Spectral form factor and power spectrum for trapped interacting rotating bosons: Crossover from integrability to quantum chaos

This study utilizes exact diagonalization of spectral form factors and power spectra to demonstrate how trapped interacting rotating bosons transition from integrable or pseudo-integrable behavior in moderate interaction regimes to strong quantum chaos consistent with the Gaussian orthogonal ensemble in strong interaction regimes, driven by the interplay between interaction strength and rotational angular momentum.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: A Dance Floor of Quantum Particles

Imagine a crowded dance floor filled with thousands of identical dancers (these are bosons). In a normal, calm room, they all move in perfect unison, holding hands and stepping to the exact same beat. This is a state called Bose-Einstein Condensation. Because they are all doing the exact same thing, the system is predictable, orderly, and "boring" from a physics perspective. We call this Integrable (or orderly).

Now, imagine two things happen to this dance floor:

1. The music gets louder and more chaotic (Strong Interaction): The dancers start bumping into each other, pushing, and shoving. They can't stay in perfect sync anymore.
2. The floor starts spinning (Rotation): The whole room begins to rotate, creating whirlpools or "vortices" where the dancers spin around a center point.

The researchers in this paper wanted to know: If we push these dancers hard enough and spin the room fast enough, does the dance floor turn into a chaotic, unpredictable mess? Or does it stay somewhat orderly?

To answer this, they used two special "cameras" to watch the dance:

1. The Spectral Form Factor (SFF): Think of this as a heartbeat monitor for the energy of the system. It tells us how the dancers' energy levels relate to one another over time.
2. The Power Spectrum: Think of this as a sound analyzer. It listens to the "noise" of the system to see if the rhythm is regular (like a metronome) or chaotic (like jazz improvisation).

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The Three Scenarios They Tested

The researchers looked at three different situations to see how the "dance" changed:

1. The Calm Room (Moderate Interaction, No Spin)

• What happened: The dancers bumped into each other a little bit, but the room wasn't spinning.
• The Result: The dancers mostly stayed in their perfect line. The "heartbeat monitor" (SFF) showed a flat line with no rhythm.
• The Meaning: The system remained Orderly (Integrable). The dancers were still too synchronized to create chaos.

2. The Spinning Room (Moderate Interaction, One Vortex)

• What happened: The room started spinning, creating a single whirlpool (a vortex).
• The Result: The dancers near the center started to get a little dizzy and drift out of line. The heartbeat monitor showed a tiny, weak rhythm.
• The Meaning: The system became Semi-Orderly (Pseudo-Integrable). It wasn't fully chaotic yet, but the perfect order was breaking down. It was like a dance where everyone is mostly in sync, but a few people are stumbling.

3. The Wild Party (Strong Interaction + Rotation)

• What happened: The music was loud (strong pushing), and the room was spinning fast, creating multiple whirlpools (multi-vortex states).
• The Result: The dancers were shoved out of their lines and scattered all over the floor. The heartbeat monitor showed a strong, steady, rising rhythm (a "linear ramp"). The sound analyzer showed a chaotic, jazz-like noise.
• The Meaning: The system became Fully Chaotic. The dancers were no longer predictable. Their movements followed the rules of Quantum Chaos, similar to how random numbers behave.

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The Key Discovery: The "Depletion" Effect

The most important finding of the paper is why the chaos happened.

In the calm state, almost all dancers were in the "main group" (the condensate).

• Interaction (bumping) pushed some dancers out of the main group.
• Rotation (spinning) created whirlpools that pulled even more dancers out of the main group.

The paper shows that chaos is born when the "main group" empties out. As long as most dancers are holding hands in a perfect line, the system is orderly. But as soon as the interactions and rotation force enough dancers to leave that line and scatter into the crowd, the system becomes chaotic.

The "Ramp" and the "Plateau"

To visualize their data, imagine a graph:

• The Dip: The moment the dancers first realize the music is changing.
• The Ramp: A steady slope going up. In a chaotic system, this ramp is long and clear. It means the dancers are interacting in a complex, unpredictable way.
• The Plateau: The flat top where the system settles.

The paper's conclusion:

• No Spin + Weak Push: No ramp. (Orderly)
• Spin + Weak Push: A tiny ramp. (Semi-orderly)
• Spin + Strong Push: A huge, clear ramp. (Chaotic)

Why Does This Matter?

This isn't just about dancing atoms. Understanding how a system moves from Order to Chaos helps scientists understand:

• Black Holes: How information gets scrambled inside them.
• Quantum Computers: How to keep them stable or how they might fail.
• Thermalization: How things heat up and reach equilibrium.

In short, the paper proves that if you take a quantum system, make the particles push each other hard, and spin the whole thing, you can force it to turn from a predictable machine into a chaotic, random mess. The "Spectral Form Factor" is the tool that lets us see that transition happening in real-time.

Technical Summary

Here is a detailed technical summary of the paper "Spectral form factor and power spectrum for trapped interacting rotating bosons: Crossover from integrability to quantum chaos" by Mohd Talib and M. A. H. Ahsan.

1. Problem Statement

The paper investigates the transition from integrability to quantum chaos in a system of harmonically trapped, interacting bosons subjected to external rotation. While the effects of interactions and rotation on superfluidity and vortex formation are well-studied, their specific role in driving the spectral statistics of the system from regular (integrable) to chaotic behavior remains under-explored.

The authors aim to characterize this crossover using two primary diagnostic tools:

• Spectral Form Factor (SFF): To analyze spectral correlations and level repulsion.
• Power Spectrum: To quantify fluctuations in energy level spacings.

The study focuses on how Bose-Einstein Condensation (BEC) depletion—caused by strong interparticle interactions and the nucleation of vortices—alters the system's statistical properties, potentially driving it toward the Gaussian Orthogonal Ensemble (GOE) universality class.

2. Methodology

Physical Model

• System: $N$ interacting bosons confined in a quasi-two-dimensional harmonic trap (cylindrically symmetric).
• Hamiltonian: The effective Hamiltonian in the laboratory frame includes kinetic energy, harmonic trap potential, and a Gaussian two-body interaction potential.
  • The system is rotated with angular velocity $\Omega$ about the $z$-axis.
  • The rotating frame Hamiltonian is defined as $\hat{H}_{rot} = \hat{H}_{lab} - \Omega \hat{L}_{z}^{lab}$.
• Parameters:
  • Particle Number ($N$): 12, 16, and 20 bosons (specifically $^{87}$Rb).
  • Interaction Regimes:
    • Moderate: $g_2 = 0.3669$ (interaction energy $\ll$ trap energy).
    • Strong: $g_2 = 3.669$ (interaction energy $\approx$ trap energy).
  • Angular Momentum States: Non-rotating ($L_z=0$), Single-vortex ($L_z=N$), and Multi-vortex states ($L_z=2N, 3N$).
• Numerical Approach:
  • Exact Diagonalization: The authors use the Davidson algorithm to obtain the lowest 100 many-body energy eigenvalues ($M=100$).
  • Unfolding: Energy levels are unfolded to remove global density variations, isolating local spectral fluctuations.

Diagnostic Tools

1. Spectral Form Factor (SFF):
   • Defined as $K(\tau) = \langle |Z(i\tau)|^2 \rangle / |Z(0)|^2$.
   • Analyzed for characteristic features: Dip (transition from short-time dynamics), Ramp (linear increase indicating level repulsion/chaos), and Plateau (saturation due to finite spectrum size).
   • A moving average in a logarithmic time window is applied to smooth fluctuations.
2. Power Spectrum ($P_k$):
   • Calculated from the Fourier transform of the cumulative number of levels (unfolded spacings).
   • Scaling Laws:
     • Integrable systems: $P_k \propto 1/k^2$ ($\alpha \approx 2$).
     • Chaotic systems (GOE): $P_k \propto 1/k$ ($\alpha \approx 1$).
     • Pseudo-integrable systems: $1 < \alpha < 2$.

3. Key Results

A. Moderate Interaction Regime

• Non-Rotating Case ($L_z=0$):
  • SFF: Exhibits a "dip-plateau" structure with no linear ramp.
  • Interpretation: The system remains integrable. This is due to the macroscopic occupation of a single-particle state (BEC), which suppresses spectral correlations.
  • Power Spectrum: Exponent $\alpha \approx 2$, confirming Poissonian statistics.
• Single-Vortex State ($L_z=N$):
  • SFF: Shows a discernible linear ramp, though the span decreases as $N$ increases.
  • Interpretation: The system enters a pseudo-integrable regime. The formation of a single vortex causes slight depletion of the condensate, introducing weak level repulsion but not full chaos.
  • Power Spectrum: Exponent $1 < \alpha < 2$(e.g.,$\alpha \approx 1.6$), consistent with pseudo-integrability.

B. Strong Interaction Regime

• Non-Rotating Case ($L_z=0$):
  • SFF: A small linear ramp emerges, indicating a crossover to pseudo-integrability.
  • Mechanism: Strong interactions scatter bosons out of the macroscopically occupied state into incoherent states, creating weak spectral correlations.
  • Power Spectrum: $1 < \alpha < 2$.
• Single-Vortex State ($L_z=N$):
  • SFF: Exhibits a pronounced linear ramp with a significant span.
  • Interpretation: The system transitions to strong quantum chaos (GOE). The combination of strong interaction and vortex nucleation significantly depletes the condensate, transferring bosons to incoherent angular momentum states.
  • Power Spectrum: Exponent $\alpha \approx 1$, confirming GOE statistics.
• Multi-Vortex States ($L_z=2N, 3N$):
  • SFF: Shows the largest linear ramp span, indicating the strongest spectral rigidity and level repulsion.
  • Interpretation: The synergistic effect of strong interactions and the nucleation of multiple vortices drives the system firmly into the chaotic regime.
  • Power Spectrum: $\alpha \approx 1$ across all $N$, confirming robust chaos.

C. Dependence on Particle Number ($N$)

• In the moderate regime, increasing $N$ reduces the linear ramp span in the single-vortex state, suggesting a tendency back toward integrability (due to stronger BEC stability).
• In the strong regime with vortices, the linear ramp persists and remains robust as $N$ increases, suggesting that the chaotic behavior is intrinsic and likely survives in the thermodynamic limit.

4. Key Contributions

1. Mapping the Crossover: The study provides a clear map of how varying interaction strength and angular momentum (vortex number) drives a trapped bosonic system from integrable $\to$ pseudo-integrable $\to$ chaotic.
2. Role of Condensate Depletion: It establishes a direct physical link between condensate depletion (caused by interactions and rotation) and the emergence of quantum chaos. The loss of macroscopic occupation of a single state is the mechanism enabling spectral correlations.
3. Validation of Tools: The paper demonstrates the consistency between SFF (time-domain) and Power Spectrum (frequency-domain) analyses in diagnosing quantum chaos in many-body bosonic systems.
4. Experimental Relevance: The parameters used (e.g., tunable interaction via Feshbach resonance, vortex generation via laser) align with current ultracold atom experimental capabilities, suggesting these chaotic signatures are observable.

5. Significance

• Fundamental Physics: The work bridges the gap between mean-field descriptions of BECs (which often assume integrability) and the complex many-body quantum chaos regime. It highlights that even in systems with high symmetry, strong interactions and rotation can induce universal chaotic behavior.
• Thermalization: By identifying the conditions for chaos (GOE statistics), the paper supports the Eigenstate Thermalization Hypothesis (ETH) in these systems, implying that such rotating, strongly interacting bosons will thermalize rapidly.
• Future Directions: The authors suggest extending this work using dynamical tools like Out-of-Time-Order Correlators (OTOCs) to further explore the connection between spectral chaos and information scrambling in these systems.

In summary, the paper concludes that interparticle interaction and vortex nucleation act as control parameters that deplete the Bose-Einstein condensate, thereby destroying integrability and driving the system toward a chaotic state characterized by GOE statistics.