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Local decoherence can continuously tune Dicke superradiance from N² peak intensity down to linear independent emission, and the fully collective boundary is a continuous phase transition in a transient observable.

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2026-07-31 19:51 UTC pith:6RPG52HF

load-bearing objection Clean scaling theory for when Dicke N^{2} survives local noise: paired variables, tunable β, and a real transient non-analyticity, with closed-form dephasing and controlled spontaneous-emission analysis backed by MC and code.

arxiv 2607.28034 v1 pith:6RPG52HF submitted 2026-07-30 quant-ph cond-mat.stat-mechphysics.optics

Scaling theory of decoherence in Dicke superradiance

Nico S. Bassler , Julian Lyne , Javier Cuerda This is my paper
classification quant-ph cond-mat.stat-mechphysics.optics
keywords Dicke superradiancedecoherencescaling theorycollective emissionlocal dephasingtransient phase transitionopen quantum systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Dicke superradiance is the classic case in which N two-level emitters radiate a short burst whose peak intensity scales as N² because collective emission builds many-body coherence. Real systems also suffer local dephasing and local spontaneous emission that fight that buildup. This paper shows that the competition is controlled by two dimensionless scaling variables, g_ξ = ξ/(NΓ) and ġ_γ = γ ln N/(NΓ). The peak intensity then obeys a generalized scaling law I★ ∝ N^β with an exponent β that can sit anywhere between 2 and 1, defining fully collective, partially collective, and independent-emitter regimes. The surface on which β drops below 2 is non-analytic in the thermodynamic limit and therefore acts as a continuous phase transition, even though the steady state remains trivial. The practical message is that simply making N larger does not guarantee quadratic scaling; the relative growth of local rates with N can push an experiment out of the fully collective window.

Core claim

Including local dephasing at rate ξ and local spontaneous emission at rate γ, the peak collective intensity of Dicke superradiance obeys I★ ∝ N^{β(g_ξ, ġ_γ)} with 1 ≤ β ≤ 2. The analytical exponent in the partially collective window is β = 1 + [1 − g_ξ − (1 + g_ξ) ln(2/(1 + g_ξ))]/ġ_γ. The boundary β = 2 is a continuous non-analyticity of this transient peak intensity in the N o ∞ limit.

What carries the argument

The hydrodynamic large-N reduction of the permutationally invariant Dicke-triangle rate equations to deterministic mean-field ODEs for the intensive variables I = ⟨S†S⟩/N² and m = M/N, initialized with the exact finite-size seed I(0) = 1/N and controlled by the two scaling variables g_ξ and ġ_γ.

Load-bearing premise

The claim rests on replacing the exact stochastic population dynamics by deterministic mean-field equations for intensive variables; near the critical boundaries, especially when spontaneous emission makes convergence only logarithmic in N, fluctuations that those equations drop can still shift the apparent exponent at finite size.

What would settle it

Measure peak collective intensity versus N at fixed g_ξ and ġ_γ (for example by scaling cavity-mediated Γ, local dephasing, and local decay together) and check whether the fitted exponent matches the predicted β(g_ξ, ġ_γ) and whether the normalized peak closes with the predicted linear or quadratic critical exponent when the fully-collective boundary is crossed.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Experiments must track how Γ(N), ξ(N) and γ(N) scale; a path that drives ġ_γ or g_ξ across the β = 2 line will lose N² scaling even as N grows.
  • In the common cavity-QED normalization that keeps single-particle rates finite, Γ ∝ 1/N so ġ_γ grows as ln N and the thermodynamic limit is independent emission.
  • The same scaling variables organize a phase diagram with fully collective (β = 2), partially collective (1 < β < 2) and independent (β = 1) regions that can be read off from trajectories on the Dicke triangle.
  • Because the non-analyticity sits in a transient peak rather than the steady state, continuous phase-transition language can apply to burst observables in open many-body systems.

Where Pith is reading between the lines

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

  • Any platform whose collective rate grows slower than linearly with N will eventually be pushed into the partially collective or independent regime by intensive local decoherence, even if the microscopic Hamiltonian is ideal.
  • The early-time amplification stage that fixes β is essentially a linear instability of the fully inverted seed; similar seed-plus-gain analyses should classify burst scaling in other collectively dissipative models (waveguide arrays, interacting bosons, etc.).
  • Finite-size rounding of the β = 2 boundary will be strongest when spontaneous emission dominates, because the controlling variable contains only ln N; large-N cavity or circuit-QED arrays are the cleanest place to resolve the linear closing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper develops a large-N scaling theory for Dicke superradiance in the presence of local dephasing (rate ξ) and local spontaneous emission (rate γ). Starting from the permutationally invariant Lindblad equation, it reduces the dynamics to deterministic mean-field ODEs for intensive variables s = S/N and m = M/N, initialized with the exact finite-N seed I(0) = 1/N. The peak collective intensity obeys the generalized scaling law I⋆ ∝ N^{β(g_ξ, ġ_γ)} with 1 ≤ β ≤ 2, where g_ξ = ξ/(NΓ) and ġ_γ = γ ln N/(NΓ). In the partially collective window the exponent is given analytically by Eq. (9). The β = 2 boundary is identified as a continuous non-analyticity of the transient peak intensity in the thermodynamic limit, with quadratic closing for pure dephasing and linear closing when spontaneous emission is present. Closed-form solutions for dephasing, early-time linearization for spontaneous emission, phase diagrams, and extensive Monte Carlo validation are provided.

Significance. If correct, the result supplies a concrete, experimentally usable criterion for when local decoherence destroys N² superradiant scaling even as N is increased. The scaling variables and the analytical exponent β are derived from the Lindblad rates rather than fitted, the dephasing case is solved in closed form, and the spontaneous-emission analysis is controlled by a stated linearization whose domain of validity is explicit. The authors ship full numerical code (permutationally invariant solvers, mean-field integrators, and Monte Carlo routines), enabling independent reproduction. Framing the β = 2 locus as a continuous transition in a transient observable is a useful conceptual addition to the literature on dissipative many-body coherence, and the cavity-QED scaling example in the conclusion gives a falsifiable platform-level prediction.

minor comments (4)
  1. [Abstract / Sec. I / Sec. IV] The terminology “continuous phase transition” for a non-analyticity of a transient peak intensity is used carefully in the introduction but could still be flagged more prominently in the abstract and conclusion, so that readers do not expect a steady-state Liouvillian transition.
  2. [Fig. 3 / Fig. A1 / App. I] Figure 3 and Fig. A1 would benefit from an explicit statement of the fitting window and number of N-points used to extract β, especially near the β = 2 line where logarithmic slow convergence is acknowledged in App. I.
  3. [Sec. III / App. F] A brief cross-reference in the main text to the alternative thermodynamic limit of fixed g_γ (App. F) would help readers who encounter that scaling in cavity-QED normalizations.
  4. [Throughout] Minor typographical inconsistencies appear in the arXiv text (e.g., “SPONT ANEOUS”, “DA T A”); these should be cleaned in production.

Circularity Check

0 steps flagged

No significant circularity: scaling exponent and phase boundaries are derived from Lindblad rates and large-N ODEs, not fitted or defined into existence.

full rationale

The load-bearing chain is self-contained and non-circular. The master equation (Eq. 1) supplies collective and local rates; permutational invariance reduces them to known Dicke-triangle rates (App. A); a standard hydrodynamic 1/N expansion yields deterministic ODEs for intensive variables (App. B, Eqs. 5 and 7); the seed I(0)=1/N is the exact finite-N collective intensity of the fully inverted Dicke state (App. C), not a free parameter; and β(g_ξ, ḡ_γ) follows by integrating the early-time linearized intensity under spontaneous emission (App. E, Eq. 9), with the dephasing-only case closed-form (App. D, Eq. 6). Scaling variables g_ξ=ξ/(NΓ) and ḡ_γ=γ ln N/(NΓ) emerge from the competition of timescales in those ODEs rather than from fits to data. Independent Gillespie/tau-leaping Monte Carlo and mean-field numerics (Fig. A1, Apps. H–I) cross-check the asymptotics. Self-citations are background or code availability, not uniqueness theorems or ansatzes that force the central claim. Nothing reduces by construction to its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

The central claim rests on the standard open-system Dicke Lindbladian plus a large-N hydrodynamic closure. No parameters are fitted to experiment. The only nontrivial modeling ingredients are permutational symmetry (exact for the chosen Liouvillian and symmetric initial state), the 1/N rate expansion, and the early-time neglect of collective back-action while the seed amplifies under spontaneous emission.

axioms (4)
  • domain assumption Dynamics are generated by the Lindblad equation with collective jump Ŝ, local dephasing σ^z_j, and local decay σ_j only (Eq. 1); no coherent Hamiltonian.
    Isolates dissipation competition; unitary or inhomogeneous terms could change trajectories on the Dicke triangle.
  • standard math Permutational symmetry reduces the state to populations p_{S,M} on the Dicke triangle with known rates (App. A).
    Exact for symmetric initial state and permutation-invariant Liouvillian; standard in the cited PIQS literature.
  • domain assumption Large-N hydrodynamics: intensive variables (s,m,I) obey deterministic ODEs from 1/N expansion of rates, with seed I(0)=1/N (Apps. B–C).
    Drops fluctuations that matter near criticality and on the independent edge; validated but not exact at finite N.
  • ad hoc to paper For γ>0, early-time amplification may be linearized by dropping I back-action on m while I ≪ g_γ(m+1/2) (App. E).
    Key step that produces the closed form for β; justified asymptotically when g_γ ~ 1/ln N but is an approximation at finite N.
invented entities (2)
  • Scaling variables g_ξ = ξ/(NΓ) and ḡ_γ = γ ln N/(NΓ) independent evidence
    purpose: Organize the thermodynamic limit so that fully, partially, and non-collective regimes occupy finite regions of a two-parameter plane.
    Dimensionless combinations derived from competing timescales, not postulated particles or forces; still the paper's main conceptual objects.
  • Transient continuous phase transition at the β=2 boundary of peak intensity no independent evidence
    purpose: Characterize non-analytic closing of the normalized peak intensity I⋆/I⋆^{(Dicke)} as N→∞.
    Authors stress it is outside the steady-state dissipative PT paradigm; evidence is mean-field non-analyticity plus finite-N approach in Monte Carlo.

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read the original abstract

The survival of many-body coherence depends on the competition between correlation buildup and decoherence. In Dicke superradiance, collective emission builds up correlations, producing a peak intensity scaling as $N^2$ for $N$ emitters. We develop a scaling theory including local dephasing and spontaneous emission and obtain fully collective, partially collective, and independent-emitter scaling regimes. The boundary of the fully collective regime defines a continuous phase transition in a transient observable. Local decoherence can prevent $N^2$ scaling despite increasing $N$.

Figures

Figures reproduced from arXiv: 2607.28034 by Javier Cuerda, Julian Lyne, Nico S. Bassler.

Figure 1
Figure 1. Figure 1: Dicke triangle, burst scalings and transitions. (a) Dynamics on the Dicke triangle, where for increasing de￾coherence strength trajectories are pushed towards smaller S–values. The inset shows the matrix elements of the local and collective processes for a given state |S, M⟩ ( ). (b) Intensity dynamics for the trajectories in (a), normalized by the peak intensity I (Dicke) ⋆ of ideal Dicke superradiance at… view at source ↗
Figure 2
Figure 2. Figure 2: Large–N dynamics. Results are shown for dephasing (top row) and spontaneous emission (bottom row) at N = 108 . (a,d) Normalized intensity as a function of rescaled time for varying scaling variables on a log-log scale. Stars (⋆) indicate the peak intensity I⋆. (b,e) Corresponding trajectories on the Dicke triangle. While dephasing changes the angle at which the trajectories enter the triangle, local sponta… view at source ↗
Figure 3
Figure 3. Figure 3: Phase diagram and Dicke-triangle trajecto￾ries. (a) Mean-field phase diagram showing the normalized intensity in the (g¯γ, gξ) plane for N = 108 . The dashed lines indicate the asymptotic phase boundaries at β = 2 (black) and β = 1 (white). In (b-d) we show corresponding represen￾tative trajectories in the three regions. Blue shading in panels (b-d) marks the region I = s 2 − m 2 < N −0.2 , corresponding t… view at source ↗

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    By utilizing spin conservation I + m2 = s2 = const., the dynamics can be reduced to a single differential equation for the inversion

    Collective decay For purely collective decay (gξ = gγ = 0) the equations reduce to ∂τ I= 2mI, ∂τ m=−I. By utilizing spin conservation I + m2 = s2 = const., the dynamics can be reduced to a single differential equation for the inversion. The same invariant follows directly from the equations of motion, since dI dm =−2m→I+m 2 = const. For the fully inverted...

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    Collective decay with dephasing For collective decay with dephasing, the equations re- duce to ∂τ I= (2m−g ξ)I, ∂τ m=−I. In this form, it is not immediately clear that these differ- ential equations can still be solved analytically. However, defining ˜m = m−g ξ/2, the equations reduce to the previ- ous case. Thus, reusing the same idea dI d ˜m =−2 ˜m→I+ ˜...

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