel method for exploring the energy evolution of nonequilibrium quantum gases. By establishing a platform for dissipation-free energy injection and the production of long-lived nonequilibrium states, the work offers direct experimental insight into energy injection and redistribution mechanisms. The agreement with the dynamic virial theorem validates the theoretical framework for strongly interacting systems out of equilibrium. Furthermore, the clarification of trap anharmonicity effects at large amplitudes is crucial for future high-precision studies. The authors suggest that this approach paves the way for investigating nonequilibrium thermodynamics and could be extended to probe energy in systems with finite interactions (BEC-BCS crossover) or under interaction modulation and quench dynamics.