---
title: "Time-frequency quantum optics"
description: "Follow photon overlap into Schmidt modes, coherent heralding and tomography through eight chapters, worked examples and three experiments."
canonical_url: "https://sandboxphysics.com/stories/time-frequency-quantum-optics/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/time-frequency-quantum-optics/"
---

FR10

# Time-frequency quantum optics

Sandbox Physics · 2026-10-28

Equal joint spectral intensities can hide different phases and different photon purities. Start with detector clicks, follow one pair amplitude through heralding and interference, then test a reconstructed state on a new measurement.

## Begin with four equally bright possibilities

Rotate one phase arrow and all four joint intensities stay fixed. The heralded photon changes from a reproducible wave packet to a mixture. Following that small example reveals why measuring intensity is only part of describing quantum light.

All phases aligned

$F_A=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$

$P=1,\quad P_{\mathrm{coin}}(0)=0$

One arrow reversed

$F_B=\frac12\begin{pmatrix}1&1\\1&-1\end{pmatrix}$

$P=\frac12,\quad P_{\mathrm{coin}}(0)=\frac14$

Equal cell brightness represents equal joint intensities. Arrow direction is complex phase. Coincidences assume independent identical heralded inputs and otherwise perfect matching. Original two-mode model figure.

Chapters one and two begin with clicks and pair amplitudes. Chapters three through six build reduced states, interference, conditional selection and tomography. The final two connect ideal measurements to conversion channels and a fixed paper calculation. Each chapter introduces the next tool when the question needs it.

[Shared textbook foundations](https://github.com/yangzichao/sandboxphysics/issues/27) · [Research route and implementation evidence](https://github.com/yangzichao/sandboxphysics/issues/40)

1. L1 · 1

   ## [What two detector clicks reveal about photon overlap](https://sandboxphysics.com/stories/time-frequency-quantum-optics/clicks-and-overlap/)

   Use beam-splitter coincidences to distinguish photon number, mode purity and overlap, then build an intuitive picture of time-frequency modes.

2. L2 · 2

   ## [Describing a photon pair with complex amplitudes](https://sandboxphysics.com/stories/time-frequency-quantum-optics/pair-amplitudes/)

   Build the low-gain pair state from pump, phase matching and collection, then replace the continuous spectrum with two orthogonal modes per arm.

3. L2 · 3

   ## [How one intensity map can hide two different purities](https://sandboxphysics.com/stories/time-frequency-quantum-optics/schmidt-and-purity/)

   Derive the reduced density matrix, Schmidt weights and purity from an equal-intensity phase example, with a worked two-mode calculation.

4. L3 · 4

   ## [How deep is the HOM dip for mixed photons?](https://sandboxphysics.com/stories/time-frequency-quantum-optics/mixed-state-hom/)

   Derive mixed-state HOM coincidences from pure-state overlap, then identify the assumptions needed to read visibility as purity.

5. L3 · 5

   ## [Preparing a partner photon by selecting one wave packet](https://sandboxphysics.com/stories/time-frequency-quantum-optics/coherent-heralding/)

   Compare coherent mode projection with unresolved intensity filtering, deriving conditional states, success probabilities and a complete purity-rate example.

6. L3 · 6

   ## [Reconstruct coherence from clicks and predict a new measurement](https://sandboxphysics.com/stories/time-frequency-quantum-optics/tomography-and-holdouts/)

   Reconstruct a two-mode state from three complementary counting bases, then test an unused coherent projection with sampling uncertainty.

7. L4 · 7

   ## [How a pulse gate approaches a single-mode measurement](https://sandboxphysics.com/stories/time-frequency-quantum-optics/conversion-kernels/)

   Connect sum-frequency transfer kernels to pump shaping, phase matching and detection effects, then explain finite-bandwidth mode cross-talk.

8. L4 · 8

   ## [Recompute a Gaussian coherence model before reading the experiment](https://sandboxphysics.com/stories/time-frequency-quantum-optics/paper-benchmarks/)

   Independently integrate a fixed 2026 preprint Gaussian density model, verify its purity formula, and distinguish that result from measured reconstruction.

## Three experiments along the way

- [Rotate a pair phase with its intensity fixed](https://sandboxphysics.com/labs/joint-spectral-phase/)
- [Compare pure and mixed inputs at a beam splitter](https://sandboxphysics.com/labs/hong-ou-mandel/)
- [Choose a herald mode and test its reconstructed partner](https://sandboxphysics.com/labs/quantum-pulse-gate/)

The new experiments use two orthogonal modes and synthetic click records. The HOM experiment treats internal-mode mixing independently from a Gaussian arrival packet and polarization. The final chapter separately integrates a continuous Gaussian density model from a versioned paper.

## Sources and calculation scope

1. [C. K. Hong, Z. Y. Ou and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Physical Review Letters 59, 2044 (1987)](https://doi.org/10.1103/PhysRevLett.59.2044)

   Original beam-splitter coincidence experiment. The course derives ideal independent single-photon overlap probabilities; synthetic records are not data from this experiment.

2. [P. J. Mosley et al., Heralded Generation of Ultrafast Single Photons in Pure Quantum States, arXiv:0711.1054v1; Physical Review Letters 100, 133601 (2008)](https://arxiv.org/pdf/0711.1054v1)

   Equation (1), Figures 1–4 and the source/collection discussion connect pump, phase matching, heralding and independent-source HOM. Figure 2 explicitly qualifies intensity-based purity by assuming no joint spectral phase variation. No crystal fit or measured points are reproduced.

3. [A. Eckstein, B. Brecht and C. Silberhorn, A Quantum Pulse Gate based on Spectrally Engineered Sum Frequency Generation, arXiv:1007.3215v2; Optics Express 19, 13770–13778 (2011)](https://arxiv.org/pdf/1007.3215v2)

   Equations (1)–(4), Figure 2 and the heralding application define the pump-times-phase-matching kernel, its Schmidt channels and ideal mode selection. Our finite two-channel Schmidt kernel is a dimensionless teaching reduction, not the paper’s PPLN parameter calculation.

4. [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](https://arxiv.org/html/2410.03606v1)

   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.

5. [A. Bhattacharjee et al., Coherent temporal filtering of multimode parametric down-conversion using a quantum pulse gate, arXiv:2608.12544v1 (12 August 2026 preprint)](https://arxiv.org/html/2608.12544v1)

   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.

6. [Sandbox Physics FR10 independent numerical benchmarks](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr10/benchmarks.json)

   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.

7. [Sandbox Physics FR10 model and source definitions](https://github.com/yangzichao/sandboxphysics/tree/main/docs/frontier/fr10)

   Defines the two orthogonal modes, coefficient convention, ideal heralding, finite two-channel Schmidt conversion kernel, synthetic counts, saved records and bounded paper calculation. These are the assumptions of the teaching models.