---
title: "How deep is the HOM dip for mixed photons?"
description: "Derive mixed-state HOM coincidences from pure-state overlap, then identify the assumptions needed to read visibility as purity."
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/time-frequency-quantum-optics/mixed-state-hom/"
last_modified: "2026-10-28"
---

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

L3 · Chapter 4

# How deep is the HOM dip for mixed photons?

Sandbox Physics · 2026-10-28

Identical mixtures cannot remove every coincidence. On different trials the photons can occupy different modes. Averaging their interference gives a coincidence probability determined by the overlap of the two density matrices.

## Before you begin

Density matrices, purity and Schmidt weights from chapter three; the balanced beam splitter from chapter one.

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

- Derive mixed-state coincidences by averaging pure-state pairs.
- Identify when visibility equals purity.
- Separate timing, internal-mode mixing and measured counts.

## Average over the possible pure-state pairs

Input one occupies $|u_a\rangle$ with probability $r_a$. Input two occupies $|v_b\rangle$ with probability $s_b$. Independence makes their joint weight the product. For each pair, the coincidence is one half times one minus squared overlap. Summing gives the trace expression.

The inputs must be independent. Correlated or entangled inputs require their joint state. We also assume one photon per input port, a balanced lossless splitter, and detection that does not resolve the internal modes being studied.

$$
P_{\mathrm{coin}}=\frac12\sum_{a,b}r_as_b\bigl(1-|\langle u_a|v_b\rangle|^2\bigr)=\frac{1-\operatorname{Tr}(\rho\sigma)}{2}
$$

The trace combines coherent overlap across both mode distributions. Pure inputs recover the squared-overlap expression.

Reading: C. K. Hong · P. J. Mosley et al.

## Identical preparation turns overlap into purity

If $\sigma=\rho$, the overlap is purity. For our equal-intensity examples, zero pair phase gives pure signal photons and zero coincidence. Half a turn gives equally mixed signals and coincidence one quarter. The pair intensity maps alone cannot predict this difference.

Two pure but orthogonal inputs have zero overlap and coincidence one half. A shallow dip can therefore reflect mixture or a mismatch between the sources. Reading visibility as purity requires independent evidence that the preparations and other assumptions match.

$$
\rho=\sigma\ \Longrightarrow\ P_{\mathrm{coin}}(0)=\frac{1-P}{2},\qquad V=\frac{P_{\mathrm{far}}-P_{\mathrm{coin}}(0)}{P_{\mathrm{far}}}=P
$$

The visibility relation also assumes a far-delay reference $P_{\mathrm{far}}=1/2$ and unchanged photon-number and background conditions during the delay scan.

Independent identical inputs with all other degrees of freedom matched.

| Pair phase | Purity | Zero-delay coincidence |
| --- | --- | --- |
| $0$ | $1$ | $0$ |
| $\pi/2$ | $3/4$ | $1/8$ |
| $\pi$ | $1/2$ | $1/4$ |

## Apply delay to the state being modeled

A delay $\tau$ multiplies spectral amplitudes by $e^{-i\omega\tau}$. This is a unitary operation. If two modes are actual frequency bins, delay changes their relative phase and can cause beating. A Gaussian envelope must follow from the chosen state.

The upgraded HOM experiment uses an explicit factorization: the two-mode mixture is an internal degree of freedom, with an independent Gaussian arrival wave packet and independent polarization. Their overlap factors multiply. This separates mixture, timing and polarization experimentally. Applying it to a continuous time-frequency state requires checking those independence assumptions.

$$
P_{\mathrm{coin}}(\tau)=\frac{1-\operatorname{Tr}[\rho\,U(\tau)\sigma U^\dagger(\tau)]}{2}
$$

This general expression supports continuous spectra and discrete frequency modes. The state and delay operator determine the actual curve.

## Keep probabilities separate from counts

Ideal probability times trial number gives expected coincidences. Mode-independent detector efficiencies scale those counts, and finite sampling adds fluctuations. Extra pairs at higher gain create more detection paths; threshold detectors merge some outcomes. The simple single-pair interpretation then needs replacement.

Ideal probabilities, expected counts and synthetic acquisition records describe different objects. Changing source or detection conditions in the HOM experiment clears the incompatible scan; changing delay can add points under the current conditions. Export records and settings before comparing preparations so physical changes remain distinguishable from sampling variation.

## Try the prediction

### [Separate mixing from mistiming](https://sandboxphysics.com/labs/hong-ou-mandel/)

1. Choose single-pair input, matched polarization and zero delay. Compare internal phase zero and half a turn.
2. Predict coincidences zero and one quarter before reading the results.
3. Hold the mixture fixed and scan delay. This delay acts on an independent Gaussian arrival packet, not propagation of the two frequency bins.

## Exercises and explanations

1. Can both inputs have purity one and still give coincidence one half?

   Read the explanation

   Yes. Orthogonal polarization or orthogonal wave packets give pure inputs with zero mutual overlap. Purity describes one preparation; HOM compares two.

2. Does halving detector efficiency double the ideal zero-delay coincidence probability?

   Read the explanation

   No. Mode-independent loss changes observed counts, not the probability conditioned on one photon entering each port. A normalization based on actual click conditions needs a detector model including missed events, background and extra pairs.

[Previous: How one intensity map can hide two different purities](https://sandboxphysics.com/stories/time-frequency-quantum-optics/schmidt-and-purity/) [Next: Preparing a partner photon by selecting one wave packet](https://sandboxphysics.com/stories/time-frequency-quantum-optics/coherent-heralding/)

## 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.