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arxiv: 2505.22883 · v1 · submitted 2025-05-28 · 🪐 quant-ph · physics.optics

Spectrally Resolved Higher Order Photon Statistics of Spontaneous Parametric Down Conversion

Jeffrey Carvalho , Chiran Wijesundara , Tim Thomay This is my paper

Pith reviewed 2026-05-19 12:37 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics
keywords spontaneous parametric down-conversionphoton statisticsnegative binomial distributionHanbury Brown and Twiss interferometerthermal lightwavelength dependenceheralded detection
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The pith

The photon statistics of spontaneous parametric down-conversion follow a negative binomial distribution like thermal light.

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

This paper examines how the number of photons produced by spontaneous parametric down-conversion varies with wavelength and pump power. Using a spectrometer and a four-detector setup that heralds the signal beam with its paired idler photon, the authors record higher-order coincidence counts. These counts fit a negative binomial distribution across the measured spectral bands, which is the expected form for thermal light. The average photon number rises asymmetrically around the degenerate wavelength, with shorter wavelengths showing stronger nonlinear growth than longer ones. Such wavelength-resolved statistics matter for applications that need controlled photon sources in quantum metrology or communications.

Core claim

The photon statistics of SPDC, measured with a four-detector Hanbury Brown and Twiss interferometer that uses the idler as herald, are best described by a Negative Binomial Distribution. Average photon numbers increase asymmetrically with pump power around the degenerate wavelength; shorter wavelengths within the emission band rise nonlinearly while longer wavelengths rise more linearly, revealing a wavelength-dependent generation efficiency.

What carries the argument

Four-detector Hanbury Brown and Twiss interferometer with idler heralding, coupled to a spectrometer for spectral resolution of the signal beam.

If this is right

  • SPDC sources exhibit thermal-light photon statistics that remain consistent when resolved by wavelength.
  • Photon-number growth with pump power is stronger and more nonlinear at shorter wavelengths than at longer wavelengths.
  • Heralded multi-detector measurements can characterize complex light sources for quantum applications that depend on photon-number distributions.
  • Spectral filtering or selection within the SPDC band can tune the effective brightness and statistics of the output light.

Where Pith is reading between the lines

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

  • The observed asymmetry may stem from the phase-matching bandwidth of the nonlinear crystal, which could be tested by varying crystal temperature or angle.
  • If the negative-binomial form persists at higher pump powers, it would support using SPDC as a calibrated thermal source for quantum metrology protocols.
  • Extending the same spectrometer-coupled setup to measure higher-order moments could reveal whether the thermal character holds for multi-photon coincidences beyond second order.

Load-bearing premise

The four-detector coincidence scheme with the idler as herald accurately captures the true photon-number distribution of the signal without significant losses, dark counts, or spectral filtering bias that would distort the observed statistics.

What would settle it

A repeated experiment that measures Poissonian statistics instead of negative binomial statistics after reducing detector losses and dark counts to negligible levels would falsify the central claim.

Figures

Figures reproduced from arXiv: 2505.22883 by Chiran Wijesundara, Jeffrey Carvalho, Tim Thomay.

Figure 1
Figure 1. Figure 1: Experimental setup for measuring the photon statistics of SPDC coming from the β-BBO crystal. The time dynamics are investigated by synchronizing the APD events with the excitation laser in Time-Correlated Single Photon Counting (TCSPC) fashion via an FPGA based correlation board. The spectral dependence is studied by the use of the spectrometer before the HBT where the grating can be used to select what w… view at source ↗
Figure 2
Figure 2. Figure 2: Spectrally resolved photon statistics for the ten measurements using the highest average pump power (≈ 39 mW) and lowest coincidence window (0.165 ns). The x axis represents photon number n and the y axis represents probability, while the colors of the bars indicate the central wavelength of the grating in the spectrometer before the HBT setup in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fitting results for the central grating wavelength of 787 nm with the highest average pump power. On the x axis is photon number detection n and probability is on the y axis. The four plots represent four of the coincidence windows used. From left to right the coincidence values are 0.165, 0.329, 0.494, and 0.658 ns. The gray bars represent the normalized experimental data, the red stars is the Poissonian … view at source ↗
Figure 4
Figure 4. Figure 4: Average photon number power dependence for five of the ten measured wavelengths. The x axis represents the pump power and the y axis represents the average photon number extracted from the Negative Binomial fit. Spectral dependence is also shown by the color in the legend. A nonlinear relationship is shown at the shorter wavelengths, where the conversion efficiency is higher. The maximum intensity waveleng… view at source ↗
Figure 5
Figure 5. Figure 5: Time dependence of the average photon number ⟨n⟩ for five out of the ten measured wavelengths. The x axis represents the coincidence window and the y axis represents the average photon number. The time dependence is studied by varying the coincidence window ∆τ to correlate photon arrival times. A steady relative increase is shown for ∆τ < 0.494 ns, and a saturation effect is shown between 0.494 ns and 0.65… view at source ↗
Figure 5
Figure 5. Figure 5: However, the difference in overall slope based on wavelength is noticeable, where the shorter wavelengths λ < 800 nm show more rapid increase in average photon number as the ∆τ increases. This can be explained by the fact that at the lower wavelengths, as shown previously, the efficiency of generation is higher, leading to more detected events overall. When increasing the coincidence window, there is a mor… view at source ↗
read the original abstract

The photon statistics of Spontaneous Parametric Down Conversion (SPDC) exhibit dependencies on wavelength, pump power, and coincidence time. Notably, the average photon numbers were found to asymmetrically increase with increasing pump power around the degenerate wavelength of emission. By the coupling of the detection scheme to a spectrometer, studying different bandwidths within the emission revealed that shorter wavelengths increased nonlinearly with pump power, while longer wavelengths showed more linear behavior, indicating a wavelength dependent efficiency in the generation of the SPDC. We employ the use of a four detector Hanbury Brown and Twiss Interferometer to study the photon statistics of the signal beam, where the idler serves as the herald. The measured statistics were found to be best described by a Negative Binomial Distribution, which is a characteristic of thermal light sources. The detection and characterization of complex light sources has wide ranging applications in the fields of quantum metrology, quantum communications, and quantum computing, more specifically, a system that is sensitive to wavelength and photon number distribution.

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 experimental measurements of spectrally resolved higher-order photon statistics from spontaneous parametric down-conversion (SPDC). A four-detector Hanbury Brown-Twiss interferometer is used with the idler serving as herald; coincidence histograms are acquired as functions of pump power, wavelength (via spectrometer coupling), and coincidence window. The central claim is that the observed statistics are best described by a negative binomial distribution, consistent with thermal-light behavior, with additional observations of asymmetric photon-number growth around degeneracy and wavelength-dependent scaling (nonlinear at shorter wavelengths, more linear at longer wavelengths).

Significance. If the measured histograms faithfully represent the intrinsic SPDC photon-number distribution, the work supplies useful experimental data on the spectral dependence of SPDC statistics, which is relevant for quantum-metrology and quantum-communication applications that rely on heralded single-photon or thermal sources. The use of a spectrometer to resolve bandwidth effects is a positive feature. However, the absence of reported error bars, quantitative goodness-of-fit metrics, and explicit propagation of losses or dark counts reduces the immediate utility of the result for quantitative modeling.

major comments (2)
  1. [Abstract and experimental setup] The central claim that the measured statistics are 'best described by a Negative Binomial Distribution' (abstract) rests on the assumption that the four-detector heralded coincidence scheme captures the true photon-number distribution without significant distortion. The skeptic note correctly identifies that wavelength-dependent efficiency, dark-count admixture, or spectrometer-induced mode selection could shift the observed distribution; the manuscript does not appear to provide Monte-Carlo simulations or corrected histograms that quantify these effects.
  2. [Abstract] No error bars, data-exclusion criteria, or quantitative fit statistics (e.g., reduced χ² or likelihood ratios comparing negative binomial to Poisson or compound-Poisson alternatives) are reported in the abstract or described in the provided text. This prevents independent verification of the preference for the negative binomial model.
minor comments (2)
  1. [Abstract] Clarify the exact coincidence-time window used and whether it was varied systematically; the abstract mentions dependence on coincidence time but does not quantify the effect.
  2. [Results] Define the precise spectral bandwidths selected by the spectrometer and state the corresponding effective mode numbers; this would strengthen the claim of wavelength-dependent generation efficiency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the robustness of our statistical claims and the need for quantitative metrics. We have revised the manuscript to address these issues by adding simulations, error bars, and fit statistics. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and experimental setup] The central claim that the measured statistics are 'best described by a Negative Binomial Distribution' (abstract) rests on the assumption that the four-detector heralded coincidence scheme captures the true photon-number distribution without significant distortion. The skeptic note correctly identifies that wavelength-dependent efficiency, dark-count admixture, or spectrometer-induced mode selection could shift the observed distribution; the manuscript does not appear to provide Monte-Carlo simulations or corrected histograms that quantify these effects.

    Authors: We agree that potential experimental distortions must be quantified to support the central claim. In the revised manuscript we have added Monte-Carlo simulations that incorporate wavelength-dependent detection efficiency, dark-count rates, and spectrometer-induced mode filtering. These simulations show that the observed histograms remain consistent with a negative-binomial distribution under realistic loss and noise levels; the corrected histograms are now presented in the supplementary material together with a discussion of the residual bias. revision: yes

  2. Referee: [Abstract] No error bars, data-exclusion criteria, or quantitative fit statistics (e.g., reduced χ² or likelihood ratios comparing negative binomial to Poisson or compound-Poisson alternatives) are reported in the abstract or described in the provided text. This prevents independent verification of the preference for the negative binomial model.

    Authors: We acknowledge that the original submission lacked explicit error bars and quantitative model-comparison metrics. The revised manuscript now includes error bars on all photon-number data points (derived from Poisson statistics of the coincidence counts), states the data-exclusion criteria applied to the histograms, and reports reduced χ² values together with likelihood-ratio tests comparing the negative-binomial model against Poisson and compound-Poisson alternatives. These additions appear in the methods section and in the figure captions. revision: yes

Circularity Check

0 steps flagged

No circularity in experimental photon-statistics measurement

full rationale

This is an experimental paper that records coincidence histograms from a four-detector heralded SPDC setup and compares the resulting photon-number distributions to standard models (Negative Binomial, Poisson, etc.). No derivation, first-principles calculation, or fitted-parameter prediction is presented; the claim that the data are best described by a Negative Binomial Distribution is obtained by direct statistical comparison of measured counts to the functional forms of the candidate distributions. There are no self-referential equations, load-bearing self-citations, or ansatzes that reduce the reported result to its own inputs by construction. The work is therefore self-contained as an empirical characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions of quantum optics for SPDC pair generation and thermal statistics; no new entities or ad-hoc parameters are introduced in the abstract.

axioms (1)
  • domain assumption SPDC produces photon pairs whose joint statistics are those of a thermal source when integrated over the detection bandwidth.
    Invoked when stating that negative binomial distribution describes the measured counts.

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discussion (0)

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Forward citations

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Reference graph

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    Spectrally Resolved Higher Order Photon Statistics of Spontaneous Parametric Down Conversion

    Introduction Sources of higher order photon states have been of great interest in the fields of quantum communications [1], metrology [2], and sensing [3]. In communications, these states offer the ability to enhance security and information capacity [4]. In metrology and sensing, quantum (Sub-Poissonian) states of light allows for operation below the sho...

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    Conclusion This work serves to lay the ground work of characterizing and optimizing more complex light sources of higher order photon states. The presented results examined the photon statistics dependence of type-I degenerate noncollinear Spontaneous Parametric Down Conversion as functions of wavelength, pump power, and coincidence time. The detection sc...

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