Quantum Statistics of Two Identical Particles and Modified Hong-Ou-Mandel Interferometer

Kicheon Kang; Won-Young Hwang

REVIEW 2 major objections 5 minor 11 references

Quantum Statistics of Two Identical Particles and Modified Hong-Ou-Mandel Interferometer

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that the probability for two identical particles to occupy different sites at thermal equilibrium is exactly $(e^{\beta\Delta}+1)/(3e^{\beta\Delta}+1)$ for bosons and $(e^{\beta\Delta}+1)/(e^{beta\Delta}+3)$ for…

desk verdict The equilibrium two-particle formula is clean and correct, but the beam-splitter-array proposal has a missing finite-temperature assumption that the current text glosses over. read the letter →

arxiv 2505.12675 v2 pith:5LUSDGIZ submitted 2025-05-19 quant-ph cond-mat.mes-hallcond-mat.stat-mechphysics.atom-ph

Won-Young Hwang , Kicheon Kang This is my paper · ORCID

classification quant-phcond-mat.mes-hallcond-mat.stat-mechphysics.atom-ph

keywords identicalparticlesquantumstatisticsHong-Ou-Mandelinterferometerthermalequilibriumdistinguishabilitybosonsfermionsunitaryinvariance

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 establishes a precise, temperature-dependent probability $P(1,1)$ for two identical particles to sit in different locations at thermal equilibrium: $(e^{\beta\Delta}+1)/(3e^{\beta\Delta}+1)$ for bosons and $(e^{\beta\Delta}+1)/(e^{\beta\Delta}+3)$ for fermions. At low temperatures these reduce to the ideal quantum values $1/3$ and $1$, and at high temperatures both approach the classical value $1/2$, so the particles become effectively distinguishable as thermal energy exceeds the internal level spacing. The authors further show that any unitary separation process that commutes with the internal Hamiltonian leaves this equilibrium distribution unchanged, which allows the distribution to be measured with a modified Hong-Ou-Mandel interferometer after the particles equilibrate. This provides a concrete, realistic experimental route to probe the transition between quantum and classical statistics of two identical particles.

What carries the argument

The key object is the internal energy-level structure of each particle, with equal spacing $\Delta$. At low temperatures only the ground internal state is occupied, so the particles remain indistinguishable and quantum statistics dominate; at high temperatures the particles occupy different internal levels and become effectively distinguishable, driving the distribution toward the classical value. The argument that the equilibrium distribution is preserved during measurement rests on the commutation relation $[H,S_T]=0$: because the single-particle Hamiltonian is proportional to the identity in the spatial basis for each internal level, any unitary scattering on the spatial degrees of freedom commutes with the internal Hamiltonian. Hence the thermal density matrix $\rho_{\rm in}=e^{-\beta H}/Z_2$ is invariant under the separation process, and the measured two-particle distribution equals the equilibrium distribution.

What would settle it

Measure the two-particle coincidence probability $P(1,1)$ for two equilibrated bosons or fermions as a function of temperature; if the measured values do not follow $(e^{\beta\Delta}+1)/(3e^{\beta\Delta}+1)$ for bosons or $(e^{\beta\Delta}+1)/(e^{\beta\Delta}+3)$ for fermions, including the intermediate mesoscopic regime, the central claim would be refuted.

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Extended reading notes

Core claim

The central discovery is an exact analytical formula for the two-particle equilibrium distribution of identical bosons and fermions when each particle has equally spaced internal energy levels. The probability $P(1,1)$ that the two particles occupy two different sites is $(e^{\beta\Delta}+1)/(3e^{\beta\Delta}+1)$ for bosons and $(e^{\beta\Delta}+1)/(e^{\beta\Delta}+3)$ for fermions, interpolating smoothly between pure quantum statistics at $kT\ll\Delta$ and classical statistics at $kT\gg\Delta$. The paper also proves that this distribution is invariant under any unitary scattering or separation process that commutes with the internal Hamiltonian, because the equilibrium density matrix is a function of that Hamiltonian and therefore also commutes with the scattering matrix. Consequently, a two-particle interferometer in which the particles first reach statistical equilibrium and then are separated unitarily will directly reveal the equilibrium distribution without distortion.

Load-bearing premise

The separation or scattering process during measurement is unitary and acts identically on all internal levels, with no energy exchange between them, so that the scattering matrix commutes with the internal Hamiltonian.

Editorial extensions

If this is right

At temperatures much smaller than the level spacing, the distribution recovers exactly the quantum-statistical values: $1/3$ for bosons and $1$ for fermions; at very high temperatures both approach $1/2$, the classical value.
The transition between quantum and classical behavior is smooth and controlled by the dimensionless ratio $kT/\Delta$, so an experiment can tune continuously between the two regimes.
The measurement scheme applies to both bosons and fermions and can be implemented with photons, electrons, or cold atoms, building on existing two-particle interference experiments.
Once the two-particle system has reached equilibrium, further beam-splitter scattering leaves the distribution unchanged, so the output of the modified interferometer directly reflects the equilibrium statistics.
The invariance argument is general: for any equilibrium state that is a function of a Hamiltonian commuting with a unitary measurement process, the measured distribution equals the equilibrium distribution.

Reading between the lines

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

The same unitary-invariance argument could be extended to equilibrium states of more than two particles, provided the Hamiltonian and the scattering process still commute.
The smooth temperature dependence of $P(1,1)$ offers a quantitative, experimentally accessible measure of 'effective distinguishability' that could be compared with other operational measures of indistinguishability.
A direct way to test the core assumption would be to engineer a state-dependent beam splitter: if $P(1,1)$ changes when the potential differs between internal levels, the assumption $[H,S_T]=0$ is violated; if it remains unchanged, the invariance is confirmed.
The result suggests that the standard dichotomy between quantum indistinguishability and classical distinguishability is incomplete, and that the crossover can be probed as a function of temperature in a single experimental platform.

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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

You should know two things about this paper. The exact partition-function calculation leading to Eq. (3) is correct and deserves to be cited: for two identical particles with equally spaced internal levels in thermal equilibrium, P(1,1) interpolates from 1/3 (bosons) or 1 (fermions) at low T to 1/2 at high T. That derivation is parameter-free and clean. The unitarity argument in Eqs. (7)–(13) is also sound: if [H,S_T]=0, the equilibrium density matrix is invariant under unitary separation, so the second proposed setup (equilibration followed by a barrier rise) measures what it claims.

The soft spot is the first experimental proposal—the dephasing beam-splitter array. The paper states after Eq. (6) that "this property is preserved for finite temperatures," meaning the array drives the system to the Eq. (3) distribution. That is asserted, not derived, and as stated it is false. The recurrence (5a) dephases only off-diagonal elements in the two-site basis. It leaves every internal-level occupation weight w_{n,m} invariant, because the beam-splitter matrix S is the same for every n and does not couple internal levels. If both particles enter in their ground internal state—the natural Hong-Ou-Mandel input—the weights remain concentrated on n=m=ground forever, and the array converges to P(1,1)=1/3, not to Eq. (3). To reach Eq. (3) via the array, the input internal populations must already be Boltzmann-distributed with e^{-β ε_n}. The paper never states or prepares that condition. This is a missing assumption in the experimental protocol, not an error in the equilibrium formula itself.

The second setup avoids this issue because equilibration happens via contact or a bath before the unitary separation; there the argument works. But the paper's general claim that both schemes directly measure the equilibrium distribution is too strong unless the internal-state preparation problem is addressed. The self-citation (Ref. 10) appears only in the discussion and is not load-bearing. The citations to existing HOM and cold-atom experiments are appropriate.

Who should read this? Anyone working on identical-particle statistics, quantum-to-classical transitions, or two-particle interferometry. The theoretical core is solid, and the proposed cold-atom experiment is close to what Kaufman et al. already demonstrated, so with a corrected beam-splitter-array story the paper would be a useful contribution.

Recommendation: send it to peer review. The flaw is significant but localized to one scheme; a competent referee can require the authors to either fix the finite-temperature claim or explicitly restrict the array proposal to pre-thermalized internal states. Not a desk reject.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the equilibrium probability P(1,1) that two identical particles occupy two different sites, when each particle carries an internal spectrum with equal spacing Delta. The authors derive the analytical formulas in Eq. (3) from the two-particle partition function, showing a crossover from purely bosonic/fermionic values at low temperature to the classical value 1/2 at high temperature. They then propose two modified Hong-Ou-Mandel setups: (a) a series of beam splitters with strong dephasing, and (b) equilibration followed by a unitary separation process. The central theoretical result is that any unitary separation process commuting with the internal Hamiltonian preserves the equilibrium distribution, which is proved in Eqs. (9)-(13).

Significance. The equilibrium formula Eq. (3) and the invariance theorem under unitary separation are clean, parameter-free results that could be useful for probing quantum statistics in cold-atom or electron interferometry, and the conceptual link between internal-level thermal occupation and effective distinguishability is interesting. However, the finite-temperature claim for the dephasing beam-splitter array is not merely underived but incorrect as stated, which affects one of the two proposed experimental schemes.

major comments (2)

[Two-particle interferometer section, after Eq. (6)] The claim that the dephasing beam-splitter array reaches the finite-temperature equilibrium of Eq. (3) is not derived and, as stated, is false. The recurrence (6) applies only to a fixed internal configuration with both particles in the same internal level. For n != m, the two particles are effectively distinguishable by their internal quantum numbers: the amplitudes for |p_n q_m> and |q_n p_m> do not interfere, and the stationary spatial distribution after dephasing is (1/4, 1/4, 1/2) for |pp>, |qq>, |pq>, not (1/3, 1/3, 1/3). Since S is independent of n, the weights of internal configurations are invariant under the recurrence. Even with Boltzmann-initialized internal populations, the array's fixed point differs from Eq. (3); for an infinite equally spaced ladder with x = e^{-beta Delta}, the array gives P_infty(1,1) = 1/2 - (1-x)/(6(1+x)) for bosons, which disagrees with Eq. (3a) at intermediate temperatures. The authors must derive the correct multi-level recurrence or withdraw the finite-temperature claim for Fig. 3(a).
[Two-particle interferometer section, Eq. (5c)] The notation in Eq. (5c) suppresses internal indices, writing rho_i as a combination of |pp><pp|, |qq><qq|, and |pq><pq|. This hides the fact that a_i, b_i, and c_i are conditioned on the internal configuration and that the Diag operation removes only spatial coherences, not the internal-level populations. Pure dephasing of the spatial path does not thermalize the internal degrees of freedom; a bath or an inelastic process is required to reach the canonical distribution of Eq. (3). As written, the protocol for Fig. 3(a) provides no mechanism for this internal thermalization, so the statement that the two particles reach statistical equilibrium is unsupported.

minor comments (5)

[Introduction] There is a typo: 'exmaple' should be 'example'.
[Eq. (5a)] The operation Diag is not defined; the authors should state whether it discards all off-diagonal elements of the density matrix in the full multi-level basis or only those in the spatial site basis.
[Eq. (6)] The derivation of the recurrence is not shown, and the phase convention for the scattering matrix S is left unspecified; the authors should justify that the recurrence depends only on R and T and not on the phases of r, r', t, t'.
[Eq. (3) and Fig. 2] The derivation of Eq. (3) implicitly assumes an infinite number of equally spaced internal levels; with a finite number N of levels, the high-temperature limit of P(1,1) is not exactly 1/2 (for bosons it is 1/2 - 1/(6N)), which should be stated explicitly.
[Before Eq. (9)] The phrase 'without energy exchange between the particles' is imprecise; the proof requires that the scattering matrix is independent of the internal level and does not couple internal levels, which should be stated directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equilibrium distribution and its invariance under unitary separation are derived from stated assumptions in a parameter-free, self-contained way.

full rationale

The central result, Eq. (3), is obtained by explicitly evaluating the two-particle canonical partition function Z2 = Tr e^{-βH} using internal levels of spacing Δ; the steps from Eqs. (2a)-(2c) to Eq. (3) are self-contained algebra and do not rely on fitting or on importing the target result. The measurement claim is supported by a direct theorem: for ρin = e^{-βH}/Z and any unitary ST commuting with H, one has [ρin, ST]=0 and therefore ρout = ST ρin ST† = ρin (Eqs. (7)-(13)). This is not circular because the invariance is derived from the stated commutator [H,ST]=0 rather than assumed. No fitted parameter is renamed as a prediction, and no load-bearing self-citation appears; Ref. [10] is cited only in a general closing remark about non-overlapping particles. The finite-temperature convergence of the beam-splitter array in Fig. 3(a) is asserted rather than derived ('It can be shown that this property is preserved for finite temperatures'), which is a missing justification or a correctness concern, but it is not a circular reduction because Eq. (3) and the unitary-invariance result do not depend on that assertion. Overall, the derivation chain has independent content and is not equivalent to its inputs.

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

The central derivation relies only on standard equilibrium statistical mechanics and the assumption of state-independent, level-preserving scattering. No free parameters are fitted. The main burden is on the experimental realizability of unitary, state-independent separation and efficient dephasing.

assumptions (5)

domain assumption The two-particle system reaches thermal equilibrium described by the canonical ensemble ρ = e^{-βH}/Z
Used throughout; the transition and measurement claim depend on the particles equilibrating before measurement. Entered in the pedagogical setup and Eq. (7).
domain assumption Each particle has an infinite ladder of equally spaced internal levels, and the spacing Δ is the same for all particles
Used to obtain the closed forms Eq. (3); real systems have finite or anharmonic levels, though the qualitative result survives.
domain assumption The beam splitter acts identically on every internal level and does not couple different internal levels
Needed for Eq. (4a) and for [H, S_T] = 0; state-dependent potentials or internal transitions would break the result.
domain assumption The dephasing in the beam-splitter array is strong enough to fully discard off-diagonal coherences
Needed for Eq. (5a) to drive the system to the equilibrium distribution; without it the unitary evolution does not increase entropy.
domain assumption The separation process after equilibration is unitary and commutes with H
Core to the measurement claim in Eqs. (10)-(13); any loss or internal-state-dependent scattering would alter the measured distribution.

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Cite this review

Pith. "Pith review of Quantum Statistics of Two Identical Particles and Modified Hong-Ou-Mandel Interferometer." pith.science (2026). https://pith.science/paper/5LUSDGIZ

@misc{pith2026250512675,
  author       = {Pith},
  title        = {Pith review of: Quantum Statistics of Two Identical Particles and Modified Hong-Ou-Mandel Interferometer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LUSDGIZ}},
  note         = {Machine review of arXiv:2505.12675}
}

read the original abstract

We propose an experimental scheme to probe the quantum statistics of two identical particles. The transition between the quantum and classical statistics of two identical particles is described by the particles having identical multiple internal energy levels. We show that effective distinguishability emerges as the thermal energy increases with respect to the energy level spacing, and the mesoscopic regime bridges quantum indistinguishability and classical distinguishability. A realistic experimental approach is proposed using a two-particle interferometer, where the particles reach statistical equilibrium before the two-particle distribution is measured. The unitarity of the scattering/separation process ensures the preservation of the equilibrium distribution and allows a direct measurement of the two-particle statistical distribution. Our results show the transition between quantum and classical behavior of the two-particle distribution, which can be directly probed by a realistic experiment.

Figures

Figures reproduced from arXiv: 2505.12675 by the authors.

**Figure 3.** FIG. 3: Proposed experimental schemes for the measurement [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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For fermions, there is only one possible state, ( |pq⟩ − |qp⟩)/ √ 2, due to the Pauli principle. These statistical properties can be found in the probability of the two particles being in two diﬀerent locations, P (1, 1). For classical particles, bosons, and fermions, this proba- bility is 1 / 2, 1 / 3, and 1, respectively. All matter obeys the laws of qu...

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C.-K. Hong, Z.-Y. Ou, and L. Mandel, Physical Review Letters 59, 2044 (1987)

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R. Liu, B. Odom, Y. Yamamoto, and S. Tarucha, Nature 391, 263 (1998)

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E. Bocquillon, V. Freulon, J.-M. Berroir, P. Degiovanni , B. Pla¸ cais, A. Cavanna, Y. Jin, and G. F` eve, Science 339, 1054 (2013)

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