Comparative Study of Indicators of Chaos in the Closed and Open Dicke Model

This paper provides a systematic comparative study of chaos indicators in closed and open Dicke models, revealing that spectral form factors can exhibit chaotic signatures even in the regular closed regime while demonstrating that the dissipative spectral form factor robustly identifies the concurrence of the superradiant phase transition and a shift to Ginibre Unitary Ensemble statistics in the open system.

The Gist

Imagine you are a detective trying to figure out if a complex machine is running smoothly (regular) or if it's about to break down into total chaos. In the world of quantum physics, this "machine" is a system of atoms interacting with light inside a box (a cavity). This specific setup is called the Dicke Model.

The scientists in this paper, Prasad Pawar and his team, set out to test different "chaos detectors" to see which ones work best for this machine, both when it's perfectly sealed (closed) and when it's leaking energy (open).

Here is the story of their investigation, explained simply:

1. The Two Worlds: Sealed vs. Leaking

• The Closed Model (The Sealed Box): Imagine a perfect, soundproof room where atoms and light bounce around forever without losing energy. The scientists want to know: Is the movement of these particles predictable (like a clock) or chaotic (like a pinball machine)?
• The Open Model (The Leaky Box): Now, imagine the room has a hole in the wall. Light leaks out, and the system loses energy. This is more realistic but much harder to analyze because the rules change.

2. The Detective Tools (Indicators of Chaos)

To solve the mystery, the team used three main tools. Think of these as different ways to listen to the machine's "heartbeat."

• Tool A: The Neighbor Check (NNSD)

  • The Analogy: Imagine a line of people standing in a hallway. In a regular system, people stand randomly; sometimes they are right next to each other, sometimes far apart (like a Poisson distribution). In a chaotic system, people are shy and avoid standing too close to their neighbors; they push each other away (like a Wigner-Dyson distribution).
  • The Finding: This tool worked perfectly. It clearly showed when the system switched from "shy neighbors" (chaos) to "random neighbors" (order).

• Tool B: The Long-Range Echo (Spectral Form Factor - SFF)

  • The Analogy: This is like shouting into a canyon and listening to the echo. In a chaotic system, the echo has a very specific shape: it starts low, rises up, and then flattens out. This is called the "Dip-Ramp-Plateau." It's the gold standard for detecting chaos.
  • The Surprise: The team found a trap! Even when the machine was running smoothly (in the regular zone), this tool still showed the "Dip-Ramp-Plateau" shape, but only if the machine wasn't infinitely large. It was a "false alarm." The tool was picking up long-range echoes that shouldn't be there in a simple system.
  • The Lesson: You can't just look at this shape and say "It's chaos!" You have to be careful. If the machine is small, this tool lies to you.

• Tool C: The Leaky Echo (Dissipative SFF - DSFF)

  • The Analogy: This is the "Leaky Box" version of the echo test.
  • The Finding: This tool was a hero. In the open, leaking system, it clearly distinguished between the smooth and chaotic states. When the system became chaotic (superradiant), the DSFF showed a beautiful, smooth "Dip-Ramp-Plateau" that matched the theoretical prediction for chaos. When it was smooth, the shape was totally different.

3. The Big Discovery: The "False Alarm"

The most interesting part of the paper is the warning about Tool B (SFF).

The scientists realized that for the "sealed" system, the "Dip-Ramp-Plateau" shape (which usually means chaos) appears even when the system is not chaotic, as long as the system isn't infinitely huge. It's like hearing a faint echo in a quiet room and thinking someone is shouting, when it's actually just the wind.

They found that you have to make the system massive (approaching an infinite number of atoms) to get rid of this false echo. Until then, the "regular" system looks suspiciously like a "chaotic" one to this specific tool.

4. The Open System Mystery

In the "leaky" system, they found that the moment the system starts leaking light (the phase transition) happens at the exact same time the system's internal statistics switch from "random" to "chaotic." It's like the machine starts leaking the exact moment it starts spinning out of control. The DSFF tool was the only one that could clearly see this switch happening.

Summary for the General Public

Think of the Dicke Model as a giant orchestra.

• Regular Mode: The musicians are playing a simple, predictable song.
• Chaotic Mode: The musicians are improvising wildly, creating a complex, unpredictable jazz jam.

The scientists tried different microphones to tell the difference.

1. Microphone 1 (Neighbor Check): Worked great. It heard the musicians pushing each other apart when they got chaotic.
2. Microphone 2 (The Echo): Made a mistake. It heard a "chaotic echo" even when the musicians were playing a simple song, unless the orchestra was absolutely massive.
3. Microphone 3 (The Leaky Echo): Worked perfectly for the open system. It clearly heard the moment the orchestra switched from a simple song to a wild jazz jam.

The Takeaway: When studying complex quantum systems, don't trust just one "chaos detector." Some tools can give you false alarms, especially if the system isn't infinitely large. The "Leaky Echo" (DSFF) seems to be the most reliable detective for open, realistic systems.

Technical Summary

1. Problem Statement

The Dicke model, describing the interaction between a bosonic field and an ensemble of two-level atoms, is a paradigmatic system for studying quantum phase transitions (QPT) and quantum chaos. It is well-established that the model undergoes a transition from a regular (normal) phase to a chaotic (superradiant) phase as the atom-field coupling strength ($g$) exceeds a critical value ($g_c$).

While the Nearest-Neighbor Spacing Distribution (NNSD) has been successfully used to characterize this transition (showing a shift from Poissonian statistics in the regular phase to Wigner-Dyson/GOE statistics in the chaotic phase), the reliability of dynamical indicators, specifically the Spectral Form Factor (SFF), remains under-explored.

• The Core Issue: It is generally assumed that a "dip-ramp-plateau" structure in the SFF (a signature of quantum chaos) only appears in chaotic regimes. However, this paper investigates whether this structure can appear in the regular regime of the Dicke model for finite system sizes, potentially leading to false positives in chaos detection.
• Open Systems: Furthermore, the behavior of chaos indicators in the open Dicke model (subject to cavity damping) is less understood. Specifically, the concurrence of the dissipative quantum phase transition and the transition in Liouvillian eigenvalue statistics (from 2D Poissonian to Ginibre Unitary Ensemble, GinUE) requires rigorous verification using dynamical tools like the Dissipative Spectral Form Factor (DSFF).

2. Methodology

The authors perform a systematic comparative study of static and dynamical chaos indicators for both closed and open Dicke models.

• Models:

  • Closed Dicke Model: Governed by the Hamiltonian $\hat{H}$. The system is analyzed in the even parity sector to reduce matrix size.
  • Open Dicke Model: Governed by a Lindblad master equation with cavity damping rate $\gamma$. The dynamics are determined by the Liouvillian super-operator $\mathcal{L}$.
  • Control: The Tavis-Cummings (TC) model (integrable cousin of the Dicke model) is used as a benchmark to rule out integrability as the cause of observed correlations.

• Indicators Analyzed:

  1. Static Indicators:
     • NNSD: Nearest-Neighbor Spacing Distribution (requires unfolding).
     • Level Spacing Ratios ($r_k$): Specifically the $k$-th order spacing ratio. Unlike NNSD, this does not require unfolding and is sensitive to long-range correlations for $k > 1$.
  2. Dynamical Indicators:
     • Spectral Form Factor (SFF): Defined as the Fourier transform of the two-point energy correlation function. The authors apply time-averaging (rectangular kernel) to mitigate oscillations.
     • Dissipative SFF (DSFF): Defined for open systems using the complex eigenvalues of the Liouvillian. The authors employ a specific unfolding and filtering procedure (based on a power-law transformation of eigenvalues) to ensure the correct universal ramp behavior for GinUE.
     • Dissipative Survival Probability (DSPF): Analyzed as an alternative dynamical indicator.

• Numerical Approach:

  • Exact diagonalization of the Hamiltonian and Liouvillian.
  • Convergence checks with respect to photon number cutoffs ($M$) and spin size ($j = N/2$).
  • Comparison against Random Matrix Theory (RMT) ensembles: GOE (Gaussian Orthogonal Ensemble) for closed chaotic systems and GinUE (Ginibre Unitary Ensemble) for open chaotic systems.

3. Key Contributions and Results

A. Closed Dicke Model: The Limitation of SFF

• NNSD and $r_1$: Confirm the expected transition from Poissonian (regular) to GOE (chaotic) statistics as $g$ crosses $g_c$.
• The SFF Anomaly: The authors discover that the SFF exhibits a dip-ramp-plateau structure even in the regular (normal) phase ($g < g_c$) for finite spin sizes ($j$).
  • This "correlation hole" appears because long-range spectral correlations persist in the regular regime for any finite $N$.
  • The structure only fully converges to the Poissonian prediction (no ramp) in the strict thermodynamic limit ($N \to \infty$).
• $k$-th Order Spacing Ratios: The study shows that higher-order spacing ratios ($r_{10}, r_{20}, r_{30}$) deviate from Poissonian predictions in the regular phase for finite $j$, confirming the presence of long-range correlations that the SFF detects.
• Conclusion on Closed Systems: The mere presence of a dip-ramp-plateau in the SFF is insufficient to declare a system chaotic. One must verify that the structure matches the specific slope and universality of the relevant RMT ensemble (GOE) and that the system is sufficiently close to the thermodynamic limit.

B. Open Dicke Model: DSFF as a Robust Indicator

• Static Statistics: The NNSD of the Liouvillian eigenvalues transitions from 2D Poissonian to GinUE as $g$ crosses the dissipative critical coupling $g_{c\gamma}$. The authors provide indirect evidence that this statistical transition coincides with the dissipative QPT.
• DSFF Performance:
  • In the superradiant (chaotic) regime ($g > g_{c\gamma}$), the DSFF displays a clear quadratic dip-ramp-plateau structure that aligns perfectly with GinUE predictions.
  • In the normal (regular) regime ($g < g_{c\gamma}$), the DSFF does not show the characteristic ramp structure. It differs significantly from both the 2D Poissonian and GinUE predictions, lacking the universal ramp.
• Significance: Unlike the SFF in the closed model, the DSFF is a faithful indicator for the open Dicke model. It clearly distinguishes between the regular and chaotic phases without the ambiguity seen in the closed case.

C. Role of Integrability

• By analyzing the Tavis-Cummings (TC) model, the authors demonstrate that the persistence of long-range correlations in the Dicke model's regular phase is not due to a lack of integrability (since the integrable TC model also shows similar SFF anomalies). This suggests the origin lies in the specific permutation invariance of the spin-cavity interaction.

4. Significance and Implications

1. Refining Chaos Detection: The paper challenges the common heuristic that a "correlation hole" in the SFF is a definitive signature of chaos. It highlights the danger of misinterpreting finite-size effects in the regular phase as chaotic behavior.
2. Robustness of DSFF: The study establishes the Dissipative Spectral Form Factor (DSFF), when properly unfolded, as a superior and reliable diagnostic tool for quantum chaos in open systems, capable of distinguishing phases where static indicators might be ambiguous.
3. Experimental Relevance: The authors discuss the feasibility of measuring these indicators in experimental setups involving ultracold atoms in optical cavities and superconducting qubits. They suggest that while preparing the specific Coherent Gibbs State (CGS) is difficult, related observables (like survival probability after quenches) could experimentally probe the correlation hole.
4. Theoretical Insight: The work provides a deeper understanding of the relationship between the Bohigas-Giannoni-Schmit (BGS) conjecture and open quantum systems, validating the extension of RMT concepts (GinUE) to dissipative dynamics.

Summary Conclusion

The paper concludes that while static indicators (NNSD) and low-order ratios are reliable, dynamical indicators like the SFF require careful interpretation regarding system size and long-range correlations. In contrast, the DSFF emerges as a robust, unambiguous indicator of the transition to chaos in open quantum systems, successfully capturing the concurrence of the dissipative phase transition and the emergence of GinUE statistics.