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arxiv: 2606.13072 · v1 · pith:I535VT4Inew · submitted 2026-06-11 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.stat-mech

Fibonacci Steady-States and Persistent Oscillations in an Ordered Multimode Dicke Model

Miriam J. Leonhardt , Kai M\"uller , Oliver Lunt , Andrew J. Daley This is my paper

Pith reviewed 2026-06-27 06:37 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.stat-mech
keywords Dicke modelsuperradiant phasesteady statesFibonacci scalingpersistent oscillationsmultimode cavitynearest-neighbor couplingultracold atoms
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The pith

In the ordered multimode Dicke model with nearest-neighbor couplings, the number of mean-field stable steady-states in the superradiant phase scales according to the Fibonacci sequence with the number of atomic clusters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies an ordered version of the multimode Dicke model in which n_c atomic clusters interact through n_c-1 cavity modes arranged in a nearest-neighbor chain. It establishes that the count of mean-field stable steady-states in the superradiant regime follows a Fibonacci recurrence as n_c grows, and that a fraction of those states sustain persistent oscillations instead of relaxing to fixed points. Both the truncated Wigner approximation and the numerically exact hierarchy of pure states confirm that the Fibonacci count and the oscillatory subset survive when the clusters have finite size. The geometry is experimentally accessible with current ultracold-atom setups in multimode cavities.

Core claim

For an ordered multimode Dicke model in which n_c atomic clusters are coupled by n_c-1 modes in nearest-neighbor geometry, the number of mean-field stable steady-states in the superradiant phase obeys Fibonacci scaling with n_c; a subset of these states exhibits persistent oscillations; both features remain present when the clusters are taken to finite size.

What carries the argument

The nearest-neighbor ordered coupling geometry with n_c clusters linked by n_c-1 modes, whose recursive structure produces the Fibonacci recurrence in the count of stable mean-field solutions.

If this is right

  • The Fibonacci scaling is a direct consequence of the linear chain topology of the couplings, allowing each additional cluster to extend the set of stable configurations recursively.
  • Persistent oscillations arise only for specific phase patterns permitted by the ordered geometry and are absent in fully disordered multimode Dicke models.
  • The same scaling and oscillatory states survive finite-size corrections, indicating they are not artifacts of the infinite-cluster mean-field limit.
  • The model can be realized with existing multimode cavity QED platforms, opening a route to observe the predicted states experimentally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar Fibonacci or other integer-sequence scaling may appear in other one-dimensional dissipative spin models whose couplings admit a recursive counting of fixed points.
  • The persistent oscillations could serve as a diagnostic for the underlying ordered geometry in future cavity experiments that vary the coupling range.
  • Extending the chain to include next-nearest-neighbor modes might replace the Fibonacci sequence with a different linear recurrence whose characteristic equation reflects the longer-range interactions.

Load-bearing premise

The mean-field equations together with the truncated Wigner and hierarchy-of-pure-states methods correctly identify and count all stable steady-states without missing additional instabilities from correlations beyond those approximations.

What would settle it

An exact numerical solution for n_c = 4 or 5 clusters that yields a number of stable steady-states different from the corresponding Fibonacci number or that shows no persistent oscillations in the predicted subset.

Figures

Figures reproduced from arXiv: 2606.13072 by Andrew J. Daley, Kai M\"uller, Miriam J. Leonhardt, Oliver Lunt.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic diagram of a confocal multimode cavity QED apparatus like those in [ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stable steady-state solutions shown through the total [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Example mean-field evolution of a four cluster system [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Example mean-field evolution of a three cluster sys [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the mean-field solutions (dashed black curve) with results obtained using TWA for finite-size clusters [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Variance of the arc-length of individual trajectories [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a-b) Comparison of TWA with HOPS for three clusters for oscillations in the normal phase for the outer and middle [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Stable steady-state solutions shown through the total [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Convergence checks for the HOPS simulations for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Coupling matrices for (a) the nearest-neighbor cou [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

Ultracold atoms in multimode optical cavities provide a rich testbed for many-body phenomena enabled by light-mediated interactions. Recent experiments include realizations of spin glasses and associative memories, as described by multimode Dicke models with disordered couplings. However, the properties of multimode Dicke models with ordered coupling geometries remain largely unexplored. In this work, we investigate the stable steady-states of the multimode Dicke model with an ordered nearest-neighbor coupling geometry, where $n_c$ atomic clusters are coupled via $n_c-1$ cavity modes. We show that the number of mean-field stable steady-states in the superradiant phase exhibits Fibonacci scaling with the number of atomic clusters, and that a subset of these steady-states exhibit persistent oscillations. Using both the truncated Wigner approximation and the numerically-exact hierarchy of pure states, we further demonstrate that these features of the stable steady-state solutions persist for finite cluster sizes. Ordered multimode Dicke models, such as the nearest-neighbor coupling geometry considered here, are accessible with current experimental technologies and point toward a broader class of strongly interacting dissipative systems with similarly rich behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the multimode Dicke model with ordered nearest-neighbor couplings, where n_c atomic clusters interact via n_c-1 cavity modes. It reports that the number of mean-field stable steady-states in the superradiant phase follows Fibonacci scaling with n_c, that a subset of these states exhibits persistent oscillations, and that both the scaling and the oscillations persist for finite cluster sizes according to truncated Wigner approximation (TWA) and hierarchy-of-pure-states (HOPS) calculations.

Significance. If the central claims hold, the work identifies an analytically tractable ordered geometry in multimode cavity QED that produces Fibonacci-structured steady-state landscapes and oscillatory attractors, providing a counterpoint to disordered multimode Dicke models and indicating experimental relevance with existing ultracold-atom platforms. The combination of mean-field analysis with both approximate and numerically exact methods is a methodological strength.

major comments (2)
  1. [§4 (Mean-field analysis)] §4 (Mean-field analysis): the recurrence that maps the nearest-neighbor coupling geometry onto Fibonacci numbers must be derived explicitly from the steady-state equations; without the closed-form relation or the explicit counting procedure, it is impossible to verify that the reported scaling is a structural consequence of the n_c-1 modes rather than a numerical coincidence.
  2. [§5 (TWA and HOPS results)] §5 (TWA and HOPS results): the manuscript must demonstrate convergence of the number of stable states with hierarchy depth in HOPS and quantify the effect of cumulant truncation in TWA on the superradiant fixed-point count; if either truncation systematically merges or eliminates states, the claim that Fibonacci scaling and persistent oscillations survive for finite n_c is not yet supported.
minor comments (2)
  1. [Abstract] Abstract: the range of n_c for which Fibonacci scaling is observed and the precise sequence (e.g., F_{n_c+2}) should be stated explicitly.
  2. [Figure captions] Figure captions: every panel showing steady-state counts or trajectories should label both the value of n_c and the method (mean-field, TWA, or HOPS) used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and will revise the manuscript accordingly to strengthen the presentation of the mean-field analysis and the finite-size results.

read point-by-point responses

  1. Referee: §4 (Mean-field analysis): the recurrence that maps the nearest-neighbor coupling geometry onto Fibonacci numbers must be derived explicitly from the steady-state equations; without the closed-form relation or the explicit counting procedure, it is impossible to verify that the reported scaling is a structural consequence of the n_c-1 modes rather than a numerical coincidence.

    Authors: We agree that an explicit derivation from the steady-state equations is required. In the revised manuscript we will expand §4 to derive the recurrence directly: the steady-state conditions on the mean-field amplitudes for the n_c-1 modes impose a chain of algebraic constraints whose number of stable solutions satisfies the Fibonacci recurrence S(n_c) = S(n_c-1) + S(n_c-2) with S(1)=1, S(2)=2. The derivation follows from partitioning the possible sign patterns of the superradiant amplitudes according to whether the additional cluster-mode pair is aligned with the preceding solution or introduces an independent stable branch. We will also include the closed-form counting procedure based on these recurrence relations. revision: yes

  2. Referee: §5 (TWA and HOPS results): the manuscript must demonstrate convergence of the number of stable states with hierarchy depth in HOPS and quantify the effect of cumulant truncation in TWA on the superradiant fixed-point count; if either truncation systematically merges or eliminates states, the claim that Fibonacci scaling and persistent oscillations survive for finite n_c is not yet supported.

    Authors: We accept that explicit convergence checks are necessary. In the revised §5 we will add data showing that the number of stable states identified by HOPS remains constant once the hierarchy depth exceeds a modest threshold for the system sizes examined, together with a quantitative comparison of TWA results at successive cumulant orders. These additions will confirm that neither truncation systematically alters the Fibonacci count or eliminates the oscillatory attractors for the finite n_c values considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims follow from model equations and numerical methods

full rationale

The paper derives the Fibonacci count of mean-field steady states directly from the recurrence structure of the nearest-neighbor ordered multimode Dicke model and verifies persistence under TWA and HOPS numerics. No self-definitional steps, no fitted inputs renamed as predictions, and no load-bearing self-citations appear in the provided text or abstract. The derivation chain is independent of its target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract.

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Reference graph

Works this paper leans on

101 extracted references · 5 canonical work pages · 1 internal anchor

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    Singletons We have found that it is useful to classify steady-state solutions by the sign pattern of their x spin components. To that end, for a given sign pattern σ= (σ 1, . . . , σnc), σ i ∈ {±1},(A4) we look for a solution with sgn(xi) = σi. For given σ, we define the counting variable qi =1 {i>1 andσ i−1=σi} +1 {i<nc andσ i+1=σi}.(A5) For a given site...

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