Dynamical aspects of steady-state subradiance: Detailed balance and its breakdown

Athreya Shankar; Jarrod T. Reilly; Murray J. Holland; Raphael Kaubruegger; Simon B. J\"ager; Walter Hahn

arxiv: 2605.15785 · v1 · pith:AOF2BNRPnew · submitted 2026-05-15 · 🪐 quant-ph · cond-mat.stat-mech

Dynamical aspects of steady-state subradiance: Detailed balance and its breakdown

Athreya Shankar , Simon B. J\"ager , Jarrod T. Reilly , Raphael Kaubruegger , Murray J. Holland , Walter Hahn This is my paper

Pith reviewed 2026-05-20 19:15 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords subradiancedetailed balanceMarkov chainbad-cavity laserdissipative phase transitionprobability currentsentropy productionself-pulsing

0 comments

The pith

In the subradiant regime of a bad-cavity laser, one phase makes the underlying Markov chain time-reversible with growing atom number while the other breaks detailed balance through circulating currents.

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

The paper examines how a dissipative phase transition in the subradiant regime of a bad-cavity laser connects to the properties of its effective Markov chain description. This chain lives in a two-dimensional space of collective angular momentum states even though the underlying quantum states can be highly entangled. In one phase the chain approaches the detailed-balance condition as atom number N increases, becoming effectively time-reversible because a collective emission process collapses the dynamics to one dimension. In the other phase time-asymmetric probability currents appear in the two-dimensional state space, accompanied by an internal entropy production rate that grows extensively with N. An observable signature of these currents is self-pulsing of the cavity output that appears as an anti-correlation dip in the intensity correlation function.

Core claim

A dissipative phase transition in the subradiant regime of the bad-cavity laser divides the steady-state behavior into two phases for the Markov chain on collective angular momentum states: one phase satisfies detailed balance with increasing atom number N and is therefore effectively time-reversible, while the other phase exhibits time-asymmetric probability currents that break detailed balance and produce a macroscopic internal entropy production rate scaling extensively with N.

What carries the argument

The two-dimensional Markov chain on collective angular momentum states that describes the bad-cavity laser dynamics, together with the detailed-balance condition and the probability currents that appear in its state space.

If this is right

Collective atomic emission reduces the Markov chain to one dimension and restores time-reversibility in one phase.
Time-asymmetric probability currents emerge in the two-dimensional state space of the other phase.
Internal entropy production grows extensively with atom number N when detailed balance is broken.
Self-pulsing appears in the cavity light output and produces a detectable anti-correlation dip in the intensity correlation function.

Where Pith is reading between the lines

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

The same reduction to collective states might allow other dissipative many-body systems to be classified according to whether their effective classical descriptions remain time-reversible.
The scaling of entropy production with system size offers a route to quantify the thermodynamic cost of maintaining steady-state subradiance in larger ensembles.
Detecting the anti-correlation dip provides a practical experimental handle on the breakdown of detailed balance without requiring full tomography of the atomic state.

Load-bearing premise

The dissipative dynamics of the bad-cavity laser can be reduced to a classical Markov chain on a two-dimensional space of collective angular momentum states.

What would settle it

Measure the second-order intensity correlation function of the cavity output and check whether an anti-correlation dip appears only in one of the two subradiant phases while the entropy production rate scales linearly with atom number N in that same phase.

Figures

Figures reproduced from arXiv: 2605.15785 by Athreya Shankar, Jarrod T. Reilly, Murray J. Holland, Raphael Kaubruegger, Simon B. J\"ager, Walter Hahn.

**Figure 2.** Figure 2: FIG. 2. Steady-state population distribution for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Population flow in steady state for the case of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Internal entropy production rate (normalized to sys [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Two-photon intensity correlation function [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

The dynamics of dissipative many-body quantum systems sometimes admit an emergent classical description in terms of a Markov chain even though the corresponding state space contains highly entangled states. In particular, a bad-cavity laser is a paradigm system whose dynamics can be formulated as a Markov chain in a two-dimensional state space spanned by collective angular momentum states. In this article, we investigate the connection between a dissipative phase transition that occurs in the subradiant regime of this system in the large atom number limit, and the properties of the underlying Markov chain. In one of the phases, the Markov chain approaches the detailed-balance condition with increasing atom number $N$ and hence becomes effectively time-reversible. This is caused by a collective atomic emission process that effectively reduces the Markov chain to one dimension. In the other phase, we find the emergence of time-asymmetric probability currents in the two-dimensional state space that signals a breakdown of detailed balance. This is accompanied by a macroscopic internal entropy production rate in the Markov chain that scales extensively with the atom number $N$. An observable consequence of the probability currents is a self-pulsing of the cavity light output in this phase, which can be detected as an anti-correlation dip in the intensity correlation function.

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.

Desk Editor's Note private letter to a colleague

The paper maps a subradiant phase transition in the bad-cavity laser onto detailed-balance restoration versus breakdown in its two-dimensional Markov chain, with the breakdown tied to observable self-pulsing.

read the letter

The core finding is that one phase of the subradiant regime drives the Markov chain toward detailed balance as N grows because a single collective emission process collapses the dynamics to one effective dimension. The other phase keeps two-dimensional currents that break detailed balance, produce entropy that scales with N, and show up as an anti-correlation dip in the cavity output intensity correlations. That split is the main new piece. Prior work on bad-cavity lasers and collective spins did not frame the transition this way in terms of time-reversibility and internal entropy production of the chain. The connection to measurable light statistics is a practical plus. The authors also keep the state space explicitly two-dimensional even though the underlying quantum states are entangled, which is a clean way to make the classical description usable. The rate scalings and entropy calculations look internally consistent on the abstract level, and the absence of free parameters in the definitions helps. The main soft spot is whether the claimed reduction to one dimension really holds in the large-N limit. The stress-test note flags that other transition channels could stay competitive; if the paper only shows dominance for one process without bounding the rest, the approach to detailed balance could be weaker than stated. I would want to see the explicit large-N asymptotics or numerical checks on the current magnitudes before accepting the time-reversibility claim at face value. Minor gaps in the abstract on error bars or finite-N corrections are easy to fix. This is aimed at people who already work on open quantum systems, dissipative phase transitions, or non-equilibrium Markov models. A reader comfortable with both collective spin models and entropy production will pick up the most. It is not a broad review paper, but the concrete link between quantum phases and classical currents is sharp enough that a serious referee should see it. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the connection between a dissipative phase transition in the subradiant regime of a bad-cavity laser and the properties of its Markov-chain description on collective angular-momentum states. In one phase the chain approaches detailed balance (and effective time-reversibility) with growing N because a single collective emission process reduces the dynamics to one dimension; in the other phase time-asymmetric probability currents appear, producing an extensive internal entropy-production rate and an observable self-pulsing signature in the cavity intensity correlation function.

Significance. The work links dissipative phase transitions in open quantum systems to nonequilibrium thermodynamic notions (detailed-balance restoration versus breakdown, macroscopic entropy production) while retaining an explicit, experimentally accessible observable. The demonstration that highly entangled states can still yield an emergent low-dimensional classical Markov chain is a useful conceptual contribution.

major comments (1)

[§3.2] §3.2 and the rate-scaling discussion following Eq. (11): the central claim that collective emission reduces the chain to an effectively one-dimensional, detailed-balance-satisfying dynamics rests on the assertion that all other transition channels are suppressed by at least 1/N. An explicit asymptotic bound or numerical check confirming that non-collective rates remain negligible uniformly across the phase (including near the transition) is needed; without it the reduction argument is not yet load-bearing.

minor comments (2)

[§4] The definition of the internal entropy production rate (Eq. (19)) is given only after the probability-current discussion; moving the definition earlier would improve readability.
[Figure 3] Figure 3 caption: the anti-correlation dip is described as 'clearly visible,' but the vertical scale and the value of the minimum should be stated numerically for quantitative comparison with experiment.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comment, which helps strengthen the central argument. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3.2] §3.2 and the rate-scaling discussion following Eq. (11): the central claim that collective emission reduces the chain to an effectively one-dimensional, detailed-balance-satisfying dynamics rests on the assertion that all other transition channels are suppressed by at least 1/N. An explicit asymptotic bound or numerical check confirming that non-collective rates remain negligible uniformly across the phase (including near the transition) is needed; without it the reduction argument is not yet load-bearing.

Authors: We thank the referee for this observation. The manuscript states that the collective emission rate scales as N while non-collective channels remain O(1), implying 1/N suppression, but we agree that an explicit uniform bound and supporting numerics would make the reduction to one-dimensional detailed-balance dynamics fully rigorous. In the revision we will add (i) an asymptotic analysis bounding the ratio of non-collective to collective transition rates by C/N for a constant C independent of the control parameter (including in a neighborhood of the transition), and (ii) numerical checks of the steady-state distribution and entropy-production rate for N up to 100 that confirm the suppression holds uniformly across the phase diagram. These additions will be placed in §3.2 and the discussion after Eq. (11). revision: yes

Circularity Check

0 steps flagged

No significant circularity; Markov chain reduction and detailed-balance analysis are derived from rate equations

full rationale

The paper starts from the quantum master equation for the bad-cavity laser, projects onto collective angular-momentum states to obtain an explicit two-dimensional Markov chain with N-dependent transition rates, and then performs an asymptotic large-N analysis of those rates. The claim that one phase is dominated by a single collective emission channel (reducing the chain to effectively one dimension and restoring detailed balance) follows from comparing the scaling of the competing rates; the entropy production and probability currents are computed directly from the same rate matrix via the standard stochastic thermodynamics formulas. None of these steps redefines a fitted quantity as a prediction or imports a uniqueness result from self-citation; the derivation remains self-contained against the underlying Liouvillian.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the reduction of the quantum dynamics to a classical Markov chain on collective angular momentum states; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The dynamics of the dissipative many-body quantum system admit an emergent classical Markov chain description in a two-dimensional state space spanned by collective angular momentum states.
Stated directly in the abstract as the foundation for analyzing detailed balance and phase transitions.

pith-pipeline@v0.9.0 · 5772 in / 1436 out tokens · 58690 ms · 2026-05-20T19:15:03.261588+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the dynamics can be formulated as a Markov chain in a two-dimensional state space spanned by collective angular momentum states... recurrence relation Eq. (9) is the condition for detailed balance... internal entropy production rate ˙Si
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

steady-state population distribution... Gaussian with mean μ = N/2 [(γ−w)/(γ+w)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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The corresponding density matrix is ˆρs a,b =|ψ a,b⟩s ⟨ψa,b|s. Then, the PIJ= 0 state can be expressed as ˆS0,0,0 = ˆρs 12 ⊗ˆρs 34 ⊗. . .⊗ˆρ s N−1,N PI ,(6) representing the fact that this state consists of a mixture ofN/2 singlet pairs. We now combine the above observations to write down the form of the states for 0< J < N/2. We propose that any general ...

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S. B. J¨ ager, T. Schmit, G. Morigi, M. J. Holland, and R. Betzholz, Lindblad master equations for quantum sys- tems coupled to dissipative bosonic modes, Phys. Rev. Lett.129, 063601 (2022)

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