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An examination of the extended Hong-Ou-Mandel effect and considerations for experimental detection

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that the extended Hong-Ou-Mandel effect is universal: for a lossless balanced 50:50 beam splitter, an odd-parity state in one input port produces a central nodal line of exact zeros along the diagonal of the joint…

desk verdict A useful diagrammatic and experimental addendum to the eHOM story, but Eq. (3) is missing the coherent-state amplitude factor, so the quantitative detection analysis needs correction before it can be trusted. read the letter →

arxiv 2501.04849 v1 pith:622PU4GH submitted 2025-01-08 quant-ph

Paul M. Alsing , Richard J. Birrittella This is my paper · ORCID
classification quant-ph MSC 81V80 PACS 42.50.-p
keywords Hong-Ou-MandeleffectextendedHOMcentralnodallinebeamsplitterinterferencecoherentstatephotonnumberparitycountingquantum
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 argues that the familiar two-photon Hong-Ou-Mandel effect generalizes far beyond two-photon inputs: on a lossless 50:50 beam splitter, any input state on one port built only from odd photon numbers forces the joint output probability for equal photon counts to vanish exactly, no matter what state is sent into the other port. This 'extended HOM' (eHOM) effect means a central nodal line of zeros runs down the diagonal of the coincidence detection distribution. The authors show the mechanism diagrammatically as a pairwise cancellation of mirror-image scattering amplitudes, each pair contributing equal magnitude and opposite sign. They then work out what an experiment would need to see the effect, using a single photon from a heralded source and a coherent-state (laser) input, and show that imperfect detection efficiency, mode functions, and time delay turn the exact zeros into measurable dips. A sympathetic reader would care because the result offers a clean, state-independent way to see a single-photon quantum effect on a bright classical beam.

What carries the argument

The mechanism is the pairwise cancellation of mirror-image scattering amplitudes for the coincidence output. Labelling A_k the amplitude in which k photons from port-1 transmit to output-1 (and the rest reflect), the total amplitude is sum_{k=0}^n A_k; for odd n the amplitudes pair as A_k + A_{n-k}, each pair having equal magnitude and opposite sign when the beam splitter is balanced, because the unitary scattering matrix carries a single -1 phase for reflection from mode-2 to mode-1. This tracing of the (-1) phase through the scattering diagrams is what converts the two-photon HOM cancellation into an n+1-fold chain of cancellations.

What would settle it

Send a single photon and a coherent state into a balanced 50:50 beam splitter with detectors capable of resolving photon number, arrange zero arrival-time difference and identical spatial/polarization modes, and count coincidences at (1,1). The central claim predicts exactly zero contribution from the interference term (the CS-only background can be measured and subtracted); observing a nonzero coincidence rate under these ideal conditions would falsify the universality of the nodal line.

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Extended reading notes

Core claim

The central claim is that the extended Hong-Ou-Mandel effect is real and universal: for a lossless, balanced 50:50 beam splitter, if the state injected into one input port has odd parity (only odd photon-number components), then the amplitude for any equal-count coincidence output |N,N> is a sum of mirror-image pairs that cancel exactly, giving P(N,N)=0 for every N, regardless of the state injected into the other port (pure or mixed). For even-parity inputs the mirror-image pairs have equal signs and an unpaired middle term survives, so no such zeros appear. For the experimentally relevant case of a single photon in one port and a coherent state in the other, the paper derives the joint detection probabilities and shows that in the idealized limit of unit efficiency, zero time delay, and perfect modal overlap, the diagonal probability P(N,N) tends to zero; realistic imperfections replace the zeros with small positive values that should still be observable with photon-number-resolving detectors at low N.

Load-bearing premise

The result collapses if the photons are not mutually indistinguishable: the derivation assumes every photon in both input ports occupies the same single spatio-temporal mode (apart from a modeled time delay), which means polarization, transverse spatial mode, and frequency-marked distinguishability are all taken to be perfect.

Editorial extensions

If this is right

  • Odd-parity inputs, such as a single photon, produce a central nodal line of exact zeros in the joint coincidence distribution for a lossless balanced beam splitter, independent of the second input state.
  • Even-parity inputs do not produce such zeros; instead the mirror-image amplitudes interfere constructively and an unpaired middle term remains.
  • For single-photon/coherent-state inputs, the diagonal probability P(N,N) vanishes in the idealized limit, so the effect is testable with photon-number-resolving detectors at low N.
  • Imperfect detector efficiency converts the exact zeros into small positive values proportional to eta^{2n}, so observation is easiest for N=1,2 coincidences.
  • At zero detection time delay the interference term vanishes regardless of the spatio-temporal mode functions, meaning the nodal structure survives for non-monochromatic photons as long as the photons are otherwise indistinguishable.

Reading between the lines

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

  • The state-independence of the central nodal line suggests a practical parity meter: any odd-photon-number state, however weak, leaves a detectable zero line against an arbitrarily bright coherent background, which could be used to certify single-photon character in noisy channels.
  • A differential measurement—recording the coincidence rate with and without the single photon—would isolate the interference term from the coherent-state background and could make the dip visible even at moderate detection efficiency.
  • The same pairwise cancellation argument should apply to other unitary two-port linear devices with equal transmission and reflection probabilities, such as directional couplers, provided the -1 phase structure is preserved.
  • The result implies that a coherent state, despite being classical-like, cannot destroy the single-photon interference node; this sharpens the distinction between the nonclassicality of the odd-parity state and the classicality of its partner state.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. Drawing on the authors' earlier work on the extended Hong-Ou-Mandel (eHOM) effect, this manuscript has two goals. First, it presents a diagrammatic account in which the vanishing of the coincidence amplitude for odd-odd Fock inputs |n,m> to a balanced lossless beam splitter is seen as pairwise cancellation of mirror-image scattering amplitudes, generalizing the two-photon HOM effect. Second, it analyzes a proposed experimental realization in which a single photon in one input port interferes with a coherent state in the other, adding imperfect detection efficiency, spatio-temporal mode functions, and detection time delays. The main quantitative results are the joint output probability P(N1,N2) for the |1,beta> input, the smeared coincidence probability under inefficient detection, the time-domain two-photon coincidence rate, and a Kelly-Kleiner photon-counting expression for the diagonal probability P_eta(N,N). The paper claims that in the ideal limit these expressions reduce to the central nodal line (CNL) of zeros.

Significance. The eHOM theorem—that an odd-parity state in one input port forces a diagonal line of zeros in the joint output distribution regardless of the other input state—is a striking and nontrivial consequence of bosonic interference, previously established by the authors in [1,2]. The present paper's diagrammatic interpretation is instructive, and its explicit treatment of detector inefficiency, mode functions, and time delays is a useful step toward an experiment. The paper is also honest about its idealizations: it assumes all photons occupy a single spatio-temporal mode and explicitly notes the degradation of the CNL under imperfect detection. However, the manuscript's new quantitative contribution is compromised by the error in Eq. (3), which affects all numerical predictions in Section III. The qualitative CNL claim is not in question, but the experimental feasibility analysis must be corrected before the paper can be recommended.

major comments (2)
  1. [Section III.A, Eq. (3) and footnote [29]] Eq. (3) is missing the factor |beta|^{2(N1+N2-1)}. Convolving Ou's result for |1,N>, P(N1,N2) = N!/(N1! N2! 2^{N+1}) (N1-N2)^2 delta_{N1+N2,N+1}, with the coherent-state photon-number distribution |c_N|^2 = e^{-|beta|^2} |beta|^{2N}/N! gives e^{-|beta|^2} |beta|^{2(N1+N2-1)} / (N1! N2! 2^{N1+N2}) (N1-N2)^2, as the paper's own footnote states. The published equation omits the |beta| factor, so all off-diagonal probabilities in Section III and the smeared-CNL sums in Eq. (4) are quantitatively incorrect. The diagonal zero is unaffected, so the qualitative CNL claim survives, but the numerical basis for the experimental feasibility discussion must be redone.
  2. [Section III.A, Eq. (4)] The second line of Eq. (4) is not equal to the first line. The first line already contains eta^{2n} in the summand; the second line inserts an additional leading factor eta^{2n} while retaining eta^{2n} inside the sum. The equality should read 2 times the off-diagonal double sum with a single eta^{2n} (or the corresponding corrected expression once Eq. (3) is fixed). As written, Eq. (4) misstates the magnitude of P_eta(n,n), although the positivity conclusion is unaffected.
minor comments (5)
  1. [Section IV.C, Eq. (14)] The second factor in the Kelly-Kleiner formula is written with a_{2,out} a^dagger_{2,out}; it should be a^dagger_{2,out} a_{2,out} to match the first factor and the normal-ordering prescription.
  2. [Section IV.B, after Eq. (12)] The manuscript should state explicitly that P^{|1,beta>}_{11,12}(t0,tau) is a two-time Glauber correlation function, not the number-resolved joint probability P(N1,N2) of Eq. (3). The nonvanishing DC term in Eq. (12a) is then understandable, but the connection to the CNL of the number-resolved distribution should be clarified at this point; it is addressed later in Section IV.C.
  3. [Section II] The paper should state at the outset that the general eHOM theorem is taken from Refs. [1,2]; the present Section II is an illustrative pairwise-cancellation argument for the outermost and penultimate pairs, with the general proof left to the references.
  4. [Throughout] There are several typographical errors that should be corrected, including 'their is no possibility' (Section II.A), 'similary' (Section II.A), 'discreetness' (Section V), and 'photon-electron' (Ref. [34], which should be 'photoelectron').
  5. [References] Reference [2] is an arXiv preprint; if a peer-reviewed version has appeared by the time of publication, it should be cited in its place.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the eHOM theorem is referenced from prior peer-reviewed work with a representative proof sketch, and the experimental formulas are independent applications of standard detection theory.

full rationale

The paper's central CNL statement is inherited from the authors' earlier papers [1,2]; Section II supplies explicit pairwise cancellation for representative amplitude pairs and refers to [2] for the full general proof. This is a normal citation of prior peer-reviewed mathematical work, not a reduction of the claim to a self-citation: the assumptions (lossless balanced beam splitter, odd-odd Fock inputs) do not include the diagonal-zero conclusion, and the cited proof has independent derivational content. The experimental sections are self-contained: Eq. (3) is a direct application of the beam-splitter transformation to |1,beta>; Eqs. (4), (12)-(20) follow from standard Glauber detection theory and the Kelly-Kleiner photon-counting formula. In particular, Eq. (20b) is a limit of an independently computed expression, so the ideal CNL is recovered rather than assumed as an input. No fitted parameter is relabeled as a prediction, no ansatz is smuggled in solely through a self-citation, and no known result is merely renamed as a new derivation. A separate non-circular correctness concern: Eq. (3) appears to omit the coherent-state amplitude factor |beta|^{2(N1+N2-1)} obtained by convolving Ou's |1,N> result with the coherent-state photon-number distribution; this affects off-diagonal probabilities and the smeared-coincidence analysis, though it does not affect the diagonal zero that constitutes the CNL.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on standard quantum optics and on the authors' previously published eHOM theorems, which are assumed rather than re-proven here.

assumptions (4)
  • domain assumption Lossless balanced beamsplitter with unitary scattering matrix S
    Used throughout; standard quantum optics model.
  • domain assumption eHOM pairwise cancellation for odd-odd Fock inputs |2n+1, 2m+1>
    Proven in refs. [1] and [2]; taken as the starting point for the diagrammatic explanation in Section II.
  • ad hoc to paper Odd-parity input at one port implies CNL for any port-2 state
    This universality claim is stated in the abstract and used in Sections III and IV; it rests on the Fock-basis completeness argument from ref. [1].
  • domain assumption Glauber detection theory and Kelly-Kleiner photon counting formula correctly model realistic detectors
    Used in Sections IV B and IV C to compute joint detection probabilities.

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Pith. "Pith review of An examination of the extended Hong-Ou-Mandel effect and considerations for experimental detection." pith.science (2026). https://pith.science/paper/622PU4GH

@misc{pith2026250104849,
  author       = {Pith},
  title        = {Pith review of: An examination of the extended Hong-Ou-Mandel effect and considerations for experimental detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/622PU4GH}},
  note         = {Machine review of arXiv:2501.04849}
}
abstract

In recent works we have explored a multi-photon extension of the celebrated two-photon Hong-Ou-Mandel (HOM) effect in which the quantum amplitudes for a two-photon input to a lossless, balanced 50:50 beamsplitter (BS) undergoes complete destructive interference. In the extended Hong-Ou-Mandel (eHOM) effect the multi-photon scattering of photons from the two input ports to the two output ports of the BS for Fock number basis input states (FS) $|n,m\rangle_{12}$ exhibit complete destructive interference pairwise within the quantum amplitudes containing many scattering components, generalizing the two-photon HOM effect. This has profound implications for arbitrary bipartite photonic input states constructed from such basis states: if the input state to one input port of the BS is of odd parity, i.e. constructed from only of odd numbers of photons, then regardless of the input state to the second 50:50 BS port, there will be a central nodal line (CNL) of zeros in the joint output probability distribution along the main diagonal for coincidence detection. The first goal of this present work is to show diagrammatically how the extended HOM effect can be seen as a succession of multi-photon HOM effects when the latter is viewed as a pairwise cancellation of mirror image scattering amplitudes. The second goal of this work is to explore considerations for the experimental realization of the extended Hong-Ou-Mandel effect. We examine the case of a single photon interfering with a coherent state (an idealized laser) on a balanced 50:50 beamsplitter and consider prospects for experimental detection of the output destructive interference by including additional effects such as imperfect detection efficiency, spatio-temporal mode functions, and time delay between the detected output photons.

Figures

Figures reproduced from arXiv: 2501.04849 by the authors.

Figure 1
Figure 1. FIG. 1. Joint output probability [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Output creation operators for mode-1 and mode-2 in terms of the input creation operators for a lossless beamsplitter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The 2-photon HOM effect with input [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The odd-odd-photon eHOM effect with input state [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The odd-odd-photon eHOM effect with input [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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

  1. PhD thesis: Modes, States, and Symmetries in quantum Optics for quantum Information and Metrology

    quant-ph 2026-07 accept novelty 7.0 of 10

    Modal structure, photon statistics, and bosonic/phase symmetries jointly determine the usable resources for photonic quantum information and metrology, with explicit gains and limits for time-frequency, HOM, and SSR settings.

Reference graph

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