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Sandbox Physics
Time-frequency quantum optics

L4 · Chapter 8

Recompute a Gaussian coherence model before reading the experiment

Sandbox Physics ·

A small calculation makes the coherent-filtering experiment easier to read. Fix one version of its Gaussian spectral-density model, verify purity by direct integration, then ask what the apparatus and reconstruction contribute beyond that model.

Freeze the version and the calculation

The August 2026 coherent-filtering preprint by Bhattacharjee and colleagues compares coherent idler heralding with intensity filtering. We fix arXiv version one. Main-text Figure 3 compares theoretical distributions, measured scans and reconstructed purity for Gaussian filters. Supplementary section four specifies a partially coherent Gaussian model that can be checked separately.

We reproduce the density-integral relation in supplementary equations twelve and thirteen. We do not digitize Figure 3 measurements or identify a dimensionless width ratio with an experimental bandwidth in terahertz. The calculation checks an explicit model before addressing a complete experimental comparison.

Reading: A. Bhattacharjee et al. · Sandbox Physics FR10 independent numerical benchmarks

Separate spectral width from coherence width

Define dimensionless frequency x=(ω−ω0)/σsx=(\omega-\omega_0)/\sigma_s and ratio r=σs/σcr=\sigma_s/\sigma_c. The parameter σs\sigma_s controls the Gaussian amplitude envelope; σc\sigma_c controls coherence decay with frequency separation. Neither may be silently replaced by an intensity full width at half maximum.

On the diagonal, the two frequencies agree and the coherence factor is one. The spectral envelope fixes normalization. Away from the diagonal, greater separation and smaller coherence width reduce matrix entries. Purity integrates their squared magnitudes, so loss of off-diagonal coherence lowers purity.

Γ(x,x′)=1πexp⁡ ⁣[−x2+x′22]exp⁡ ⁣[−r2(x−x′)22]\Gamma(x,x')=\frac{1}{\sqrt\pi}\exp\!\left[-\frac{x^2+x'^2}{2}\right]\exp\!\left[-\frac{r^2(x-x')^2}{2}\right]
This dimensionless form of supplementary equation twelve has explicit normalization ∫Γ(x,x) dx=1\int\Gamma(x,x)\,dx=1.

Rotate the double integral into two Gaussian integrals

Set u=(x+x′)/2u=(x+x')/\sqrt2 and v=(x−x′)/2v=(x-x')/\sqrt2. A rotation preserves the area element. Squaring the density makes the exponent −u2−(1+2r2)v2-u^2-(1+2r^2)v^2. Each direction integrates independently. The scaling rule for a Gaussian integral yields the expression below.

The independent script instead integrates squared density in the original frequency coordinates with trapezoid weights, then divides by its numerical trace squared. The first grid runs from minus six to six with 240 intervals; the second runs from minus eight to eight with 480 intervals. Across five ratios, the maximum absolute error is about one part in ten trillion.

P=∬∣Γ(x,x′)∣2 dx dx′=11+2r2P=\iint|\Gamma(x,x')|^2\,dx\,dx'=\frac{1}{\sqrt{1+2r^2}}
This is supplementary equation thirteen, independently checked here by direct quadrature. Infinite coherence width gives unit purity.
B=ΔΩ ΔT=1+r2,P=12B2−1B=\Delta\Omega\,\Delta T=\sqrt{1+r^2},\qquad P=\frac{1}{\sqrt{2B^2-1}}
These evaluations use supplementary equations fifteen and sixteen under the same Gaussian assumptions. We do not independently integrate the chronocyclic Q-function marginals in this reproduction. Applying them requires the stated Gaussian form and coherence assumptions.
00.5100.511.52
Vertical: purityHorizontal: spectral/coherence width ratio
Line: published Gaussian-model formula. Dots: original direct quadrature at five dimensionless width ratios. They are model calculations, not experimental measurements.
Original deterministic calculations retained with the generating script.
Spectral/coherence width ratioPurityMarginal bandwidth product
01.0000001.000000
0.50.8164971.118034
10.5773501.414214
20.3333332.236068

Reading: A. Bhattacharjee et al. · Sandbox Physics FR10 independent numerical benchmarks

Ask where each panel in Figure 3 comes from

The theoretical distribution is a forward calculation. The measured distribution comes from coincidence scans with mode probes. The density matrix and purity come from reconstruction with positivity and normalization constraints. A Gaussian bandwidth product provides intuition; the paper estimates purity through reconstruction.

The supplement also shows how a finite scan window can make distinct reconstructed states produce very similar measured distributions. A good image fit may leave state ambiguity. The holdout in chapter six makes that issue tangible in two modes. A continuous experiment also needs sufficient measurement coverage and calibrated effects.

Reading: A. Bhattacharjee et al.

Carry the calculation toward current research

Programmable multi-output sorting asks how reliably an instrument separates a whole mode set. The work introduced in the 2024 preprint was subsequently published in Optics Express. Its detector effects, output resolution and input encodings provide the next reading task. Insert calibrated effects into conditional-state predictions and test unused modes.

Higher gain introduces extra pairs. Drift mixes preparations during long scans, and loss may depend on mode. Each requires a larger state or measurement model. The two-mode examples separate phase, conditioning and counts so that an extension can identify which assumption fails first.

Reading: L. Serino et al.

Try the prediction

Test a conditional-state prediction on an unused mode

  1. Choose a source and coherent selector, then predict purity and success.
  2. Acquire synthetic counts, reconstruct from three bases and compare the holdout.
  3. Admit the second converted mode, acquire again and distinguish ideal-model, device and sampling effects.

Exercises and explanations

  1. What are purity and bandwidth product when spectral and coherence widths agree?

    Read the explanation

    The ratio is one, giving purity 1/31/\sqrt3 and bandwidth product 2\sqrt2. Substitution in the bandwidth-product relation agrees, within the same Gaussian assumptions.

  2. Does thirteen-digit numerical agreement establish the purity of the experimental device?

    Read the explanation

    It establishes agreement for this specified Gaussian calculation. Measurements and uncertainty analysis must determine whether an experimental state follows the model, the instrument is calibrated and the scan window is adequate.

Sources and calculation scope

  1. L. Serino et al., Programmable time-frequency mode-sorting of single photons with a multi-output quantum pulse gate, arXiv:2410.03606v1; DOI 10.1364/OE.544206

    Section 2 and Figure 2 specify the transfer kernel and the ideal/projective versus nonideal POVM description. Sections 3–4 explain detector tomography and output resolution. This work has an Optics Express journal DOI; the version read here is the 2024 preprint.

  2. A. Bhattacharjee et al., Coherent temporal filtering of multimode parametric down-conversion using a quantum pulse gate, arXiv:2608.12544v1 (12 August 2026 preprint)

    Main equations (1)–(5), Figures 1–4 and supplementary sections 1–7 were read for conditional filtering and reconstruction. The bounded reproduction is supplementary equations (12)–(13); (15)–(16) are evaluated analytically. We do not reproduce the measured Figure 3 points, tomography algorithm or full device.

  3. Sandbox Physics FR10 independent numerical benchmarks

    Original direct quadrature of the normalized Gaussian spectral density and separate matrix/eigenvalue calculations for the phase example. The adjacent reproduce-benchmarks.mjs script records grid windows, resolutions and errors. No historical measurements are included.

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