Negative absolute temperature attractor in a dense photon gas

D. Kharenko; F. Frezza; F. Mangini; K. Stefanska; M. Ferraro; M. Gervaziev; S. Babin; S. Wabnitz; V. Couderc; W.A. Gemechu

REVIEW 4 major objections 5 minor 1 cited by

Negative absolute temperature attractor in a dense photon gas

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that increasing input power in a few-mode step-index fiber inverts the mode population from the fundamental to the highest-order mode, which it interprets as the optical temperature passing through infinity to a negative…

desk verdict Solid experiments on modal inversion, but the negative-temperature attractor is not supported: the sign flip is forced by a fixed-U_L calibration and an unproved equation of state. read the letter →

arxiv 2505.21163 v1 pith:VQKRMEFD submitted 2025-05-27 physics.optics

M. Ferraro , F. Mangini , K. Stefanska , W.A. Gemechu , F. Frezza , V. Couderc , M. Gervaziev , D. Kharenko

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S. Babin S. Wabnitz

This is my paper

classification physics.optics

keywords negativeabsolutetemperaturephotongasmultimodefiberfew-modemodepopulationinversionspatialbeamfreezingRayleigh-JeansdistributionKerrnonlinearity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to show that a dense gas of photons inside a few-mode optical fiber behaves like a non-ideal, Van der Waals gas, and that this has a dramatic consequence: as the input laser power grows, the equilibrium distribution of light among the fiber modes inverts, with the highest-order mode taking over from the fundamental mode. The authors interpret this as the optical temperature of the photon gas increasing until it crosses infinity and flips sign, landing in a negative-absolute-temperature regime that approaches zero from below. They call this final state "spatial beam freezing" and argue that it is a stable attractor, so at high power the output beam shape is essentially independent of injection conditions and external perturbations. The significance is practical as well as fundamental, because it offers an all-optical way to control laser beam shape for high-power sources, nonlinear imaging, and optical communications.

What carries the argument

The carrying object is the modified equation of state $U_L + U_{NL} - \mu P = M T$, paired with the Rayleigh-Jeans mode distribution $|c_i|^2 = -T/(\mu + \beta_i)$, where $\beta_i$ are the propagation constants of the guided modes. The new term is the nonlinear potential energy $U_{NL} = \frac{\gamma}{2}\frac{P^2}{M}$, which is quadratic in total power and inversely proportional to the number of modes, so it acts like a density-dependent interaction energy of a Van der Waals gas. Holding $U_L$ fixed at its low-power value while solving the resulting equation for $T/P$ makes the temperature a nonlinear function of power: it rises, diverges at the equipartition condition, and then approaches $0^-$ as the highest-order mode becomes the only populated state. The sign flip at $T = \pm\infty$ is not a singularity of the model, because the Lagrange multiplier conjugate to momentum conservation is $1/T$.

What would settle it

Recompute the linear internal energy from the experimentally measured mode occupancies at each power level instead of carrying the lowest-power value through all powers, then solve the paper's Eq. (8) with that updated value; if the temperature no longer flips sign or the $T = 0^-$ branch disappears, the negative-temperature attractor would be an artifact of the fixed-energy assumption.

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Extended reading notes

Core claim

The paper's central claim is that adding a Kerr-type potential energy to the equation of state of a photon gas in a few-mode fiber changes the power dependence of the optical temperature so drastically that the mode population spontaneously inverts. In the weakly nonlinear ideal-gas regime, the Rayleigh-Jeans distribution with conserved power and momentum gives an optical temperature whose sign is fixed by the linear internal energy; the paper's modified equation of state, $U_L + U_{NL} - \mu P = M T$, with $U_{NL} = \frac{\gamma}{2}\frac{P^2}{M}$, makes $T$ a nonlinear function of power. As power rises, $T$ increases, diverges at the equipartition point $U_L = U_{L,c}$, then returns from $T = 0^-$, which corresponds to the highest-order mode being the only populated state. In the experiment, the measured mode occupancy follows the Rayleigh-Jeans form at every power, with the highest-order guided modes carrying about 90% of the power at the highest input powers, and the same freezing is observed at other wavelengths where the number of guided modes differs.

Load-bearing premise

The whole temperature-versus-power curve is computed while holding the linear part of the internal energy fixed at its lowest-power value, even though the measured mode populations redistribute strongly with power; if that energy value actually changes, the inferred temperatures, the sign flip, and the negative-temperature branch would not follow.

Editorial extensions

If this is right

At low power the output is dominated by the fundamental mode; above a threshold the same fiber outputs a beam dominated by the highest-order guided modes, so beam quality degradation appears as a predictable thermodynamic transition rather than an uncontrolled defect.
The $T = 0^-$ state is stable: changing the launch offset, bending the fiber, or squeezing it does not change the final beam, which is what makes the effect usable as a form of spatial mode locking.
Because freezing occurs at each wavelength slice of a spectrally broadened pulse, the mechanism extends to polychromatic beams, with the frozen mode being the highest-order mode guided at that wavelength.
The inversion happens at power levels well below the self-focusing damage threshold, so it is accessible in ordinary step-index optical fibers.
Regardless of whether the low-power gas starts with positive or negative temperature, raising the density always drives it toward $T = 0^-$ rather than away from it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The model's key quantitative step is to keep the linear internal energy $U_L$ fixed at its lowest-power value while solving for the temperature at every power; because the measured mode populations redistribute strongly with power, recomputing $U_L$ from the measured distribution at each power could shift or remove the sign flip.
A testable extension is to check whether the critical power for the inversion scales as predicted with the mode count $M$, since the added potential energy is proportional to $P^2/M$; varying the fiber core or wavelength to change $M$ would directly probe the density mechanism.
The attractor behavior may be a more general phenomenon than this specific fiber model: any nonlinearity that adds a positive, power-dependent term to the internal energy could push a Rayleigh-Jeans gas toward $T = 0^-$, so similar inversion should appear in other multimode nonlinear systems.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has a credible experimental discovery—power-induced modal inversion in a few-mode fiber, robust to offset injection—but the headline claim of a negative absolute temperature attractor is not supported. The thermodynamic analysis has a load-bearing calibration step that forces the sign flip.

What's new: the observation that high power drives the output modal distribution from the fundamental to the highest-order modes, with the highest-order pair holding about 90% of power, is new and well documented. The cut-back, power conservation, and offset experiments are the right checks. The polychromatic freezing across spectral slices is a nice bonus. The authors are also honest: they explicitly state that Eq. (7), the modified equation of state, cannot be derived from the conservation laws.

Where it falls down: the determination of T in Methods. They compute U_L once from the lowest-power mode occupancy and keep it constant for all powers. But the experiment's own observable—modal redistribution—means U_L changes substantially. For a fixed input profile rescaled in power, U_L should scale with P, not stay fixed. Holding U_L constant forces U_L/P to drift toward zero as P grows, which pushes the system across the critical value in Eq. (5) and produces the divergence of T and the eventual T→0− branch. In other words, the sign flip is baked into the calibration. The equation of state is also a guess, not derived, and γ is fitted to the highest-power point, which is exactly where the attractor is claimed. Removing the fixed-U_L assumption would give different temperatures, and it is entirely possible the sign flip disappears or at least shifts. Since T is not measured, only inferred from this model, the negative-temperature claim is not established. The modal inversion could be real physics explained by something else (e.g., nonlinear mode coupling), but the paper does not make that case.

Bottom line: the experiments are worth engaging with, and a serious referee should see this. The paper needs major revision: either derive the equation of state, measure T directly, or at minimum recompute temperatures with a self-consistent U_L(P) from the measured mode occupancies. As it stands, I would not cite the negative-temperature claim, but I would cite the empirical freezing effect.

Recommendation: send to peer review, with a note to the authors that the fixed-U_L step is the central issue.

Referee Report

4 major / 5 minor

Summary. The paper reports experiments in a few-mode step-index fiber (SMF-28 operated below cutoff) in which increasing input power causes the output mode occupancy to redistribute from the fundamental mode to the highest-order modes, with the highest-order LP21 modes carrying about 90% of the power at the largest powers. The authors interpret this redistribution thermodynamically: they retain the Rayleigh-Jeans distribution, add a Kerr-type potential energy U_NL = γP²/(2M), modify the equation of state to U_L + U_NL − μP = MT, and infer from this model that the optical temperature rises with power, diverges at an equipartition point, and then approaches zero from below, defining a negative-absolute-temperature attractor. Supporting experiments include cut-back tests for thermalization, power-conservation checks, offset-injection robustness measurements, and observations at several wavelengths.

Significance. If the thermodynamic interpretation is correct, the paper would be a notable experimental demonstration of a negative optical temperature in a multimode waveguide and of a robust, all-optically controlled modal inversion, with potential interest for beam shaping and high-power fiber lasers. The experimental core is valuable: the modal decomposition is state-of-the-art, the cut-back and power-conservation checks support the claim of a thermalized, lossless propagation, and the offset-injection data provide a convincing demonstration of the robustness of the high-order-mode dominance. However, the central quantitative claim—that the system reaches a negative temperature approaching zero from below—is not directly measured but is inferred from a model that contains an unproved equation of state and a questionable calibration procedure. Those issues are load-bearing and require substantial revision before the negative-absolute-temperature attractor claim can be accepted.

major comments (4)

[Methods, 'Determination of thermodynamic parameters'] The procedure of computing U_L once from the lowest-power mode occupancy and then 'keeping it as a constant for all power levels' when solving Eq. (8) is inconsistent with the definition U_L = −Σβ_i|c_i|² in Eq. (2). The measured modal redistribution from LP01 to LP21 directly implies that U_L/P changes substantially with power; for an input profile that is simply rescaled in power, U_L/P should remain near −β_1, whereas fixing U_L drives U_L/P toward zero as P grows. This forced crossing of the critical value −Σβ_i/M in Eq. (5) is what produces the temperature divergence and the subsequent T→0⁻ branch in the authors' Fig. 2d and Fig. 3b. Since the central claim of a negative-absolute-temperature attractor depends on this branch, the authors must either recompute T using a power-dependent U_L(P) obtained from the measured occupancies or provide a physical justification for why U_L, as used in Eq. (8), should be the low-power linear energy rather than the instantaneous kinetic energy.
[Main text, 'Real photon gas'] Equation (7), U_L + U_NL − μP = MT, is acknowledged in the manuscript to 'cannot be directly derived' from the Rayleigh-Jeans law and the conservation laws of P and U. This relation is not a minor technicality: it is the equation used to convert the measured mode occupancies into the optical temperature values plotted in Fig. 3b and to locate the T→0⁻ attractor. Because the equation of state is not derived, the quantitative temperature axis, the critical power at which T diverges, and the strength of the attractor are all model-dependent. The authors should either derive Eq. (7) from a controlled approximation of the nonlinear Hamiltonian or treat the model as a purely phenomenological fit and clearly separate the directly measurable quantities (mode occupancies, U_L) from the inferred thermodynamic parameters.
[Fig. 3 and Methods, 'Determination of thermodynamic parameters'] The nonlinear coefficient γ is obtained by fitting Eq. (8) to the highest-power data point, and the same point is the one that most strongly determines the negative-temperature branch. Consequently, the agreement between the theoretical curve and the experimental occupancies at the highest power in Fig. 3a is not an independent validation of the model. The authors should provide a sensitivity analysis showing how the inferred temperatures and the predicted critical powers vary with γ and with the choice of the reference power used to fix U_L, including confidence intervals on γ from the fit.
[Main text, 'Robustness of spatial beam freezing'] The statement that the negative-temperature attractor occurs 'no matter the sign of the temperature at low power' is supported experimentally by the offset-injection measurements in Fig. 4, but those measurements are shown at only two power levels and are compared with the model only qualitatively. The theoretical statement of a universal attractor to T→0⁻ in Fig. 2c,d is computed from the unproved equation of state (7) and the fixed-U_L procedure. To make the attractor claim quantitative, the authors should report for the offset data the model-inferred temperatures as a function of power and verify that they indeed converge to the same negative near-zero value regardless of the low-power sign of T.

minor comments (5)

[Abstract and Main text] The paper uses 'demonstrate' for the negative absolute temperature regime, but T is always inferred from the model rather than directly measured. The wording should be softened, e.g., 'consistent with' or 'inferred from the thermodynamic model,' and the distinction between measured mode occupancies and inferred thermodynamic parameters should be made explicit in the abstract.
[Fig. 2 caption / Fig. 3] The normalized nonlinear coefficient γ takes the value 0.05 in Fig. 2 but 2.04 in Fig. 3, with no statement of units or normalization. Please specify how γ is normalized, and report the fitted value with a confidence interval.
[Methods, 'Experiments'] The manuscript reports average input powers (e.g., 0.1 mW, 3.3 mW, 4 mW) for femtosecond pulses at 100 kHz repetition rate, but the thermodynamic description and the nonlinear potential energy depend on the instantaneous power or pulse energy. The authors should report peak powers or pulse energies and justify using average power in the thermodynamic fits.
[Eq. (8)] The sign conventions in the denominator of Eq. (8) are easy to misread; a short derivation in the Methods section showing how Eq. (8) follows from Eqs. (3) and (7) would improve clarity.
[Main text, 'Real photon gas'] The text states that U_NL remains about two orders of magnitude lower than U_L while also describing the gas as 'dense.' This apparent tension should be clarified: the non-ideal nature of the gas is invoked through the equation of state even when the potential energy is small compared with the linear energy.

Circularity Check

2 steps flagged · score 6.0 of 10

The negative-temperature branch is calibrated, not predicted: γ is fitted to the highest-power point and U_L is held fixed, forcing the sign flip and the T→0− attractor.

fitted input called prediction [Methods, 'Determination of thermodynamic parameters' (pages 8-9)]
"Specifically, we extracted the value of γ via nonlinear fit of eq. (8) at the highest input power. At this point, we fixed the value of γ and used eq. (8) to determine T corresponding to each input power. Within this procedure, the values of the thermodynamic parameters are determined by just two sets of experimental data, i.e., those at the highest and the lowest power."

The high-power negative-temperature state is not an independent prediction: the free parameter γ in the phenomenological potential (6) is fitted so that eq. (8) reproduces exactly the highest-power data point—the same point at which the highest-order modes dominate and the paper claims T→0−. The theoretical T(P) curve and the assignment of a negative, near-zero temperature to state E are therefore imposed by the fit rather than derived from the data. The intermediate-power agreement provides some independent support, but the existence of the negative-temperature branch at high power is calibrated in, not predicted.
self definitional [Methods, 'Determination of thermodynamic parameters' (page 9)]
"At first, we computed UL by using eq. (2) from the data at the lowest power. Such a value of UL is kept as a constant for all power levels, when determining the thermodynamic parameters with eq. (8)."

Holding U_L fixed as a constant while P grows makes U_L/P decrease monotonically through the critical value U_L,c/P, which is precisely the mechanism that produces the temperature divergence and the subsequent T→0− branch. But the measured modal occupancies redistribute strongly with power, so the linear internal energy -Σβ_i|c_i|^2 computed from the measured occupancies is not constant. The sign flip and the negative-temperature attractor are thus a consequence of the definitional choice to treat U_L as a fixed parameter rather than as the measured power-dependent quantity, not a consequence of the data alone.

full rationale

The empirical phenomenon—a power-induced inversion from the fundamental to the highest-order mode—is real and independently observed, and the authors are transparent that eq. (7) 'cannot be directly derived' from the RJ law and the conservation laws. However, the central negative-absolute-temperature claim does not stand independently of the fitting procedure. U_L is fixed at the lowest-power value, so U_L/P decreases with P and forces a crossing of the critical value that drives the sign flip; γ is fitted to the highest-power point, exactly where the model is claimed to yield T→0−. Thus the high-power branch of the T(P) curve is determined by the two anchor points used for calibration. The intermediate-power data and the offset-injection robustness measurements do provide genuine, non-circular support, so the paper is only partially circular. The main unsupported step is the assumption that the Rayleigh-Jeans law and the modified equation of state (7) remain valid in the dense regime; that is a correctness risk more than a circularity, but it is compounded by the fit at the highest power.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central thermodynamic interpretation rests on two unproven postulates: the validity of the Rayleigh-Jeans law in the nonlinear regime and the specific equation of state with the phenomenological potential energy. The potential energy form introduces a fitted coefficient gamma. The experimental observation of modal inversion does not require these postulates, but the negative-temperature claim does.

free parameters (1)

gamma (normalized nonlinear coefficient) = 2.04 (units not stated; fitted via eq. (8) at the highest input power)
Introduced in U_NL = gamma P^2/(2M) and fitted to the experimental mode occupancy at the highest input power (Methods). All subsequent T(P) values, the sign flip, and the negative-temperature branch depend on this fit.

assumptions (5)

ad hoc to paper The Rayleigh-Jeans distribution |c_i|^2 = -T/(mu + beta_i) remains valid in the strongly nonlinear regime.
The paper states 'we keep the same statistics as in the weakly nonlinear regime' (Real Photon Gas section), despite the ideal-gas theory being linear while the observed redistribution is nonlinear. This is assumed without derivation.
ad hoc to paper The modified equation of state U_L + U_NL - mu P = M T (eq. 7) holds.
The authors explicitly write that eq. (7) 'cannot be directly derived' from the RJ law and conservation laws, yet the negative temperature attractor is a direct consequence of this equation.
ad hoc to paper The potential energy takes the form U_NL = gamma P^2/(2M).
Chosen as 'the simplest law' scaling quadratically with power and inversely with mode number; no derivation from the nonlinear Schrodinger equation or mode overlap integrals is provided.
domain assumption The photon gas reaches thermal equilibrium over the fiber length.
The cut-back experiment (Supplementary Note 1) is used to support thermalization, but the theory assumes an equilibrium distribution; incomplete thermalization or a different equilibrium would invalidate the inferred T.
domain assumption Power and the total Hamiltonian are conserved, with no losses or cladding leakage.
Supported by Supplementary Note 2, but still assumed in the model. Any residual loss or leakage would alter the thermodynamic parameters and the inferred temperatures.

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Pith. "Pith review of Negative absolute temperature attractor in a dense photon gas." pith.science (2026). https://pith.science/paper/VQKRMEFD

@misc{pith2026250521163,
  author       = {Pith},
  title        = {Pith review of: Negative absolute temperature attractor in a dense photon gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQKRMEFD}},
  note         = {Machine review of arXiv:2505.21163}
}

read the original abstract

Statistical mechanics permits to connect the macroscopic properties of matter with the laws governing the evolution of its microscopic constituents. Such an approach has been very successful for systems of particles governed by either classical or quantum mechanics. In a classical gas, different thermodynamic laws apply to the weakly or strongly interacting particles of an ideal or real gas, respectively. Here, we demonstrate that a similar situation occurs for a gas of photons, which is contained in a finite-dimensional box such as a multimode waveguide. We use a few-mode system provided by a standard step-index fiber operated below cutoff, which permits to prepare a high-density gas of photons. We show that, owing to the attractive potential energy contribution to the photon energy induced by the nonlinear Kerr effect, the mode population exhibits a spontaneous inversion from the fundamental to the highest-order mode, as the input laser beam power grows larger. This inversion of the mode power distribution leads to a stable attractor for the output beam, and is associated with a progressive increase of the optical temperature until a flip of its sign leads to a new regime of negative absolute temperatures. Our work demonstrates the ability to all-optically control the shape of laser beams, which is a prerequisite for applications in high-power laser sources, nonlinear imaging, and optical communication systems.

Figures

Figures reproduced from arXiv: 2505.21163 by the authors.

**Figure 2.** FIG. 2. (a) Propagation constant vs. mode number. The black dashed horizontal line corresponds to the value of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. a) Modal distribution inversion in a 15 m long optical fiber. The image pairs in the insets correspond to the measured [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Robustness of spatial beam freezing in a 50 m long optical fiber. On the top: variation of the output near field vs. the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Spatial beam freezing with varying number of modes in a 50 m long optical fiber. (a) Output spectrum for 5 mW [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Example of beam evolution along the fiber, for an average input power of approximately 4 mW for all the cases. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7. Experimental evidence of power conservation. Output vs. input power. The dashed line is a linear fit, with linear [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures
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Guided spatio-temporal light is predicted to undergo complete (2+1)D Bose–Einstein condensation at positive temperature (anomalous dispersion, lowest mode) and at negative temperature (normal dispersion, highest mode)...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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