---
title: "Quantum projection noise: better detection still leaves a random count"
description: "Which fluctuations remain when a fixed number of identically prepared atoms are measured with an ideal detector?"
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language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/quantum-projection-noise-population-counts/"
last_modified: "2026-10-18"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Quantum projection noise: better detection still leaves a random count

Sandbox Physics · 2026-10-18

Which fluctuations remain when a fixed number of identically prepared atoms are measured with an ideal detector?

## Prepare again before counting again

Itano and colleagues studied population fluctuations after preparing trapped ions in superpositions of two internal states. Their 1993 experiments included a single mercury ion and groups of beryllium ions. The fluctuation size depended on the prepared state, rather than being a fixed detector background.

The relevant ensemble is repeated preparation followed by readout. Repeatedly detecting the same already projected state without restoring the preparation asks a different question. The number of fluorescence photons used to identify a state is also distinct from the number of independently prepared atomic outcomes.

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy

## State the independent-atom contract

Let the fixed atom number be the number accepted before state readout, and let every independent atom have the same excited-state probability. With ideal binary classification, the excited count has the binomial variance below. The estimated fraction divides that count by the fixed total.

This is a reference for independent preparation and detection, not a universal noise floor for every correlated quantum state. Atom-number variation, common drive fluctuations, readout errors and entanglement require additional terms or a different probability model. Counting correlations must be tested rather than removed by naming the noise quantum.

$$
\operatorname{Var}(K)=Np(1-p),\qquad \sigma_{\hat p}=\sqrt{\frac{p(1-p)}{N}}
$$

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy

## Enumerate a small example before scaling it

For four independent atoms with equal probabilities, the sixteen equally likely bit strings give excited counts from zero through four with multiplicities one, four, six, four and one. Directly summing those outcomes gives mean two and variance one; no noisy detector is needed to produce the distribution.

For one hundred independent atoms at equal probabilities, the count standard deviation is five and the fraction standard deviation is 0.05. Four hundred atoms reduce the fraction value to 0.025. These invented examples compare single preparations with the same probability; they do not establish an averaging law for a drifting clock.

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy

## Shared fluctuations can imitate excess atomic noise

Now invent a preparation error shared by every atom in a trial: the excitation probability is either 0.4 or 0.6 with equal chance. The average probability is still one half. For one hundred atoms, conditional binomial noise contributes variance 24 and trial-to-trial movement of the mean contributes 100, giving total variance 124.

The independent fixed-probability reference would give only 25. This calculation uses the variance decomposition into within-trial and between-trial contributions. More detected photons may improve state classification while leaving that common preparation error untouched. A variance-versus-population scan therefore needs control and detector calibrations.

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy

## A quiet population is not automatically a better clock

At zero or unit excitation probability, the ideal binomial count is deterministic. This alone does not choose the best frequency-sampling point: population slope and noise both vary. In the ideal Ramsey, projection-noise-only limit their ratio can remain finite as an extremum is approached; additional detector noise changes the optimization.

One population basis also cannot distinguish every coherent superposition from a mixture with identical diagonal populations. A second phase-sensitive measurement is needed for that distinction. The paper’s 1995 erratum corrected an attribution concerning interpretations of quantum mechanics; it did not turn the experiment into a discriminator between those interpretations.

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy W. M. Itano et al. (1995), Erratum: Quantum projection noise, PRA 51, 1717

## Read the estimate together with the count model

The Ramsey Lab retains finite count records under its declared sampling model. The qubit Lab supplies a separate comparison between state preparation and measurement bases. Neither recreates the original ion-trap detector or certifies a laboratory clock as projection-noise limited.

For any claimed improvement, retain the accepted atom count, preparation settings, state-classification calibration and frequency response. Then ask which variance the intervention reduces. An uncertainty statement becomes useful when the recorded trial and the probability model describe the same experiment.

Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy

## Continue with the models

These are existing teaching Labs. Their simulated records are separate from the historical observations discussed above.

- [Compare finite Ramsey population records](https://sandboxphysics.com/labs/ramsey-frequency/)
- [Compare states in more than one basis](https://sandboxphysics.com/labs/qubit-state/)

## Sources and reading notes

We consulted Itano et al.’s 1993 paper in a readable coauthor-hosted PDF and checked the 1995 erratum. The NIST PDF exposed no extracted text here, and local NIST retrieval returned 403. The binomial enumeration and common-noise example below are independent teaching calculations, not fits to the mercury or beryllium data.

1. [W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copy](https://www.magnoliagilligan.org/files/pubs/pdfs/itano_1993_quantum_projection_noise.pdf)
2. [W. M. Itano et al. (1995), Erratum: Quantum projection noise, PRA 51, 1717](https://journals.aps.org/pra/pdf/10.1103/PhysRevA.51.1717)

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