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Theory of the monochromatic advanced-wave picture and applications in biphoton optics

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Pith's one-line read The paper claims that the spatial wave function of a photon pair from spontaneous parametric down-conversion is exactly the impulse response of a classical optical setup: propagate backward from one photon, multiply by the pump and…

desk verdict A clean formalization of Klyshko's advanced-wave picture for monochromatic light, with a solid core derivation and a genuinely load-bearing phenomenological no-detection extension that needs more work. read the letter →

arxiv 2412.02088 v2 pith:I3DMOT64 submitted 2024-12-03 quant-ph physics.optics

Yi Zheng , Jin-Shi Xu , Chuan-Feng Li , Guang-Can Guo This is my paper · ORCID
classification quant-phphysics.optics
keywords spontaneousparametricdown-conversionadvanced-wavepicturebiphotonwavefunctionquantumimagingwithundetectedphotonsopticalreciprocityspatialcorrelationsholographytwo-photoninterference
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 formalizes the advanced-wave picture for monochromatic light: it claims that the spatial wave function of a photon pair produced by spontaneous parametric down-conversion is obtained by treating one photon's position as a point source, propagating its field backward through the optical setup, multiplying the field at each point by the pump amplitude times the crystal's nonlinear response, and propagating forward to the other photon's position. This makes the biphoton wave function the impulse response function of a classical optical system, so bulk-crystal thickness, polarization, postselection, and even the case where one photon is never detected can be handled as ordinary linear optics. The payoff is that quantum imaging protocols, including ghost imaging, imaging with undetected photons, and polarization-entangled holography, can be unfolded into classical setups in which spatial resolution and field of view follow from simple impulse-response and filtering formulas.

What carries the argument

The load-bearing identity is Eq. (12), $\psi(r_1,r_2)=\int dr\,h_1(r,r_1)\,\chi^{(2)}(r)U_p(r)\,h_2(r_2,r)$, which expresses the biphoton wave function as back-propagation, multiplication, and forward-propagation. It is derived from the low-gain SPDC Hamiltonian together with optical reciprocity, which lets backward-traveling advanced waves be computed with the same impulse response functions as forward light. In the unfolded picture, a point source at one detection position travels backward through the setup, is re-emitted at each point with amplitude $\chi^{(2)}U_p$, and travels forward to the other detection position, so a bulk crystal becomes a transverse spatial filter with a sinc-shaped coherence transfer function and the shifted sine-integral point-spread function $\mathrm{Ssi}[n_o k|\rho|^2/(2L)]$. Postselection, bucket detection, and no detection are different rules for superposing advanced-wave sources, and polarization enters by giving the susceptibility and the Green's functions polarization indices.

What would settle it

Measure the interference visibility as a function of the amplitude transmittance $T$ of a real, semi-transparent object placed in the undetected arm of a two-crystal interferometer. The paper predicts an extra incoherent background $(1-T^2)/2$ from the object and visibility exactly $T$; without the advanced-wave-from-absorption rule the visibility would be $2T/(1+T^2)$. A measured curve matching the second expression at any $T$ would falsify the no-detection extension.

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Extended reading notes

Core claim

The central result is Eq. (12): the biphoton wave function is $\psi(r_1,r_2)=\int dr\,h_1(r,r_1)\,\chi^{(2)}(r)U_p(r)\,h_2(r_2,r)$ (up to which photon is taken as the advanced one). Here $h_1$ and $h_2$ are the classical impulse responses of the linear optical systems seen by the two photons, $\chi^{(2)}(r)$ is the position-dependent second-order susceptibility, and $U_p(r)$ is the pump field. The paper claims this identity is an equality, not an analogy: the conditional state of one photon is exactly what a classical field does after backward propagation, multiplication by the pump-nonlinearity product, and forward propagation. From this follow the shifted-sine-integral form of the bulk-crystal biphoton state, the treatment of pure-state postselection and bucket detection as different superpositions of advanced-wave sources, the rule that an undetected photon makes absorbing regions emit incoherent advanced waves of strength $1-|T|^2$, and quantitative resolution and field-of-view formulas for quantum imaging with undetected photons and polarization-entangled holography.

Load-bearing premise

The load-bearing premise is that when one of the two photons is never detected, an absorbing object emits a backward-propagating 'advanced wave' whose strength is set solely by the absorption coefficient; the paper states this rule as phenomenological, without a microscopic derivation, so the predicted background and visibility in undetected-photon imaging stand or fall with it.

Editorial extensions

If this is right

  • A bulk SPDC crystal acts in the unfolded setup as a spatial filter whose transfer function is a sinc of the squared transverse momentum, so crystal thickness enters every resolution and field-of-view formula as a classical filtering width.
  • In quantum imaging with undetected photons using momentum correlation, the spatial resolution at the object plane is $\sqrt{2\ln 2}\,\lambda_i f_1/(\pi w)$, independent of the signal wavelength, while the field of view is set by the narrower of the two crystal-induced sinc widths.
  • In the position-correlation version, the image magnification is $M_4M_6$, the field of view is $\sqrt{2\ln 2}\,wM_2$, and the two interfering beams are perfectly matched only when $M_2M_4=1$.
  • A partially absorbing object in the undetected path contributes an incoherent background of strength $1-|T|^2$, which is what makes the measured interference visibility equal to $T$ rather than $2T/(1+T^2)$.
  • The free-space evolution of a double-Gaussian biphoton state becomes a Gaussian beam in the unfolded setup: its correlation center stays fixed for short propagation distances and anticorrelates for longer ones, with a spherical phase curvature corresponding to propagation distance $2z$.

Reading between the lines

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

  • The paper leaves implicit that the same fold-unfold logic applies to any low-gain nonlinear process whose Hamiltonian is a product of pump amplitude and two creation operators, such as four-wave mixing, so the advanced-wave picture could be used as a design tool in those systems as well.
  • If the no-detection advanced-wave rule survives direct test, quantum imaging with undetected photons becomes a direct absorption probe: the measured visibility-versus-$T$ curve is a spectrometric channel that does not require detecting the light that interacted with the object.
  • Because the biphoton wave function is identified with a classical impulse response, a thin SPDC crystal fed by a designed classical system should produce a biphoton state proportional to that system's impulse response; classical Fourier-optics design libraries could therefore be recycled to engineer two-photon spatial states.
  • The quasimonochromatic appendix predicts a time-domain signature for pulsed-pump two-photon interference: advanced-wave components whose backtracking time does not overlap the pump wave at the crystal produce no retarded wave, so the coincidence rate should track the pump temporal envelope; this is a direct, testable consequence that the main text only sketches.
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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

3 major / 5 minor

Summary. The paper develops a monochromatic version of Klyshko's advanced-wave picture (AWP) for spontaneous parametric down-conversion. Starting from the standard low-gain SPDC Hamiltonian, it obtains the biphoton wave function psi(r1,r2) = integral dr h1(r,r1) chi^(2)(r) U_p(r) h2(r2,r), interprets this as a classical propagation, multiplication, propagation sequence, and applies the result to bulk-crystal SPDC states, quantum imaging with undetected photons (QIUP), polarization-entanglement quantum holography, ghost imaging, and a qualitative discussion of the Hong-Ou-Mandel effect. For postselected point detection, the central formula is a formalization of standard results and reproduces the known Ssi correlation. For the no-detection case, the paper introduces a phenomenological source from absorbing regions and uses it to obtain the QIUP visibility.

Significance. Equation (12) is a clean and potentially useful formalization. The derivation from the SPDC Hamiltonian via reciprocity is transparent, has no fitted parameters, and the bulk-crystal calculation in Sec. III A reproduces the known sine-integral biphoton correlation. The applications give explicit closed-form estimates for spatial resolution and field of view, which are checked against standard analyses in the literature. The no-detection branch, however, is the least developed part: it is explicitly phenomenological and carries the quantitative QIUP visibility prediction. If that branch is placed on a microscopic footing, this would be a valuable reference for biphoton spatial optics and for designing nonlinear devices.

major comments (3)
  1. [Appendix D 3, Eqs. (D6)-(D7); Sec. IV A] The no-detection branch of the AWP rests on the phenomenological replacement a^dag_{k,r} = integral dr' h(r',r)(a^dag_{r'} + gamma(r') b^dag_{r'}) with gamma = sqrt(alpha/n). This is asserted rather than derived from a microscopic model of absorption. The added b^dag term is precisely what converts the two-field interferometric visibility 2T/(1+T^2) in Sec. IV A into the claimed value T through the 1-T^2 background; tracing over the undetected idler from the unmodified state of Eq. (10) gives J2(r2,r2') = integral dr1 psi(r1,r2) psi*(r1,r2') with no gamma^2 term. I therefore do not regard the no-detection reduction as established to the same standard as Eq. (12). A derivation from a standard quantization of absorbing dielectrics (for example, Langevin-noise operators) or a direct comparison with an unapproximated field-theoretic calculation of the QIUP visibility is needed before the claim can be taken as general.
  2. [Appendix D 3, Eqs. (D6)-(D7); Sec. IV A] The derivation of the no-detection mutual coherence function assumes that a closed surface S covering the setup emits advanced waves with unit intensity, while absorptive regions inside S emit with intensity gamma^2, and that all these advanced waves superpose incoherently. This incoherent-emission assumption is load-bearing for the QIUP background, yet it is introduced heuristically and is not derived from the linearity that underpins Eq. (12). In particular, the paper does not specify how S is modeled as a physical object (a detector, an absorber, or a mathematical boundary) and why its advanced-wave source strength is exactly unity. The claim that the object emits light of intensity 1-|T|^2 in Eq. (78) is a direct consequence of this heuristic, so the quantitative QIUP result should be presented as a prediction of the phenomenological model, not as a theorem of the AWP framework.
  3. [Sec. III A, Eq. (21) and Appendix E] The derivation of the bulk-crystal wave function evaluates the integral over z inside the crystal as a Cauchy principal value at z = 0. Since the original integral over the crystal includes z = 0 and the paraxial Green's function is singular there, the principal-value prescription is not automatic. The result agrees with the known momentum-space answer, so I do not doubt the final formula, but the paper should explain why the principal value is the physically appropriate regularization, since a different contour would change the phase and the result.
minor comments (5)
  1. [Sec. II A, Eq. (2)] The Green's function for propagation in a homogeneous medium contains a factor n in the amplitude; given that the paper defines photon wave functions through field amplitudes, a brief remark on how this factor is reconciled with the n-dependent classical intensity would avoid confusion.
  2. [Sec. III A, footnote 72] The derivation relies on two perfect 4f systems with unit magnification and infinite aperture; a sentence on how finite apertures would modify Eq. (21) would help readers who want to apply the result to actual experiments.
  3. [Sec. IV A, Eq. (31)] Several expressions in the QIUP analysis use proportionality signs that omit constant phases and normalization factors; since the visibility is determined by interference, the paper should state explicitly that the constant phase C absorbs all omitted setup phases.
  4. [Appendix H, footnote 97] The quasi-monochromatic treatment uses rectangular bandpass filters and footnote 97 acknowledges that this violates causality; the main-text discussion of the Hong-Ou-Mandel effect should therefore be labeled as heuristic rather than as a derivation, especially since the monochromatic core of the paper does not need this appendix.
  5. [Fig. 2] The caption of Fig. 2 appears to contain a garbled phrase ('Mask approx or'); please check the final figure and caption for typographical errors.

Circularity Check

0 steps flagged · score 0.0 of 10

Core AWP derivation is self-contained; no circularity found, though the no-detection extension in Appendix D3 is a phenomenological assumption rather than a derived result.

full rationale

The central formula, Eq. (12), is derived, not assumed: it follows from the standard first-order SPDC Hamiltonian, Eq. (9), the two-photon state Eq. (10), the linear-propagation identity Eq. (6), and the optical reciprocity h(r,r0)=h(r0,r). No parameter is fitted to the biphoton data being explained. The bulk-crystal result, Eq. (21), reproduces the independent momentum-space result, Eq. (18), including the Ssi correlation function, so it is a cross-check against the standard formalism rather than a circular reduction. The QIUP resolution and field-of-view calculations in Sec. IV A are explicit paraxial propagation calculations from the same formula, and the authors compare them with the standard literature rather than fitting them. The only epistemically soft point is Appendix D3's no-detection extension, where the paper explicitly says 'Phenomenologically' and introduces the internal-excitation operator b̂† with γ=√(α/n). This is a heuristic model, not a microscopic derivation, and it is load-bearing for the QIUP visibility background 1−T². However, it is not circular: γ is fixed by the absorption coefficient α and the Beer's-law energy loss 1−T², not by the target visibility T, and the visibility T is invoked as an external benchmark (citations [45,48,76]). The self-citations [13] and [24] report the authors' own experimental measurements and are not used to justify the derivation; there is no uniqueness theorem or ansatz smuggled in via self-citation. Thus the derivation chain is self-contained against external benchmarks, and the no-detection assumption should be weighed as a correctness risk rather than as circularity.

Assumptions & free parameters 1 free parameters · 7 assumptions · 1 invented entities

The monochromatic AWP core rests on standard quantum-optics assumptions (low-gain SPDC, classical pump, paraxial propagation, reciprocity). The extensions to no detection and quasimonochromatic pulses rely on two explicitly flagged heuristic models, which account for the ad_hoc entries. No data are fitted; the only hand-chosen numerical constant is the 0.455 Gaussian-matching coefficient borrowed from earlier work.

free parameters (1)
  • sinc-to-Gaussian width-matching coefficient = 0.455
    Adopted from Ref. [21] so that sinc(ax^2) and exp(−0.455ax^2) have the same 1/e^2 width. This is an approximation choice, not fitted to data, and affects only the double-Gaussian propagation discussion.
assumptions (7)
  • domain assumption Low-gain, first-order perturbation theory describes SPDC; the pair creation amplitude is proportional to χ(2)(r)Up(r) at a single point (Eqs. 9-11).
    Standard and stated in Sec. II D; higher-order and pump-depletion effects are neglected.
  • domain assumption The pump beam is a classical, monochromatic, undepleted coherent field with amplitude Up(r).
    Stated in Sec. II D; the paper explicitly limits itself to monochromatic pump, signal, and idler.
  • standard math Optical reciprocity Eq. (8) holds for all linear, passive optical elements used, including birefringent crystals with walk-off (Sec. II C).
    The AWP formula Eq. (12) uses h(r,r0)=h(r0,r); the authors note exceptions such as the Faraday effect and handle e-light walk-off with a sign flip (footnote 62).
  • domain assumption Paraxial approximation for all Green's functions and angular-spectrum propagation (Eqs. 2-3, A1).
    Used throughout; the authors note in Sec. V that more realistic diffraction theories could be chosen.
  • ad hoc to paper Two perfect 4f systems with unit magnification and infinite aperture are used to place the detection points at the crystal back focal plane (Sec. III A, footnote 72).
    This construction lets the AWP derivation of Eq. (21) avoid backward-propagating SPDC contributions; it is an idealized optical system.
  • ad hoc to paper An undetected photon in absorbing media is described by an internal excitation operator b̂† with source strength γ=√(α/n) (Eq. D5).
    Introduced 'phenomenologically' in Appendix D 3; the object's 1−T² advanced-wave intensity follows from this model and is used in the QIUP visibility analysis.
  • ad hoc to paper Quasimonochromatic treatment assumes rectangular bandpass filters and ignores the causality violation this creates (Appendix H, footnote 97).
    The paper explicitly states 'This issue is ignored here', so the HOM discussion is illustrative rather than a complete polychromatic theory.
invented entities (1)
  • Internal excitation operator b̂†_r
    purpose: Models advanced-wave emission from absorbing regions when a photon is undetected (Eq. D5).
    It is a phenomenological bookkeeping operator calibrated to reproduce the classical absorption factor 1−T²; the paper provides no independent falsifiable prediction tied to it.

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Cite this review

Pith. "Pith review of Theory of the monochromatic advanced-wave picture and applications in biphoton optics." pith.science (2026). https://pith.science/paper/I3DMOT64

@misc{pith2026241202088,
  author       = {Pith},
  title        = {Pith review of: Theory of the monochromatic advanced-wave picture and applications in biphoton optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3DMOT64}},
  note         = {Machine review of arXiv:2412.02088}
}
read the original abstract

Klyshko's advanced-wave picture (AWP) is mainly interpreted by replacing the nonlinear crystal producing biphotons via spontaneous parametric down-conversion (SPDC) by a mirror in quantum imaging protocols with thin crystals, where the biphotons are perfectly correlated in position at the crystal. To better explain the biphoton spatial states produced by arbitrary crystals and pump beams, we develop a formal theory of AWP with monochromatic lights that the conditional wave function of one photon is calculated by propagation, multiplication, and another propagation. The case of more general photon postselection or no detection and the inclusion of polarization are studied. Then, we explain the form of the biphoton state from SPDC with a bulk crystal and its free-space propagation. By treating the biphoton wave function as an impulse response function of a classical optical setup, we analyze quantum imaging with undetected photons and quantum holography with polarization entanglement, where properties like the spatial resolution can be concisely deduced. This method can be employed to design nonlinear materials or novel quantum imaging techniques. Finally, we discuss Klyshko's original proposal beyond monochromatic lights with the Hong-Ou-Mandel effect as an example.

Figures

Figures reproduced from arXiv: 2412.02088 by the authors.

Figure 1
Figure 1. FIG. 1. The principle of monochromatic AWP. The biphoton wave [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. In the AWP-unfolded setup, if the pump beam walk-off can [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) In classical optics, free-space propagation is a convolu [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The setup of Mach-Zehnder-type quantum imaging with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The setup of the simplified quantum holography with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. In the unfolded setup of ghost imaging, the crystal plane [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The wave packets traveling backward (blue, advanced, be [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Works this paper leans on

112 extracted references · 72 canonical work pages · cited by 1 Pith paper

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    double Klyshko picture

    Essential case After biphoton creation, according to Eq. (6), ψ(r1, r2) = Z dr0χ(2)(r0)Up(r0)h1(r1, r0)h2(r2, r0) = Z drh1(r, r1)χ(2)(r)Up(r)h2(r2, r) = Z drh2(r, r2)χ(2)(r)Up(r)h1(r1, r). (12) Here, the down-converted photons undergo linear processes after creation, so h1(r, r0) and h2(r, r0) are calculated from the first-order susceptibilities of the sp...

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    Case of pure-state postselection and the linearity in A WP Sometimes, one photon is postselected to a pure state (for example, being collected by a single-mode fiber [69]). Let- ting the postselected wave function defined on a certain plane S1 be ψ1(r), the annihilation operator ˆar1 is replaced by R S1 drψ1(r)ˆa† r † = R S1 drψ∗ 1(r)ˆar, and we can super...

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    Case including polarization If different polarizations are considered (for example, using two NLCs with different optical axis directions can produce polarization-entangled biphotons [36]), the pump beam is de- scribed by Up,H (r) and Up,V (r). Letting σ = H, Vbe the index of polarization, the biphoton state is |Ψ⟩ = X σp,σ1,σ2 Z drχ(2) σpσ1σ2 (r)Up,σp (r...

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    (27) After the Fourier lens f3, the amplitude at the camera plane is Us3′(ρ) = Z dρ′Us2′(ρ′) exp − i2πρ λsf3 · ρ′ ∝ sinc πLλ+|ρ|2 2λ2sf 2 3 e− i2πρi λs f3 ·ρ

    QIUP using momentum correlation In the unfolded setup of QIUP using momentum correla- tion, a point source emits the i light at ρi of the second crystal Ui2(ρ) = δ(ρ − ρi), which immediately creates the reflected s light whose amplitude is Us2′(ρ) ≈ Ssi 2π|ρ − ρi|2 Lλ+ Up(ρi). (27) After the Fourier lens f3, the amplitude at the camera plane is Us3′(ρ) = ...

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    With the point source at ρi, the amplitude of the createds light from the second crystal is still Eq

    QIUP using position correlation In QIUP using position correlation, we denote the absolute values of the magnification ratios of the three 4f systems as M2 = f2/f1, M4 = f4/f3, and M6 = f6/f5. With the point source at ρi, the amplitude of the createds light from the second crystal is still Eq. (27), which we denote byUs4′(ρ) in this case. So, at the camer...

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    Case of partially coherent pump beam If the classical pump beam is partially coherent in trans- verse space, it must be quasimonochromatic. Similar as the density matrix in quantum mechanics, the mutual coherence function (at the same time) can be decomposed into an inco- herent superposition of various coherent light fields [17] Jp(r, r′) = X j Upj(r)U ∗...

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    So, in the AWP, the state of photon 2 when photon 1 is postselected to r1 [letting r′ 1 = r1 in Eq

    = Tr ˆar1 ˆar2 ˆρˆa† r′ 1 ˆa† r′ 2 = X j Z drχ(2)(r)Upj(r)h1(r1, r)h2(r2, r) × Z dr′χ(2)(r′)U ∗ pj(r′)h∗ 1(r′ 1, r′)h∗ 2(r′ 2, r′) = X j ψj(r1, r2)ψ∗ j (r′ 1, r′ 2), (D3) 12 where ψj(r1, r2) is the biphoton wave function from the pump beam component Uj(r). So, in the AWP, the state of photon 2 when photon 1 is postselected to r1 [letting r′ 1 = r1 in Eq. ...

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    With the initial state |Ψ⟩ [Eq

    Case of bucket detection In the case of bucket detection of photon 1 at a surface region S1, the photon 2 states with different photon 1 posi- tions superpose incoherently, and the resulting state is gen- erally mixed. With the initial state |Ψ⟩ [Eq. (10)], the con- ditional state after photon 1 bucket detection is ˆρbuc =R S1 dr1ˆar1 |Ψ⟩⟨Ψ|ˆa† r1, and th...

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    If we are only interested in the intensity of photon 2, the result isR S1 dr1|ψ(r1, r2)|2

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