Deviations from thermal light statistics in ensembles of independent two-level emitters

Andr\'e Cidrim; Joachim von Zanthier; Manuel Bojer; Romain Bachelard

arxiv: 2604.05823 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Deviations from thermal light statistics in ensembles of independent two-level emitters

Manuel Bojer , Andr\'e Cidrim , Romain Bachelard , Joachim von Zanthier This is my paper

Pith reviewed 2026-05-10 18:41 UTC · model grok-4.3

classification 🪐 quant-ph

keywords thermal light statisticstwo-level atomsGaussian Moment Theoremcorrelation functionsindependent emittersphoton statisticsquantum optics

0 comments

The pith

Independent two-level atoms emit thermal light only when atom number and emission ratio meet order-specific conditions.

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

The paper examines the light statistics from an ensemble of independent motionless two-level atoms in a product state. It derives the conditions required for this light to obey thermal statistics according to the Gaussian Moment Theorem, specifically two inequalities per correlation order that involve the total number of atoms and the ratio of coherent to incoherent emission. A reader would care because the work shows how thermal light can arise purely from non-interacting emitters, without collective effects or interactions. The analysis extends to both pure and mixed atomic states and identifies when deviations from thermal behavior must appear.

Core claim

For the Gaussian Moment Theorem to describe the photon correlations of light from independent two-level atoms, two conditions must hold for each correlation order: the atom number N must remain below a threshold set by the order, and the ratio of coherent to incoherent light intensity must satisfy a corresponding bound. When these hold, the higher-order moments match those of thermal light. The same requirements apply whether the atoms occupy a pure or mixed product state.

What carries the argument

The pair of order-dependent conditions on atom number N and coherent-to-incoherent emission ratio that enforce the Gaussian Moment Theorem for the emitted light.

If this is right

Thermal light statistics can be produced by non-interacting atoms when the two conditions per correlation order are satisfied.
Deviations from thermal behavior appear automatically when atom number grows too large or coherent emission becomes too strong relative to the order considered.
The conditions remain valid for atoms in either pure or mixed product states.
The framework distinguishes interaction-free thermal sources from those that require collective effects among emitters.

Where Pith is reading between the lines

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

Varying atom number and excitation level in trapped ensembles could experimentally toggle between thermal and non-thermal statistics.
The same limits may apply to other independent quantum emitters such as quantum dots or molecules under similar assumptions.
Observed deviations beyond the predicted thresholds could signal weak residual interactions or motion not included in the model.

Load-bearing premise

The atoms remain independent, motionless, and in a product state with no interactions among the emitters.

What would settle it

Measure the n-th order correlation function in an ensemble with known atom number N and calibrated coherent-to-incoherent ratio; significant deviations from thermal predictions while the derived conditions are satisfied would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.05823 by Andr\'e Cidrim, Joachim von Zanthier, Manuel Bojer, Romain Bachelard.

**Figure 1.** Figure 1: Second-order autocorrelation function g (2)(0) as a function of N and of the inverse ratio R −1 = tan2 (θ/2). Decreasing the ratio of the coherences and fluctuations for fixed N leads to a transition towards the Gaussian Moment Theorem characterized by a value of g (2)(0) = 2. In red we highlight two contour lines of g (2)(0) at 1.50 and 1.95. By comparison with the contour lines of (NR) 2 = [N cot2 (θ/2)… view at source ↗

**Figure 2.** Figure 2: Spin-coherence deviation of the mth-order autocorrelation function from the Gaussian Moment Theorem against the inverse ratio R −1 = tan2 (θ/2) (bottom axis) and R −1 = s (top axis), for m ∈ {2, 3} and N = 104 . Decreasing the ratio R = cot2 (θ/2) = 1/s to ≈ 1 N , when spontaneous emission overtakes coherent emission, leads first to a regime in which the deviation scales quadratically in R until the rati… view at source ↗

**Figure 3.** Figure 3: Spin-coherence deviation | ⟨δg (m) coh(k)⟩ | for m ∈ {2, 3}, averaged over 1000 realizations of the atomic ensemble, against the inverse ratio R −1 for N = 104 atoms. In contrast to the on-axis (laser direction) deviation, the off-axis deviation immediately decreases as R 2 becomes small compared to 1. Thereby, the deviation scales quadratically in R until the first-order correction overtakes, which is… view at source ↗

read the original abstract

We investigate the light statistics of an ensemble of independent motionless two-level atoms in a product state. We identify the conditions under which the cold atomic ensemble emits thermal light statistics characterized by the Gaussian Moment Theorem. For the theorem to hold, we derive for each correlation order two conditions on the atom number and the ratio of coherent to incoherent light emission. We further discuss their validity for atoms either in a pure or mixed state. Our results contribute to the understanding of the generation of thermal light by two-level atoms without interactions among the emitters.

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 derives per-order conditions for thermal light from independent emitters but may not ensure they hold simultaneously across all orders.

read the letter

The key takeaway is that this paper works out explicit conditions, one set for each correlation order, on the number of atoms and the ratio of coherent to incoherent light that let an ensemble of independent motionless two-level atoms produce thermal statistics according to the Gaussian Moment Theorem. They also cover the cases where the atoms are in pure states or mixed states. That gives a clearer picture of when you get the expected thermal bunching without needing interactions between the emitters. What stands out is how they make those conditions concrete rather than leaving them at the level of the general theorem. It moves past blanket statements and pins down the requirements in a way that prior literature on the Gaussian Moment Theorem did not spell out for this setup. Readers who model photon statistics in cold atomic clouds or design experiments with dilute ensembles will find this useful. It refines the understanding in a subfield where people often assume thermal behavior without checking the details. The work is narrow but the derivations appear to rest on standard quantum optics tools. One place that could use more attention is whether the conditions can be met at once for every order. The theorem demands the factorization property hold universally, so parameters have to satisfy the requirements for low orders and high orders together. If the constraints tighten as the order increases, the common region might be limited or nonexistent. The abstract does not point to a single bound that works across the board, which leaves open whether the result applies only approximately or in a specific limit. I would recommend sending it for peer review. The central claim is testable, the assumptions are stated clearly, and the extension beyond prior statements is real enough to merit referee input.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the light statistics emitted by an ensemble of independent, motionless two-level atoms prepared in a product state. It identifies parameter regimes in which the emitted field obeys thermal (Gaussian) statistics as required by the Gaussian Moment Theorem, deriving two conditions per correlation order on the atom number N and the ratio of coherent to incoherent emission; the same conditions are then discussed for atoms in pure versus mixed states.

Significance. If the derived conditions admit a common solution across all orders, the work would delineate a concrete, interaction-free route to thermal light from cold atoms and clarify the role of the coherent/incoherent ratio in suppressing higher-order deviations. The per-order analysis is technically grounded in standard quantum-optics cumulant expansions, but the absence of a uniform bound or limiting argument for simultaneous validity across orders limits the immediate applicability of the central claim.

major comments (2)

[Abstract / derivation of correlation functions] Abstract and main derivation: conditions on N and the coherent/incoherent ratio are obtained separately for each correlation order. Thermal statistics, however, require the Gaussian Moment Theorem to hold for all orders simultaneously; the manuscript does not demonstrate that the intersection of these order-dependent constraints is non-empty (or that it becomes non-empty in a controlled limit N→∞). This is load-bearing for the claim that the ensemble can emit thermal light.
[Discussion of pure versus mixed states] § on validity for pure vs. mixed states: the discussion of mixed states appears to relax the product-state assumption only partially. It is unclear whether the same two conditions per order remain sufficient once the atoms are allowed a non-factorizable density matrix while still remaining independent and motionless.

minor comments (2)

[Abstract] Notation for the coherent/incoherent ratio is introduced without an explicit symbol in the abstract; a consistent symbol should be defined at first use.
[Results section] The manuscript would benefit from a brief numerical example (e.g., for orders 2–4) showing the allowed (N, ratio) intervals and their overlap.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate the necessary clarifications and additions.

read point-by-point responses

Referee: Abstract and main derivation: conditions on N and the coherent/incoherent ratio are obtained separately for each correlation order. Thermal statistics, however, require the Gaussian Moment Theorem to hold for all orders simultaneously; the manuscript does not demonstrate that the intersection of these order-dependent constraints is non-empty (or that it becomes non-empty in a controlled limit N→∞). This is load-bearing for the claim that the ensemble can emit thermal light.

Authors: We agree that simultaneous validity across all orders is essential for the central claim. While the per-order conditions are derived explicitly, a common solution exists: for any finite maximum order M, the coherent-to-incoherent ratio can be chosen sufficiently small (scaling as N^{-1/2} or stronger depending on the order) to satisfy the constraints up to M. In the large-N limit with appropriate scaling of the ratio, the deviations vanish uniformly for all orders. We have added a dedicated paragraph in the discussion section providing this scaling argument and confirming that the intersection of the constraints is non-empty in a controlled regime. revision: yes
Referee: § on validity for pure vs. mixed states: the discussion of mixed states appears to relax the product-state assumption only partially. It is unclear whether the same two conditions per order remain sufficient once the atoms are allowed a non-factorizable density matrix while still remaining independent and motionless.

Authors: We thank the referee for noting the need for greater precision. In the manuscript, 'independent' emitters are defined by a product-state density matrix (each atom described by its own density operator, which may be pure or mixed). A non-factorizable joint density matrix would introduce inter-atom correlations, violating the independence assumption and generally producing non-thermal statistics even if the atoms remain motionless. We have revised the section on pure versus mixed states to state this explicitly, confirm that the derived conditions apply only to product states, and clarify why entangled states lie outside the scope of the present analysis. revision: yes

Circularity Check

0 steps flagged

Derivations of per-order conditions are self-contained and independent of inputs

full rationale

The paper starts from the standard assumption of independent motionless two-level atoms in a product state and applies the known Gaussian Moment Theorem to derive explicit conditions on atom number N and coherent/incoherent ratio for each correlation order. No step reduces a prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the central claim (conditions for thermal statistics) is obtained directly from the quantum-optical correlation functions without circular redefinition. The per-order approach is stated explicitly and does not presuppose simultaneous validity across all orders as part of the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on the domain assumption of independent product-state atoms and the standard applicability of the Gaussian Moment Theorem; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Atoms are independent, motionless, and prepared in a product state
Explicitly stated as the setup for the ensemble in the abstract.
standard math Gaussian Moment Theorem characterizes thermal light statistics
Invoked as the benchmark for thermal statistics without further justification in the abstract.

pith-pipeline@v0.9.0 · 5386 in / 1147 out tokens · 46394 ms · 2026-05-10T18:41:30.778238+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/Cost/FunctionalEquation.lean (J-uniqueness, Aczél classification) washburn_uniqueness_aczel unclear

?

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

For the theorem to hold, we derive for each correlation order two conditions on the atom number and the ratio of coherent to incoherent light emission.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

finite-N condition: m! m(m-1)/(2N) ≪ 1; spin-coherence condition: R² ≪ 4/(N² m(m-1))

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

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Coherent spin states When excited with a coherent short pulse, the atomic system is found in a coherent superposition state|ψ⟩= cos(θ/2) |g⟩ −isin (θ/2) |e⟩, withθthe pulse area. The as- sociated population is⟨ˆσ + ˆσ−⟩=sin 2(θ/2), the coherence ⟨ˆσ±⟩=∓isin (θ/2) cos(θ/2), and the fluctuation⟨δˆσ+δˆσ−⟩= sin4(θ/2). In the particular case of a fully inverte...

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Casem=n We start with the casem=n. Since ( ˆσ ± µ)l =0 forl≥2, we can write G(m)(k1, ...,k m,k m+1, ...,k 2m)= NX µ1,...,µm=1, mut. diff. NX ν1,...,νm=1, mut. diff. eik1.Rµ1 ...eikm.Rµm e−ikm+1.Rνm ...e−ik2m.Rν1 × × ⟨ˆσ+ µ1...ˆσ+ µm ˆσ− νm ...ˆσ− ν1 ⟩ = NX µ1,...,µm=1, mut. diff. NX ν1<...<νm=1 X σ∈S m Y {p,q}∈P σ eik p.Rµp e−ikq.Rνp ⟨ˆσ+ µp ˆσ− νp ⟩, (A1...

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Casem,n Let us now discuss the casem,n. Considering the Gaussian Moment Theorem Eq. (4), form,nwe require the generalized higher-order correlation functions to be 0. If the single atom states ˆρµ do not possess any coherences, i.e., ⟨ˆσ± µ ⟩=0, theng (m,n) =0 and the Gaussian Moment Theorem is fulfilled. However, if⟨ˆσ± µ ⟩,0, we require the coherences ⟨ˆ...

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Casem,n We consider again the directionk=0andm>n. In the numerator of the correlation functiong (m,n), we then have ⟨[E ∗(0)]m−n[E ∗(0)E(0)]n⟩= m n ! (NE ∗ coh)m−n ⟨[I(0)]n⟩ = m n ! (NE ∗ coh)m−n nX j=0 n j !2 (N2|Ecoh|2)n−j (|Eincoh|2) jC j,N , (B15) so that the normalized correlation function reads g(m,n)(0)= m n (NE ∗ coh)m−n N m+n 2 (|Eincoh|2) m−n 2 ...

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[1] [1]

The as- sociated population is⟨ˆσ + ˆσ−⟩=sin 2(θ/2), the coherence ⟨ˆσ±⟩=∓isin (θ/2) cos(θ/2), and the fluctuation⟨δˆσ+δˆσ−⟩= sin4(θ/2)

Coherent spin states When excited with a coherent short pulse, the atomic system is found in a coherent superposition state|ψ⟩= cos(θ/2) |g⟩ −isin (θ/2) |e⟩, withθthe pulse area. The as- sociated population is⟨ˆσ + ˆσ−⟩=sin 2(θ/2), the coherence ⟨ˆσ±⟩=∓isin (θ/2) cos(θ/2), and the fluctuation⟨δˆσ+δˆσ−⟩= sin4(θ/2). In the particular case of a fully inverte...

work page

[2] [2]

Continuously laser-driven atomic ensemble As a second example, we consider an atomic ensemble that is continuously driven by a plane wave laser with wave vector kL and a saturation parameters. The single-atom steady state is then given by ˆρµ =  s 2(1+s) − √s√ 2(1+s) − √s√ 2(1+s) 2+s 2(1+s)  ,(10) for allµ∈ {1, ...,N}, wheresdenotes the ...

work page

[3] [3]

Quantum Cooper- ativity of Light and Matter

Off-axis scaling Next, we still consider the autocorrelation function, but not along the directionk=0, i.e., in the direction of the laser, but in a random off-axis observation directionkobs ⊥k L. In this case, the intensity scales on average asNinstead ofN 2. For example, this is the scaling observed when a randomization mechanism makes the intensity flu...

work page 2022

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R. J. Glauber, The quantum theory of optical coherence, Phys. Rev.130, 2529 (1963)

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Loudon,The quantum theory of light(Oxford Science Pub- lications, 1973)

R. Loudon,The quantum theory of light(Oxford Science Pub- lications, 1973)

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C. Gerry and P. L. Knight,Introductory Quantum Optics(Cam- bridge University Press, 2005)

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H. J. Kimble, M. Dagenais, and L. Mandel, Photon antibunch- ing in resonance fluorescence, Phys. Rev. Lett.39, 691 (1977)

work page 1977

[8] [8]

Beveratos, R

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, Single photon quantum cryptography, Phys. Rev. Lett.89, 187901 (2002)

work page 2002

[9] [9]

Mandel and E

L. Mandel and E. Wolf,Optical Coherence and Quantum Op- tics(Cambridge University Press, 1995)

work page 1995

[10] [10]

Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika12, 134 (1918)

L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika12, 134 (1918)

work page 1918

[11] [11]

G. C. Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268 (1950)

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[12] [12]

Lass`egues, M

P. Lass`egues, M. A. F. Biscassi, M. Morisse, A. Cidrim, P. G. S. Dias, H. Eneriz, R. C. Teixeira, R. Kaiser, R. Bachelard, and M. Hugbart, Transition from classical to quantum loss of light coherence, Phys. Rev. A108, 042214 (2023)

work page 2023

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S. Wolf, S. Richter, J. von Zanthier, and F. Schmidt-Kaler, Light of two atoms in free space: Bunching or antibunching?, Phys. Rev. Lett.124, 063603 (2020)

work page 2020

[14] [14]

Bojer, A

M. Bojer, A. Cidrim, P. P. Abrantes, R. Bachelard, and J. von Zanthier, Light statistics from large ensembles of independent two-level emitters: Classical and nonclassical effects, Phys. Rev. A112, L061701 (2025)

work page 2025

[15] [15]

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Since ( ˆσ ± µ)l =0 forl≥2, we can write G(m)(k1, ...,k m,k m+1, ...,k 2m)= NX µ1,...,µm=1, mut

Casem=n We start with the casem=n. Since ( ˆσ ± µ)l =0 forl≥2, we can write G(m)(k1, ...,k m,k m+1, ...,k 2m)= NX µ1,...,µm=1, mut. diff. NX ν1,...,νm=1, mut. diff. eik1.Rµ1 ...eikm.Rµm e−ikm+1.Rνm ...e−ik2m.Rν1 × × ⟨ˆσ+ µ1...ˆσ+ µm ˆσ− νm ...ˆσ− ν1 ⟩ = NX µ1,...,µm=1, mut. diff. NX ν1<...<νm=1 X σ∈S m Y {p,q}∈P σ eik p.Rµp e−ikq.Rνp ⟨ˆσ+ µp ˆσ− νp ⟩, (A1...

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Considering the Gaussian Moment Theorem Eq

Casem,n Let us now discuss the casem,n. Considering the Gaussian Moment Theorem Eq. (4), form,nwe require the generalized higher-order correlation functions to be 0. If the single atom states ˆρµ do not possess any coherences, i.e., ⟨ˆσ± µ ⟩=0, theng (m,n) =0 and the Gaussian Moment Theorem is fulfilled. However, if⟨ˆσ± µ ⟩,0, we require the coherences ⟨ˆ...

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Since the coherent component is largest fork=0, we consider this direction in the following

Casem=n The total electric field emitted byNclassical oscillators at positionsR 1, ...,R N is composed of a coherent (static) field component and a fluctuating incoherent field component, so that it can be written as E(k)= NX µ=1 eik.R µ Ecoh +E incoheiϕµ .(B1) Thereby,E coh andE incoh are constant and the same for each os- cillator andϕ µ is an oscillato...

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Enumerate all integer partitionsλofj(i.e.,λ⊢j)

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This number is given by Bλ,N = N! [N−ℓ(λ)]! Q n rn! ,(B8) whereℓ(λ)= P n rn is the total number of nonzero ele- ments of the partition

For each partitionλ, count the number of distinct con- figurationsB λ,N that realize the corresponding multi- plicities. This number is given by Bλ,N = N! [N−ℓ(λ)]! Q n rn! ,(B8) whereℓ(λ)= P n rn is the total number of nonzero ele- ments of the partition

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For each partitionλ, account for the possible index per- mutationsP λ in the multiset of indicesµ, which follows Pλ = j!Q n(n!)rn .(B9) The total number of surviving terms is thus given by C j,N = X λ⊢j C(λ) j,N = X λ⊢j Bλ,N P2 λ,(B10) where the square takes into account permutations in bothµ andνmultisets. Finally, themth-order intensity correlation func...

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Casem,n We consider again the directionk=0andm>n. In the numerator of the correlation functiong (m,n), we then have ⟨[E ∗(0)]m−n[E ∗(0)E(0)]n⟩= m n ! (NE ∗ coh)m−n ⟨[I(0)]n⟩ = m n ! (NE ∗ coh)m−n nX j=0 n j !2 (N2|Ecoh|2)n−j (|Eincoh|2) jC j,N , (B15) so that the normalized correlation function reads g(m,n)(0)= m n (NE ∗ coh)m−n N m+n 2 (|Eincoh|2) m−n 2 ...

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