🌀 nonlinear sciences

Exact diagonalization study of energy level statistics in harmonically confined interacting bosons

This paper employs exact diagonalization to demonstrate that while harmonically confined interacting bosons exhibit regular (Poisson) or weakly chaotic statistics in moderate interaction regimes, they transition to strong chaotic behavior characterized by GOE distributions in strong interaction regimes, with rotation further enhancing this chaos.

Original authors: Mohd Talib, M. A. H. Ahsan

Published 2026-01-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohd Talib, M. A. H. Ahsan

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 inside a giant, invisible bowl. On this floor, we have groups of identical dancers (bosons) who are all trying to move to the same beat. The bowl is shaped like a flat pancake (a "quasi-2D plane"), and the music is a steady, rhythmic hum (the "harmonic trap").

The dancers have two main things influencing their moves:

The Bowl's Shape: The walls of the bowl push them toward the center. This is the "trap energy."
The Dancers' Personal Space: The dancers don't like to bump into each other. They have a repulsive force, like invisible bubble wrap, that pushes them apart when they get too close. This is the "interaction energy."

The scientists in this paper wanted to know: When these dancers move, is their pattern orderly and predictable, or is it chaotic and random?

To figure this out, they didn't just watch the dance; they looked at the "energy levels" (the specific steps or notes the dancers can take). They used a special mathematical toolkit to see if the gaps between these steps were random or if they followed a strict rule.

The Two Main Scenarios

The researchers tested two different "mood" settings for the dance floor:

1. The "Calm" Dance (Moderate Interaction)

The Setup: The dancers are polite. The force pushing them apart is weak compared to the force of the bowl keeping them in.
The Result (No Spinning): When the bowl isn't spinning, the dancers move in a very orderly, predictable way. Their steps follow a "Poisson distribution."
- Analogy: Imagine a line of people waiting for a bus. They stand at random intervals, but they don't care about each other. Sometimes two people stand close, sometimes far apart. There is no "level repulsion" (they don't actively avoid each other). This is a regular, non-chaotic system.
The Result (Spinning): If you start spinning the bowl slowly (creating a single vortex), the dancers get a little more jittery. They start to show signs of "weak chaos." They aren't fully random yet, but they aren't perfectly orderly either.

2. The "Wild" Dance (Strong Interaction)

The Setup: The dancers are very pushy. The force pushing them apart is now just as strong as the bowl's walls.
The Result (No Spinning): Suddenly, the dance floor becomes chaotic. The steps no longer look random; they look like a complex, chaotic system.
- Analogy: Now, the dancers are actively avoiding each other. If one person takes a step, the others immediately shift to avoid bumping into them. This is called "level repulsion." The pattern of steps now matches the "GOE distribution" (Gaussian Orthogonal Ensemble), which is the mathematical fingerprint of chaos.
The Result (Spinning): When you spin the bowl while the dancers are being pushy, the chaos goes into overdrive. The system becomes strongly chaotic.

The Twist: How Many Dancers?

The researchers also changed the number of dancers (12, 16, or 20).

In the Calm scenario, adding more dancers actually made the system more orderly (more like the random bus line).
In the Wild scenario, adding more dancers made the chaos fluctuate. Sometimes it got more chaotic, sometimes it settled back down a bit, but it generally stayed in the chaotic zone.

The "Spin" Factor

The paper found that rotation is the ultimate chaos amplifier.

Even when the dancers were just being moderately pushy, spinning the bowl made them act more chaotically.
When the dancers were already being very pushy, spinning the bowl made the chaos even stronger.
They even tested spinning the bowl very fast (creating 2 or 3 vortices). In these cases, the system was purely chaotic, regardless of how many dancers were on the floor.

The Tools They Used (Simplified)

To measure this, the scientists used four different "rulers":

Nearest-Neighbor Spacing (NNSD): Measuring the distance between one step and the very next one.
Ratio of Spacings: Comparing the distance between step A and B, to the distance between B and C. (This is a clever trick that avoids some mathematical errors).
Long-Range Rulers (Dyson-Mehta & Level Number Variance): These looked at the pattern over a long stretch of steps to see if the whole dance floor was rigid or flexible.

The Bottom Line

The paper concludes that the behavior of these trapped atoms is a tug-of-war between the trap (which wants order) and the interaction (which creates complexity).

Weak interaction + No spin = Orderly (Regular).
Strong interaction OR Spin = Chaotic.
Strong interaction + Spin = Maximum Chaos.

Essentially, the study shows that by simply changing how hard the particles push each other or how fast the system spins, you can switch a quantum system from being a predictable clockwork machine to a wild, chaotic storm. This helps scientists understand how "quantum chaos" emerges in the real world, specifically in ultra-cold gases like those made of Rubidium atoms.

Technical Summary: Exact Diagonalization Study of Energy Level Statistics in Harmonically Confined Interacting Bosons

Problem Statement
The paper investigates the statistical properties of energy spectra in quantum many-body systems, specifically focusing on harmonically confined interacting bosons in a quasi-2D plane. While Random Matrix Theory (RMT) provides a framework for distinguishing between regular (integrable) and chaotic (non-integrable) systems via the Bohigas-Giannoni-Schmit (BGS) and Berry-Tabor conjectures, the specific effects of varying two-body interaction strength, particle number ( $N$ ), and rotation on trapped interacting bosons had not been fully explored. Previous studies often utilized approximate methods or focused on specific limits. This work aims to fill that gap by performing an exact diagonalization study to understand how the interplay between interaction energy and trap energy, combined with rotation, drives the transition from regular to chaotic spectral behavior.

Methodology
The authors employ an exact diagonalization approach to solve the many-body Hamiltonian for $N = 12, 16,$ and $20$ bosons of $^{87}\text{Rb}$ .

System Model: The bosons are confined in a quasi-2D harmonic trap with an anisotropy parameter $\lambda_z = \omega_z/\omega_\perp = 4$ . The interaction is modeled by a repulsive Gaussian potential with a range $\sigma = 0.1 a_\perp$ . The system is analyzed in both non-rotating ( $L_z = 0$ ) and rotating frames (single-vortex state $L_z = N$ , and higher angular momenta $L_z = 2N, 3N$ ).
Interaction Regimes: Two distinct regimes are examined:
1. Moderate Interaction: Interaction energy is small compared to trap energy ( $g_2 = 0.3669$ , corresponding to $a_{sc} = 1000 a_0$ ).
2. Strong Interaction: Interaction energy is comparable to trap energy ( $g_2 = 3.669$ , corresponding to $a_{sc} = 10000 a_0$ ).
Statistical Tools: The study analyzes the lowest 100 energy levels using both short-range and long-range correlation measures:
- Short-Range: Nearest-Neighbor Spacing Distribution (NNSD) $P(s)$ and the distribution of the ratio of consecutive level spacings $P(r)$ . The authors emphasize $P(r)$ as it avoids the unfolding procedure required for $P(s)$ , thereby minimizing systematic errors. The Brody distribution is used to fit the data, with parameter $b$ interpolating between Poisson ( $b=0$ ) and Gaussian Orthogonal Ensemble (GOE, $b=1$ ).
- Long-Range: The Dyson-Mehta $\Delta_3(L)$ statistic and the level number variance $\Sigma^2(L)$ .

Key Contributions and Results

Non-Rotating Case ( $L_z = 0$ ):
- Moderate Interaction: The system exhibits regular behavior. The NNSD and $P(r)$ distributions align with the Poisson distribution, indicating a lack of level repulsion. The Brody parameter $b$ decreases systematically with increasing $N$ (from $0.33$ at $N=12$ to $0.04$ at $N=20$ ), suggesting that spectral correlations diminish as the number of bosons increases in this regime. Long-range statistics ( $\Delta_3$ and $\Sigma^2$ ) also follow Poisson behavior for small energy intervals.
- Strong Interaction: The system transitions to chaotic behavior. The distributions align with the GOE, signifying strong level repulsion. However, the degree of chaos is modulated by $N$ . For $N=16$ , the system shows the strongest chaotic signatures ( $b=0.85$ , $\langle r \rangle \approx 0.538$ ), while for $N=12$ and $N=20$ , the system reverts to a weakly chaotic regime with some regularity. The $\Delta_3(L)$ statistic saturates at significantly lower values in the strong regime compared to the moderate regime, confirming the transition to a non-integrable system.
Rotating Case (Single-Vortex State $L_z = N$ ):
- Moderate Interaction: Rotation induces a transition from regular to weakly chaotic behavior. The system exhibits signatures of chaos with a degree of regularity (Brody parameters $b \approx 0.52 - 0.64$ ). The mean gap ratio $\langle r \rangle$ is close to $0.5$, intermediate between Poisson and GOE.
- Strong Interaction: The system displays strong chaotic behavior consistent with GOE statistics ( $b \approx 1.0$ , $\langle r \rangle \approx 0.55$ ). The rotation is found to enhance the chaotic nature of the system compared to the non-rotating case in the same interaction regime.
Higher Angular Momenta ( $L_z = 2N, 3N$ ):
- In the strong interaction regime, systems with $L_z = 2N$ and $L_z = 3N$ exhibit strong signatures of quantum chaos, with both short-range ( $P(r)$ ) and long-range ( $\Delta_3$ ) statistics aligning with the GOE distribution.

Significance and Claims
The authors claim that their study provides a comprehensive understanding of how interaction strength, particle number, and rotation govern the spectral correlations in trapped interacting bosons.

Transition Mechanism: The interplay between interaction energy and trap energy governs the transition from regular (Poisson) to chaotic (GOE) behavior.
Role of Rotation: Rotation is identified as a factor that enhances chaotic behavior in both moderate and strong interaction regimes.
Methodological Robustness: The study validates the use of the ratio of consecutive level spacings $P(r)$ as a reliable tool for short-range correlation analysis, as it avoids the potential errors associated with the unfolding procedure required for NNSD.
Universality: The results support the universal applicability of Random Matrix Theory in describing spectral correlations in quantum many-body systems, specifically ultracold Bose systems.

The paper concludes modestly, suggesting that these findings advance the understanding of quantum chaos in ultracold Bose systems and noting that future work could extend this analysis to spectral form factors and extended Bose-Hubbard models with infinite-range interactions.