---
title: "Joint Spectral Phase & Photon Purity Interactive Lab | Sandbox Physics"
description: "Can identical joint intensities hide different photon purities? Joint intensity keeps each amplitude’s size and discards its phase."
canonical_url: "https://sandboxphysics.com/labs/joint-spectral-phase/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/joint-spectral-phase/"
---

FR10-A · The phase a spectrum misses

# Joint Spectral Phase & Photon Purity

**Can identical joint intensities hide different photon purities?** Joint intensity keeps each amplitude’s size and discards its phase. Those phases decide whether the two possible herald outcomes leave the same signal mode. Equal intensities can therefore hide either a pure signal photon or a mixed one.

Turn one pair amplitude through a full circle. The four intensities stay equal, but the heralded photon changes from one coherent mode to an equal mixture.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The model has two orthonormal collected spectral bins on each arm and a normalized pure, low-gain photon-pair state. A herald detector ignores the idler bin. The phase diagram represents joint-state amplitudes, not an apparatus or propagation. HOM predictions assume independent identical heralded states, one photon per input, a balanced lossless splitter and matched polarization. Multipair emission, loss and detector noise are omitted.

Ideal model predictions

## Intensity stays still. Purity changes.

### Joint intensity and reduced-state coherence

$\rho_s=FF^\dagger$

### A complete phase turn

$\mathcal P=\operatorname{Tr}(\rho_s^2),\qquad C_0=\frac{1-\mathcal P}{2}$

Read the coefficient matrix

$F(\theta)=\frac12\begin{pmatrix}1&1\\1&e^{i\theta}\end{pmatrix},\qquad |F_{jk}|^2=\frac14$

Rows name signal bins; columns name idler bins. Ignoring the idler outcome adds the two possible signal states as a mixture. The two eigenweights are the probabilities of the orthogonal Schmidt modes.

Physics tutorial

## Joint Spectral Phase & Photon Purity

**Background** A photon pair can occupy several frequency combinations at once. A joint-intensity measurement counts how often each combination occurs. It loses the relative phase between their amplitudes.

**Why it matters** A click in the idler arm heralds a signal photon. If the detector does not resolve frequency, that click can leave several different signal modes. Their mixture determines how well two independent heralded photons interfere.

Start with the essentials

- **Focus question**: Can identical joint intensities hide different photon purities?
- **One-sentence intuition**: Equal joint intensities do not determine whether different herald outcomes leave the same signal mode.

### Core mathematical model

#### The phase hidden from intensity

$$
F(\theta)=\frac12\begin{pmatrix}1&1\\1&e^{i\theta}\end{pmatrix},\qquad |F_{jk}|^2=\frac14
$$

Rows label two orthonormal signal bins, and columns label two orthonormal idler bins. Each cell has the same probability for every phase. Only the lower-right amplitude changes direction in the complex plane.

#### Ignoring the idler frequency

$$
\rho_s=FF^\dagger=\frac14\begin{pmatrix}2&1+e^{-i\theta}\\1+e^{i\theta}&2\end{pmatrix}
$$

Each possible idler outcome contributes one column of the coefficient matrix. Because the outcome is unread, their density matrices add. At zero phase the columns describe the same signal state; at half a turn they describe orthogonal signal states.

#### From Schmidt weights to interference

$$
\begin{aligned}\lambda_\pm&=\frac{1\pm|\cos(\theta/2)|}{2}\\\mathcal P&=\lambda_+^2+\lambda_-^2=\frac{3+\cos\theta}{4}\\K&=\frac1{\mathcal P},\qquad C_0=\frac{1-\mathcal P}{2}\end{aligned}
$$

The weights are the squared singular values of the pair amplitude. For independent identical mixed single photons at a balanced splitter, zero-delay coincidence depends on their state overlap, which here equals purity.

### Common difficulties

#### A joint phase differs from a local phase

**Typical misconception** A phase plate on either photon can turn just one joint amplitude.

**Better mental model** A local signal phase rotates a whole row, and a local idler phase rotates a whole column. Such local unitary changes preserve Schmidt weights. Rotating only one cell changes the joint state.

#### A mixed photon still carries exactly one photon

**Typical misconception** Low purity means there is sometimes no photon or more than one.

**Better mental model** This model always conditions on one generated pair. Purity describes uncertainty about the signal mode after the idler frequency is ignored. Vacuum, multipair emission and loss need a larger model.

### Run the experiment

1. 01

   #### Start with one signal mode

   Choose One shared mode. Predict whether the idler frequency can tell the two signal preparations apart.

   **What to observe:** All arrows align. Both herald outcomes leave the same signal state, the largest Schmidt weight is one, and ideal zero-delay HOM coincidence vanishes.

2. 02

   #### Keep the intensity and lose purity

   Choose Two equal modes and compare the four intensity cells with the previous preset.

   **What to observe:** Every intensity remains one quarter. The two signal states are now orthogonal; both Schmidt weights are one half. Purity is one half and ideal HOM coincidence is one quarter.

3. 03

   #### Find what changes under phase reversal

   Compare A hidden quarter turn with Reverse the phase. Then drag the gold clock through a full turn.

   **What to observe:** Opposite phases have the same purity and HOM coincidence, but the imaginary coherence changes sign. A phase-sensitive mode projection can distinguish them.

Physical conclusion

A joint spectrum needs phase information before it can predict heralded purity.

## Related experiments and articles

Time-frequency quantum optics · Reading path

### From photon clicks to pulse-mode selection

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Describing a photon pair with complex amplitudes**](https://sandboxphysics.com/stories/time-frequency-quantum-optics/pair-amplitudes/)
2. [Article **How one intensity map can hide two different purities**](https://sandboxphysics.com/stories/time-frequency-quantum-optics/schmidt-and-purity/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)