Quench induced collective excitations: from breathing to acoustic modes

This paper investigates interaction-quench-induced collective excitations in harmonically trapped two-dimensional Bose-Einstein condensates using numerical and analytical methods, revealing a breakdown of scale invariance at low energies and characterizing trap-associated acoustic modes at high energies to provide an experimentally accessible spectroscopy of these many-body states.

The Gist

Imagine a crowd of people (atoms) all holding hands and moving in perfect unison inside a giant, invisible bowl. This is a Bose-Einstein Condensate (BEC), a state of matter where thousands of atoms act like a single "super-atom."

This paper is about what happens when you suddenly shout at this crowd. In physics terms, this "shout" is called a quench. It's an abrupt change in how strongly the atoms interact with each other (like suddenly making everyone hold hands tighter or looser).

The researchers wanted to see how the crowd reacts to this sudden change. They found that the crowd doesn't just react in one way; it depends on how you look at the reaction. They discovered two distinct "languages" the crowd speaks, depending on whether you are looking at the big picture or zooming in close.

Here is the breakdown of their findings using simple analogies:

1. The Two Regimes: The "Breathing" vs. The "Sound"

The researchers looked at the crowd's reaction in two different ways:

• Low Energy (The Big Picture / The "Breathing" Mode):
  Imagine the whole crowd in the bowl expanding and contracting together, like a giant lung breathing in and out.

  • The Old Theory: Scientists used to think this "breathing" followed a very strict, perfect rhythm based on a mathematical symmetry (like a clock ticking perfectly). They thought the rhythm would always be a simple multiple of the bowl's size.
  • The New Discovery: The researchers found that if you look closely at the edges of the crowd or if the "shout" (quench) is very loud, that perfect rhythm breaks. The breathing gets messy. Instead of the perfect clock, the crowd starts behaving like a fluid (like water in a bucket). The rhythm changes to match the rules of hydrodynamics (how fluids flow) rather than the perfect mathematical symmetry.
  • The Takeaway: The "perfect symmetry" only works if you ignore the messy details of the edges. In the real world, the edges matter, and the rhythm changes.

• High Energy (The Zoomed-In View / The "Sound" Mode):
  Now, imagine zooming in on a tiny patch of the crowd. Instead of the whole group breathing, you see individual ripples or sound waves traveling through the people.

  • The Problem: Previous theories assumed the crowd was spread out evenly everywhere (like a flat, infinite ocean). But in reality, the crowd is squished into a bowl, so it's denser in the middle and thinner at the edges.
  • The New Discovery: The researchers realized that to predict the speed of these sound waves correctly, you have to account for the "bowl" (the trap). They created a new formula that acts like a renormalized chemical potential.
  • The Analogy: Think of it like driving a car. If you assume the road is perfectly flat (the old theory), you calculate your speed one way. But if the road has a hill in the middle (the trap), your speed changes. The researchers figured out exactly how to adjust the speed calculation to account for that "hill," fixing a discrepancy between what experiments saw and what old theories predicted.

2. Why Do the Waves Fade Away? (The "Leaky Bucket")

One of the most interesting findings is that these high-energy sound waves don't last forever; they die out (decay).

• The Analogy: Imagine you throw a stone into a pond. The ripples travel outward. But if the pond is actually a small, shaped bowl, the ripples eventually hit the edge, scatter, and lose their energy.
• The Physics: Because the atoms are trapped in a bowl, they aren't in a perfect, infinite space. The "sound waves" eventually run out of the dense crowd and hit the empty space at the edge of the trap. Once they leave the main group, the wave disappears. The researchers calculated exactly how long these waves last based on how fast they are moving and how big the crowd is.

3. The "Spectroscopy" Aspect

The paper concludes that by listening to these different rhythms (the breathing and the sound waves) and measuring how long they last, scientists can use the BEC as a spectroscope.

• The Analogy: Just as a musician can tell what instrument is playing by the sound it makes, physicists can "listen" to the collective excitations of the atoms to understand the hidden properties of the quantum state. If the rhythm is off, it tells them about the interactions; if the wave dies quickly, it tells them about the trap's shape.

Summary

In short, this paper is a guidebook for understanding how a quantum crowd reacts when you suddenly change the rules of the game.

1. At low energy: The crowd's "breathing" is messier than we thought because the edges of the trap break the perfect symmetry.
2. At high energy: The "sound waves" travel at speeds that depend heavily on the shape of the trap, not just the atoms themselves.
3. The result: By fixing the math to include the "bowl" shape, the researchers finally made the theory match the real-world experiments perfectly.

It's a bit like realizing that to predict how a drum sounds, you can't just look at the drumhead; you have to understand the shape of the drum shell holding it, too.

Technical Summary

1. Problem Statement

The paper addresses the dynamics of quantum many-body systems far from equilibrium, specifically focusing on Bose-Einstein Condensates (BECs) confined in harmonic traps. While theoretical models often assume idealized, homogeneous systems with contact interactions, real experiments involve:

• Intrinsic inhomogeneity due to trapping potentials.
• Finite interaction ranges (especially near Feshbach resonances) that break strict scale invariance.
• Dimensional constraints (2D systems) where compressibility and viscosity change near the BKT transition.

The authors aim to bridge the gap between idealized theories (which predict specific symmetries like SO(2,1) conformal symmetry) and experimental realities by studying interaction quenches (abrupt changes in interaction strength $g_0 \to g_1$) in a 2D harmonically trapped BEC. They specifically investigate the breakdown of scale invariance at low energies and the role of trap effects on high-energy acoustic modes.

2. Methodology

The study employs a dual approach combining numerical simulations and analytical derivations:

• Numerical Model:

  • The system is modeled using the Gross-Pitaevskii (GP) equation in a harmonic trap.
  • Quantum fluctuations are simulated by adding noise seeds $\chi(r, t)$ to the wavefunction.
  • The system is studied in the Thomas-Fermi regime ($\mu/\hbar\omega_0 \gg 1$).
  • Interaction Quench: The interaction strength is changed from $g_0$ to $g_1 = g_0 g_x$.
  • Analysis: The authors perform Fast Fourier Transforms (FFT) on particle density $\rho(r, t)$ and wavefunction perturbations $\delta\psi(r, t)$ to extract excitation frequencies and spectral distributions.

• Analytical Framework:

  • Low Energy: Comparison between Stringari's hydrodynamic theory and Pitaevskii-Rosch's SO(2,1) conformal symmetry predictions.
  • High Energy: Linearization of the GP equation to derive a modified Bogoliubov dispersion relation that accounts for the trap.
  • Structure Factor: Derivation of the dynamic structure factor $S_q(k, t)$ post-quench to connect theory with experimental observables.

3. Key Contributions and Results

A. Low-Lying Excitations: Breakdown of Scale Invariance

• The Conflict: Idealized 2D BECs with contact interactions possess SO(2,1) conformal symmetry, predicting breathing modes at frequencies $\omega_n = 2n\omega_0$ (even integers). Hydrodynamic theory (Stringari) predicts a different spectrum: $\omega_n = \omega_0 \sqrt{2n^2 + 2nl + 2n + l}$.
• The Finding: The authors demonstrate that scale invariance breaks down at short observational length scales (due to finite cut-offs inherent in simulations and experiments).
  • At short length scales (small $r$) or weak quenches, the system exhibits a hybrid spectrum. While the dominant modes follow the hydrodynamic prediction (Eq. 2), additional modes appear.
  • At large length scales or strong quenches ($g_x$ large), the system effectively restores conformal symmetry, and the even-integer modes ($\omega_n \approx 2n\omega_0$) become dominant.
• Significance: This explains why experiments sometimes observe deviations from the pure conformal symmetry predictions, attributing it to the interplay between observation scale and interaction strength.

B. High-Energy Excitations: Trap-Modified Acoustic Modes

• The Problem: Previous theoretical models for high-momentum ($k$) excitations assumed a homogeneous density, failing to match experimental data (specifically regarding the "quench-up" scenario in Ref. [20]).
• The Solution: The authors derive a renormalized global chemical potential, $\mu_{\text{eff}}$, which averages the trapping potential over the condensate density:
  $$\mu_{\text{eff}} \approx \frac{2}{3}\mu$$
• The Result:
  • The high-$k$ excitations follow a modified Bogoliubov dispersion relation:
    $$\omega_k = \sqrt{\frac{\mu_{\text{eff}}}{m}k^2 + \frac{\hbar^2}{4m^2}k^4}$$
  • This modified theory successfully reconciles numerical simulations with experimental data for both quench-up and quench-down scenarios, resolving discrepancies where standard homogeneous theory failed (particularly for quench-up).
• Finite Lifetime: The study explains the observed decay of high-$k$ oscillations. Because the trap breaks translation invariance, momentum $k$ is not a perfect quantum number. An excited mode disperses into other eigenmodes and propagates out of the condensate region.
  • The lifetime is estimated as $T_s = r_0 / v_k$, where $v_k$ is the group velocity. Numerical results confirm this scaling ($T_s \propto r_0$ and $T_s \propto v_k^{-1}$).

4. Significance

• Experimental Relevance: The paper provides a comprehensive framework for interpreting "interaction quench" experiments in 2D BECs. It clarifies that the observed collective modes are not purely conformal or purely hydrodynamic but depend on the observation scale and quench strength.
• Resolution of Discrepancies: By introducing the renormalized chemical potential $\mu_{\text{eff}}$, the authors resolve long-standing discrepancies between theory and experiment regarding high-energy acoustic modes in trapped systems.
• Spectroscopy Tool: The analysis of frequencies and damping rates serves as a built-in spectroscopy tool for characterizing many-body states, offering insights into the equation of state and the nature of interactions in non-equilibrium regimes.
• Methodological Advance: The work demonstrates how to rigorously incorporate trap effects and finite-size cut-offs into the study of scale-invariant systems, moving beyond idealized homogeneous approximations.

In conclusion, this paper establishes that realistic trapped BECs exhibit a hybrid collective mode structure at low energies and require trap-modified dispersion relations at high energies, providing a robust theoretical foundation for future non-equilibrium quantum simulations.