Choose exactly what counts as a trial
Use Itano and colleagues’ 1993 paper, Quantum projection noise: Population fluctuations in two-level systems, together with its 1995 erratum. The useful starting question is what gets prepared, what gets detected, and what is repeated. Their experiments used a single mercury ion and groups of beryllium ions to study fluctuations between two internal states.
One trial starts with a new preparation. A pulse prepares an atom so that measurement may find it in either internal state; fluorescence then identifies the outcome. Prepare again, measure again, and the outcomes fluctuate even when the detector classifies every state correctly.
Keep two counts separate in your notes. The many fluorescence photons used to identify one atomic state improve that identification. The number of freshly prepared atoms determines how many independent state outcomes have been sampled. More photons from an already measured atom improve the first of these counts.
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
Build the simplest distribution before fitting data
Imagine four independent atoms, all prepared with equal chances of the two outcomes. There are sixteen equally likely sequences. The excited count can be zero, one, two, three or four, with one, four, six, four and one sequences respectively. The mean count is two and its variance, the squared spread averaged over the sixteen sequences, is one.
For a fixed accepted total and a common independent excitation probability, the excited count follows a binomial distribution. The expression below gives its variance and the standard deviation of the measured fraction. The hat marks a fraction estimated from the outcomes. These symbols refer to accepted atoms before classification, with perfect state identification assumed.
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
Change the sample size and predict the scale
At even odds, one hundred independent atoms give a count standard deviation of five. Divide by the hundred atoms and the fraction has a standard deviation of 0.05. Four hundred atoms reduce that to 0.025. Four times the sample size halves this uncertainty.
These are constructed examples at a fixed probability, useful for checking an analysis. When examining data, compare the observed variance at different prepared populations with the expected curved dependence: small near a certain outcome and largest around even odds. Detector background has to be calibrated so that it can be distinguished from this state-dependent spread.
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
Add one shared disturbance and watch the prediction change
Suppose a preparation fluctuates between excitation probabilities of 0.4 and 0.6 with equal chance. Every atom in a given trial shares that setting. The average probability remains one half, but now a whole group tends to move upward or downward together.
For one hundred atoms, the binomial variance within a trial is 24. The change of the group mean from trial to trial adds another 100, giving a total variance of 124. At a fixed probability the variance would have been 25. This teaching calculation separates variation inside each trial from variation between trials.
A claim of projection-noise-limited operation therefore needs checks on shared drive fluctuations, atom-number changes and classification errors. Entangled or otherwise correlated atomic states also require a probability model that includes their correlations. The independent-atom formula is the reference for the preparation just defined.
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
Translate count noise into the frequency being measured
A clock turns an excitation fraction into a frequency by using the slope of a resonance. Check the slope as well as the fluctuation: for an ideal unit-contrast fringe whose maximum reaches excitation probability one, both the projection noise and the slope approach zero near that maximum, and their limiting ratio needs care. Extra detector noise changes the useful operating point. An especially quiet count is informative only together with its frequency response.
For the next paper, record the accepted atom total, the fresh-preparation cycle, the state-classification calibration, the observed variance and the resonance slope. Measuring a population in only one basis also leaves some superpositions and statistical mixtures indistinguishable; another phase-sensitive measurement can distinguish them. The 1995 erratum concerns an attribution about interpretations of quantum mechanics, so retain that correction when reading the historical discussion.
Read the evidence: W. M. Itano et al. (1993), Quantum projection noise: Population fluctuations in two-level systems, PRA 47, 3554–3570 — coauthor-hosted copyW. M. Itano et al. (1995), Erratum: Quantum projection noise, PRA 51, 1717