---
title: "Describing a photon pair with complex amplitudes"
description: "Build the low-gain pair state from pump, phase matching and collection, then replace the continuous spectrum with two orthogonal modes per arm."
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  zh-CN: "https://sandboxphysics.com/zh/stories/time-frequency-quantum-optics/pair-amplitudes/"
last_modified: "2026-10-28"
---

[Time-frequency quantum optics](https://sandboxphysics.com/stories/time-frequency-quantum-optics/)

L2 · Chapter 2

# Describing a photon pair with complex amplitudes

Sandbox Physics · 2026-10-28

A pair source determines more than whether a pair appears. It also sets the possible ways to share frequency between the photons. The joint spectral amplitude stores these possibilities with their phases. Heralding and interference start from that amplitude.

## Before you begin

Chapter one, probability and complex arrows. Integrals here mean sums over a continuous range of frequencies.

[Review the preceding chapter](https://sandboxphysics.com/stories/time-frequency-quantum-optics/clicks-and-overlap/)

- Separate pair-generation probability from the normalized single-pair state.
- Identify the roles of pump, phase matching and collection.
- Connect matrix indices to orthogonal frequency modes.

## Specify the events being retained

Spontaneous parametric down-conversion shares pump energy between two output fields. We call their paths signal and idler. At low gain the chance of producing a pair per pump pulse is small. We first condition on exactly one pair. Most pulses still contain vacuum; two-pair and higher contributions require a photon-number model.

The joint spectral amplitude, or JSA, is $F(\omega_s,\omega_i)$ after specified spatial and polarization collection. The subscripts label the two arms. Normalization makes the probabilities for all frequency combinations add to one. It does not set the pair rate.

$$
|\Psi\rangle=\iint F(\omega_s,\omega_i)|\omega_s\rangle_s|\omega_i\rangle_i\,d\omega_s\,d\omega_i,\qquad \iint|F|^2\,d\omega_s\,d\omega_i=1
$$

This is the conditional single-pair state. Absolute count rates also need generation probability, loss, efficiency and repetition rate.

Reading: P. J. Mosley et al. · A. Bhattacharjee et al.

## Contributions from the crystal must add coherently

The pump spectrum restricts the sum of the two output frequencies. Pair creation can occur throughout the crystal. Contributions from those positions also need compatible phases to add constructively. Phase matching describes this condition. A finite crystal produces a finite matching width and a phase. Collection optics then select the spatial and polarization amplitudes that reach the experiment.

A scalar propagation approximation organizes these factors as the product below. Here $\alpha$ is the complex pump envelope, $\Phi$ is the phase-matching amplitude and $C$ represents the specified collection operation. These are amplitudes, so multiplying intensity pictures alone loses information.

$$
F\propto\alpha(\omega_s+\omega_i)\,\Phi(\omega_s,\omega_i)\,C(\omega_s,\omega_i),\qquad \Phi\propto\operatorname{sinc}\!\left(\frac{\Delta kL}{2}\right)e^{i\Delta kL/2}
$$

$L$ is length and $\Delta k$ is the longitudinal wave-vector mismatch including the chosen quasi-phase-matching grating term. The convention is $\operatorname{sinc}x=\sin x/x$. The expression assumes the stated propagation and collection approximation.

Reading: P. J. Mosley et al.

## Replace the continuum with orthogonal modes

Choose two orthogonal frequency modes in each arm. They might be normalized wave packets in nonoverlapping frequency intervals. Rows label signal modes; columns label idler modes. Each matrix entry is a complex amplitude. Its squared magnitude is the probability of that mode pair.

The coefficient $F_{jk}$ is a projection of the continuous JSA onto mode functions. It is not simply a value at one frequency point. A sampled continuous integral needs quadrature weights absorbed into its discrete coefficients. Our two-mode experiment instead defines orthogonal discrete modes directly, with dimensionless coefficients.

$$
|\Psi\rangle=\sum_{j,k=0}^{1}F_{jk}|j\rangle_s|k\rangle_i,\qquad \sum_{j,k}|F_{jk}|^2=1
$$

Four complex numbers specify this single-pair model. They expose phase and conditioning without a full crystal-propagation calculation.

## Four equally bright entries

Set all four amplitudes to one half. Each mode pair has probability one quarter. Now rotate only the lower-right arrow through an angle $\theta$. Every intensity stays fixed. The next chapter follows this change through to single-photon purity.

Pump, material and collection geometry determine which amplitudes a source can realize. Fixing one specified pair amplitude lets us ask how different herald measurements change the partner state. This separates the physical roles of source design and measurement selection.

$$
F(\theta)=\frac12\begin{pmatrix}1&1\\1&e^{i\theta}\end{pmatrix}
$$

Phase is in radians. Multiplying the whole matrix by one common phase changes no prediction. Rotating one entry changes coherent sums between rows.

## Try the prediction

### [Match phase arrows to the intensity cells](https://sandboxphysics.com/labs/joint-spectral-phase/)

1. Choose the equal-phase preset and check that the four intensities sum to one.
2. Rotate only the lower-right phase and predict which intensity map changes.
3. Follow the signal coherence and explain how its state can change with fixed joint intensities.

## Exercises and explanations

1. Must two sources with the same normalized coefficient matrix deliver equal detected pair rates?

   Read the explanation

   No. The matrix describes mode probabilities conditioned on producing and collecting one pair. Repetition rate, pair probability, path loss and detector efficiency affect the number detected per second.

2. How does multiplying every coefficient by $i$ differ from multiplying only the lower-right coefficient by $i$?

   Read the explanation

   The first adds a common phase to the whole state and leaves all probabilities and density matrices unchanged. The second changes a relative phase that coherent measurements can detect.

[Previous: What two detector clicks reveal about photon overlap](https://sandboxphysics.com/stories/time-frequency-quantum-optics/clicks-and-overlap/) [Next: How one intensity map can hide two different purities](https://sandboxphysics.com/stories/time-frequency-quantum-optics/schmidt-and-purity/)

## Sources and calculation scope

1. [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.

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)](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.