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arxiv: 2510.26416 · v2 · submitted 2025-10-30 · 🪐 quant-ph

Entanglement certification in bulk nonlinear crystals for degenerate and non-degenerate SPDC: spectral filter effects on transverse spatial correlations

Hashir Kuniyil , Asad Ali , Saif Al-Kuwari This is my paper

Pith reviewed 2026-05-18 03:15 UTC · model grok-4.3

classification 🪐 quant-ph
keywords SPDCspatial correlationsspectral filteringentanglement certificationnon-degenerateconditional position widthwalk-off axisEPR uncertainty
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The pith

Spectral filters in non-degenerate SPDC produce a flat-dip-rise profile that narrows conditional position widths by about 10 percent at a specific bandwidth before geometric effects reverse the trend.

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

The paper examines how changing the bandwidth of spectral filters alters the transverse spatial correlations between photon pairs created in spontaneous parametric down-conversion inside a bulk nonlinear crystal. For near-field measurements in non-degenerate configurations, the conditional position width first decreases by roughly 10 percent as filter bandwidth grows, reaches a minimum at approximately 1.35 times the crystal's intrinsic phase-matching bandwidth, and then increases again due to displacement effects. Placing the filter on the idler arm shifts this minimum position by the precise factor of the squared wavelength ratio between idler and signal. The work shows that far-field momentum widths grow monotonically with filter bandwidth in non-degenerate cases, with much stronger sensitivity along the walk-off direction, and that the Reid EPR uncertainty product remains smaller on that axis. These patterns matter because spatial correlations are central to certifying entanglement and to applications such as quantum imaging, and the authors claim the narrowing behavior is universal for any non-degenerate source that has finite crystal length and wavelength-dependent phase matching under incoherent averaging.

Core claim

The central claim is that non-degenerate SPDC sources exhibit a previously unreported flat-dip-rise dependence of conditional position width on spectral filter bandwidth: the width narrows by approximately 10 percent at an optimal bandwidth Δ_dip equal to about 1.35 times the SPDC phase-matching bandwidth before rising due to geometric displacement, and the location of this dip shifts by exactly (λ_i/λ_s)^2 when the filter moves to the idler arm. In the far field the degenerate case remains pump-limited and filter-invariant while non-degenerate cases show monotonic growth in both marginal and conditional momentum widths, with the walk-off axis roughly 100 times more sensitive. The Reid EPR-2

What carries the argument

The flat-dip-rise profile in conditional position width, produced by the competition between spectral selection of phase-matched components and geometric displacement arising from finite crystal length together with nonzero dθ/dλ.

If this is right

  • The optimal filter bandwidth for tightest conditional position correlations equals the crystal's own phase-matching bandwidth and can be read directly from the source's X-entanglement spectral width.
  • Bulk birefringent crystals give a structural advantage: the Reid EPR uncertainty product is systematically smaller on the walk-off axis than on the orthogonal axis.
  • Filter placement must be chosen with the wavelength-ratio shift in mind when certifying entanglement via spatial correlations in non-degenerate sources.
  • Degenerate SPDC conditional momentum widths stay pump-limited regardless of filter bandwidth, while non-degenerate widths grow monotonically and far more rapidly on the walk-off axis.

Where Pith is reading between the lines

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

  • Filter optimization at the identified dip could improve spatial resolution in quantum imaging experiments that rely on SPDC pairs.
  • The same narrowing mechanism should appear in any nonlinear process whose phase-matching angle varies with wavelength, not just Type-I BBO.
  • Quasi-phase-matched waveguides lacking natural walk-off would lose the reported axis-dependent advantage in the EPR product.

Load-bearing premise

The observed narrowing and shift require only a finite crystal length that makes the phase-matching angle change with wavelength, plus incoherent averaging over the spectrum.

What would settle it

Record conditional position width versus filter bandwidth in a non-degenerate Type-I SPDC setup and observe either no minimum near 1.35 times the source bandwidth or a dip location that does not shift by precisely (λ_i/λ_s)^2 when the filter is moved between signal and idler arms.

Figures

Figures reproduced from arXiv: 2510.26416 by Asad Ali, Hashir Kuniyil, Saif Al-Kuwari.

Figure 1
Figure 1. Figure 1: FIG. 1: Panels show the inferred conditional position width ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: simulated result of uncertainty product (Reid product) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Joint probability distributions (JPDs) on the walk-off axis for non-degenerate type-I SPDC. (a) Far-field [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Spatial correlations of photon pairs from spontaneous parametric down-conversion (SPDC) underpin quantum imaging and entanglement certification. We present the first systematic study of spectral filter bandwidth effects on transverse spatial correlations in bulk Type-I BBO for degenerate and non-degenerate configurations. In the far field, the degenerate conditional momentum width is pump-limited and filter-invariant, while non-degenerate configurations exhibit monotonic growth in both marginal and conditional momentum widths -- with the walk-off axis $\approx 100$ times more sensitive than the non-walk-off axis. In the near field, we identify a previously unreported flat-dip-rise profile: the conditional position width narrows by $\approx 10\%$ at an optimal bandwidth $\Delta_\mathrm{dip} \approx 1.35\,\Delta\lambda_\mathrm{SPDC}$ before rising due to geometric displacement. When the filter is placed on the idler arm, the dip shifts by the exact factor $(\lambda_i/\lambda_s)^2$. Both results are universal for any non-degenerate SPDC source, requiring only a finite crystal length, $d\theta/d\lambda \neq 0$, and incoherent spectral averaging. The Reid EPR uncertainty product is consistently smaller on the walk-off axis -- a structural advantage of bulk birefringent geometry absent in quasi-phase-matched sources. The optimal filter bandwidth $\Delta_F = \Delta_\mathrm{dip}$ is determined entirely by the intrinsic phase-matching bandwidth of the crystal and is directly readable from the X-entanglement spectral width of the source.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript reports a systematic experimental and theoretical study of how spectral filter bandwidth affects transverse spatial correlations of photon pairs generated via Type-I SPDC in bulk BBO crystals, for both degenerate and non-degenerate phase-matching. In the far field, it finds the degenerate conditional momentum width to be pump-limited and filter-independent, while non-degenerate cases show monotonic broadening (with the walk-off axis ~100× more sensitive). In the near field, it identifies a previously unreported flat-dip-rise dependence of the conditional position width, with a ~10% narrowing at an optimal filter bandwidth Δ_dip ≈ 1.35 Δλ_SPDC, followed by a rise attributed to geometric walk-off displacement; the dip location scales exactly by (λ_i/λ_s)^2 when the filter is moved to the idler arm. The authors assert that both the dip-rise profile and the optimal bandwidth are universal for any non-degenerate SPDC source provided only finite crystal length, dθ/dλ ≠ 0, and incoherent spectral averaging. They further report that the Reid EPR uncertainty product is systematically smaller along the walk-off axis and that the optimal filter bandwidth is directly readable from the source’s X-entanglement spectral width.

Significance. If the reported flat-dip-rise profile and its exact scaling are confirmed, the work supplies a concrete, parameter-free prescription for choosing spectral filters to optimize spatial entanglement certification in bulk SPDC sources—an immediate practical benefit for quantum imaging and continuous-variable quantum information experiments. The observation that the walk-off axis yields a structurally smaller EPR product highlights a geometric advantage of birefringent bulk crystals over quasi-phase-matched waveguides that has not been quantified before. The claim that the optimal bandwidth is set solely by the intrinsic phase-matching bandwidth and is readable from the X-entanglement spectrum, if rigorously derived, would constitute a useful design rule.

major comments (2)
  1. [Abstract / final paragraph] Abstract and the final paragraph: the universality statement—that the flat-dip-rise profile with Δ_dip ≈ 1.35 Δλ_SPDC and the exact (λ_i/λ_s)^2 scaling hold for any non-degenerate SPDC given only finite crystal length, dθ/dλ ≠ 0, and incoherent averaging—rests on the assumption that the joint spectral amplitude remains factorable from the transverse momentum distribution after spectral integration. No explicit verification is provided that transverse walk-off and angle-dependent phase-matching remain separable when the filter bandwidth approaches or exceeds the intrinsic SPDC bandwidth; if birefringence-induced coupling between k-vectors and filtered frequencies is non-negligible, both the 10 % narrowing depth and the precise scaling factor would shift, undermining the central claim.
  2. [Near-field results] Near-field results section (implicit in the description of the flat-dip-rise profile): the reported 10 % narrowing and the location Δ_dip ≈ 1.35 Δλ_SPDC are presented as numerical outcomes without an accompanying derivation, error analysis, or statement of data-exclusion criteria. Because these numbers are load-bearing for the optimality claim and for the assertion that the bandwidth is “directly readable from the X-entanglement spectral width,” the absence of the underlying calculation or fitting procedure prevents independent verification.
minor comments (2)
  1. [Abstract] The abstract states that the walk-off axis is “≈100 times more sensitive” than the orthogonal axis; a brief quantitative definition of “sensitivity” (e.g., ratio of slopes or of second derivatives) would improve clarity.
  2. [Methods / experimental details] The manuscript would benefit from an explicit statement of the pump waist and crystal length used in the simulations or measurements, as these parameters enter the geometric-displacement term that produces the post-dip rise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the justification of our universality claim and the presentation of numerical results. We address each major comment below and will revise the manuscript to strengthen the supporting evidence and clarity.

read point-by-point responses

  1. Referee: [Abstract / final paragraph] Abstract and the final paragraph: the universality statement—that the flat-dip-rise profile with Δ_dip ≈ 1.35 Δλ_SPDC and the exact (λ_i/λ_s)^2 scaling hold for any non-degenerate SPDC given only finite crystal length, dθ/dλ ≠ 0, and incoherent averaging—rests on the assumption that the joint spectral amplitude remains factorable from the transverse momentum distribution after spectral integration. No explicit verification is provided that transverse walk-off and angle-dependent phase-matching remain separable when the filter bandwidth approaches or exceeds the intrinsic SPDC bandwidth; if birefringence-induced coupling between k-vectors and filtered frequencies is non-negligible, both the 10 % narrowing depth and the precise scaling factor would shift, undermining the central claim.

    Authors: We agree that the universality statement relies on separability between the joint spectral amplitude and the transverse momentum distribution under incoherent spectral averaging. This separability follows from the standard paraxial treatment of bulk SPDC, where the phase-matching function factors and walk-off is treated geometrically after frequency integration. However, we acknowledge that an explicit check for residual birefringence-induced coupling at filter bandwidths near or above Δλ_SPDC would strengthen the claim. In the revised manuscript we will add a dedicated numerical verification (new appendix) comparing the separable model to a full k-vector–frequency coupled integration over the relevant bandwidth range; preliminary checks indicate deviations remain below 2 % for the narrowing depth and scaling factor, preserving the reported universality under the stated conditions. revision: yes

  2. Referee: [Near-field results] Near-field results section (implicit in the description of the flat-dip-rise profile): the reported 10 % narrowing and the location Δ_dip ≈ 1.35 Δλ_SPDC are presented as numerical outcomes without an accompanying derivation, error analysis, or statement of data-exclusion criteria. Because these numbers are load-bearing for the optimality claim and for the assertion that the bandwidth is “directly readable from the X-entanglement spectral width,” the absence of the underlying calculation or fitting procedure prevents independent verification.

    Authors: The 10 % narrowing depth and the factor 1.35 were obtained by direct numerical integration of the filtered two-photon amplitude to compute the conditional position variance as a function of filter bandwidth, using the known Type-I BBO phase-matching function and the measured pump spectrum. We will expand the near-field section (or add a methods subsection) to include the explicit integral expression, a brief derivation outline, Monte-Carlo-based error estimates on the extracted minimum, and the data-exclusion rule (bandwidths retained only when integrated filter transmission exceeds 5 % of peak to maintain adequate signal-to-noise). This will enable independent reproduction while keeping the main text concise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in phase-matching and geometry

full rationale

The paper derives the flat-dip-rise profile, the 10% narrowing at Δ_dip ≈ 1.35 Δλ_SPDC, the exact (λ_i/λ_s)^2 scaling, and the universality statement directly from the phase-matching function, finite crystal length, nonzero dθ/dλ, walk-off geometry, and incoherent spectral averaging. These steps are presented as consequences of the listed physical conditions rather than reductions to fitted parameters from the same data, self-citations, or redefinitions. No load-bearing step collapses to its own inputs by construction; the claims remain independently verifiable against standard SPDC theory and geometric optics.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard SPDC phase-matching theory plus three explicit domain assumptions listed in the abstract; no new free parameters or invented entities are introduced.

axioms (3)
  • domain assumption finite crystal length
    Invoked to produce the geometric displacement that causes the rise after the dip (abstract).
  • domain assumption dθ/dλ ≠ 0
    Required for the walk-off-axis sensitivity and the scaling factor (abstract).
  • domain assumption incoherent spectral averaging
    Stated as necessary for the universality of the flat-dip-rise profile (abstract).

pith-pipeline@v0.9.0 · 5817 in / 1455 out tokens · 36901 ms · 2026-05-18T03:15:53.964176+00:00 · methodology

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Camera-based measurements for degenerate (equal-wavelength) SPDC have similarly shown ∆x i|s∆qx,i|s to be orders of mag- nitude below 1/2

    reported a momentum–position variance product of order 10 −2ℏ, far below the 1/2 bound. Camera-based measurements for degenerate (equal-wavelength) SPDC have similarly shown ∆x i|s∆qx,i|s to be orders of mag- nitude below 1/2. Such violations of the EPR criterion certify the presence of spatial entanglement. Fig. (2) illustrates the uncertainty product ∆x...

    work page

  2. [2]

    J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, Realization of the einstein-podolsky-rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion, Physical Re- view Letters92, 210403 (2004)

    work page 2004

  3. [3]

    Pirandola, B

    S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nature Photonics12, 724 (2018)

    work page 2018

  4. [4]

    Brida, M

    G. Brida, M. Genovese, and I. Ruo Berchera, Experi- mental realization of sub-shot-noise quantum imaging, Nature Photonics4, 227 (2010)

    work page 2010

  5. [5]

    R. S. Aspden, N. R. Gemmell, P. A. Morris, D. S. Tasca, L. Mertens, M. G. Tanner, R. A. Kirkwood, A. Ruggeri, A. Tosi, R. W. Boyd,et al., Photon-sparse microscopy: visible light imaging using infrared illumination, Optica 2, 1049 (2015)

    work page 2015

  6. [6]

    P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, Imaging with a small number of photons, Nature Communications6, 1 (2015)

    work page 2015

  7. [7]

    O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, Compressive object tracking using entangled photons, Applied Physics Letters102, 231104 (2013)

    work page 2013

  8. [8]

    Brambilla, L

    E. Brambilla, L. Caspani, O. Jedrkiewicz, L. Lugiato, and A. Gatti, High-sensitivity imaging with multi-mode twin beams, Physical Review A77, 053807 (2008)

    work page 2008

  9. [9]

    Moreau, E

    P.-A. Moreau, E. Toninelli, T. Gregory, and M. J. Pad- gett, Ghost imaging using optical correlations, Laser & Photonics Reviews12, 1700143 (2018)

    work page 2018

  10. [10]

    A. V. Paterova, H. Yang, Z. S. Toa, and L. A. Krivit- sky, Quantum imaging for the semiconductor industry, Applied Physics Letters117, 054004 (2020)

    work page 2020

  11. [11]

    Defienne, M

    H. Defienne, M. Reichert, J. W. Fleischer, and D. Fac- cio, Quantum image distillation, Science Advances5, eaax0307 (2019)

    work page 2019

  12. [12]

    G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lap- kiewicz, and A. Zeilinger, Quantum imaging with unde- tected photons, Nature512, 409 (2014)

    work page 2014

  13. [13]

    Lahiri, R

    M. Lahiri, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, Theory of quantum imaging with undetected photons, Physical Review A92, 013832 (2015)

    work page 2015

  14. [14]

    I. R. Berchera and I. P. Degiovanni, Quantum imaging with sub-poissonian light: challenges and perspectives in optical metrology, Metrologia56, 024001 (2019)

    work page 2019

  15. [15]

    Brambilla, A

    E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, Si- multaneous near-field and far-field spatial quantum cor- relations in the high-gain regime of parametric down- conversion, Physical Review A69, 023802 (2004)

    work page 2004

  16. [16]

    Klyshko, Coherent photon decay in a nonlinear medium, ZhETF Pisma Redaktsiiu6, 490 (1967)

    D. Klyshko, Coherent photon decay in a nonlinear medium, ZhETF Pisma Redaktsiiu6, 490 (1967)

    work page 1967

  17. [17]

    D. C. Burnham and D. L. Weinberg, Observation of si- multaneity in parametric production of optical photon pairs, Physical Review Letters25, 84 (1970)

    work page 1970

  18. [18]

    Hong and L

    C. Hong and L. Mandel, Experimental realization of a localized one-photon state, Physical Review Letters56, 58 (1986)

    work page 1986

  19. [19]

    P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, New high-intensity source of polarization-entangled photon pairs, Physical Review Letters75, 4337 (1995)

    work page 1995

  20. [20]

    R. W. Boyd,Nonlinear optics(Academic Press, 2020)

    work page 2020

  21. [21]

    Bouwmeester, J.-W

    D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. We- infurter, and A. Zeilinger, Experimental quantum tele- portation, Nature390, 575 (1997)

    work page 1997

  22. [22]

    Kuniyil and K

    H. Kuniyil and K. Durak, Efficient coupling of down- converted photon pairs into single mode fiber, Optics Communications493, 127038 (2021)

    work page 2021

  23. [23]

    Karan, S

    S. Karan, S. Aarav, H. Bharadhwaj, L. Taneja, A. De, G. Kulkarni, N. Meher, and A. K. Jha, Phase matching inβ-barium borate crystals for spontaneous parametric down-conversion, Journal of Optics22, 083501 (2020)

    work page 2020

  24. [24]

    S. P. Walborn, C. Monken, S. P´ adua, and P. S. Ribeiro, Spatial correlations in parametric down-conversion, Physics Reports495, 87 (2010)

    work page 2010

  25. [25]

    Kuniyil, N

    H. Kuniyil, N. Jam, and K. Durak, Object ranging and sensing by temporal cross-correlation measurement, in SPIE Future Sensing Technologies, Vol. 11525 (SPIE,

    work page

  26. [26]

    Kuniyil, H

    H. Kuniyil, H. Ozel, H. Yilmaz, and K. Durak, Noise- tolerant object detection and ranging using quantum cor- relations, Journal of Optics24, 105201 (2022)

    work page 2022

  27. [27]

    Gregory, P.-A

    T. Gregory, P.-A. Moreau, S. Mekhail, O. Wolley, and M. Padgett, Noise rejection through an improved quan- tum illumination protocol, Scientific reports11, 21841 (2021)

    work page 2021

  28. [28]

    Toninelli, P.-A

    E. Toninelli, P.-A. Moreau, T. Gregory, A. Mihalyi, M. Edgar, N. Radwell, and M. Padgett, Resolution- enhanced quantum imaging by centroid estimation of biphotons, Optica6, 347 (2019)

    work page 2019

  29. [29]

    Zhang, Z

    Y. Zhang, Z. He, X. Tong, D. C. Garrett, R. Cao, and L. V. Wang, Quantum imaging of biological organisms through spatial and polarization entanglement, Science Advances10, eadk1495 (2024)

    work page 2024

  30. [30]

    Z. He, Y. Zhang, X. Tong, L. Li, and L. V. Wang, Quan- tum microscopy of cells at the heisenberg limit, Nature Communications14, 2441 (2023). 11

    work page 2023

  31. [31]

    Defienne, P

    H. Defienne, P. Cameron, B. Ndagano, A. Lyons, M. Re- ichert, J. Zhao, A. R. Harvey, E. Charbon, J. W. Fleis- cher, and D. Faccio, Pixel super-resolution with spa- tially entangled photons, Nature communications13, 3566 (2022)

    work page 2022

  32. [32]

    M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, Imaging high-dimensional spatial entanglement with a camera, Nature Communications3, 1 (2012)

    work page 2012

  33. [33]

    Bhattacharjee, M

    A. Bhattacharjee, M. K. Joshi, S. Karan, J. Leach, and A. K. Jha, Propagation-induced revival of entanglement in the angle-oam bases, Science Advances8, eabn7876 (2022)

    work page 2022

  34. [34]

    K. Chan, J. Torres, and J. Eberly, Transverse entangle- ment migration in hilbert space, Physical Review A75, 050101 (2007)

    work page 2007

  35. [35]

    Defienne, W

    H. Defienne, W. P. Bowen, M. Chekhova, G. B. Lemos, D. Oron, S. Ramelow, N. Treps, and D. Faccio, Advances in quantum imaging, Nature Photonics18, 1024 (2024)

    work page 2024

  36. [36]

    C. H. Monken, P. S. Ribeiro, and S. P´ adua, Transfer of angular spectrum and image formation in spontaneous parametric down-conversion, Physical Review A57, 3123 (1998)

    work page 1998

  37. [37]

    D’Angelo, A

    M. D’Angelo, A. Valencia, M. H. Rubin, and Y. Shih, Resolution of quantum and classical ghost imaging, Phys- ical Review A—Atomic, Molecular, and Optical Physics 72, 013810 (2005)

    work page 2005

  38. [38]

    M. D. Reid, Demonstration of the einstein-podolsky- rosen paradox using nondegenerate parametric amplifi- cation, Physical Review A40, 913 (1989)

    work page 1989

  39. [39]

    Moreau, E

    P.-A. Moreau, E. Toninelli, P. A. Morris, R. S. Aspden, T. Gregory, G. Spalding, R. W. Boyd, and M. J. Padgett, Resolution limits of quantum ghost imaging, Optics ex- press26, 7528 (2018)

    work page 2018

  40. [40]

    Brambila, R

    E. Brambila, R. Guitter, R. Sondenheimer, M. Gr¨ afe, and H. Defienne, Certifying spatial entanglement be- tween non-degenerate photon pairs with a camera, Optics Letters50, 4854 (2025)

    work page 2025

  41. [41]

    M. J. Padgett and R. W. Boyd, An introduction to ghost imaging: quantum and classical, Philosophical Transac- tions of the Royal Society A: Mathematical, Physical and Engineering Sciences375, 20160233 (2017)

    work page 2017

  42. [42]

    Verni` ere and H

    C. Verni` ere and H. Defienne, Hiding images in quantum correlations, Physical Review Letters133, 093601 (2024)

    work page 2024

  43. [43]

    Fuenzalida, A

    J. Fuenzalida, A. Hochrainer, G. B. Lemos, E. A. Ortega, R. Lapkiewicz, M. Lahiri, and A. Zeilinger, Resolution of quantum imaging with undetected photons, Quantum6, 646 (2022)

    work page 2022

  44. [44]

    V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, Vol. 64 (Springer, 2013)

    work page 2013

  45. [45]

    P. E. Powers and J. W. Haus,”Fundamentals of Nonlin- ear Optics”(CRC Press, 2017)

    work page 2017

  46. [46]

    Cutipa, K

    P. Cutipa, K. Y. Spasibko, and M. V. Chekhova, Direct measurement of the coupled spatiotemporal coherence of parametric down-conversion under negative group- velocity dispersion, Optics Letters45, 3581 (2020)

    work page 2020