Applications of maximum likelihood estimations for analyzing photon counts in few atom experiments

L. Sanchez; M. F. Andersen; M. Weyland; P. Ruksasakchai

REVIEW 3 major objections 5 minor 41 references

Applications of maximum likelihood estimations for analyzing photon counts in few atom experiments

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Maximum likelihood fits of model fluorescence distributions that account for light-assisted collision loss recover atom-number proportions in a tight optical tweezer to few-percent accuracy from about 600 test events where thresholds fail.

desk verdict Useful model-based MLE recipe for tight-tweezer atom counting, with an in-sample validation that deserves an external test before you trust the numbers elsewhere. read the letter →

arxiv 2412.17434 v3 pith:BDYNWR55 submitted 2024-12-23 physics.atom-ph

M. Weyland , L. Sanchez , P. Ruksasakchai , M. F. Andersen This is my paper · ORCID

classification physics.atom-ph

keywords maximumlikelihoodestimationopticaltweezersatomnumberdistributionlight-assistedcollisionsphotoncountingfluorescencebootstraperrorfew-atomexperiments

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

When several atoms share a tight optical tweezer, the light used to detect them causes rapid atom loss through light-assisted collisions. That loss makes the recorded photon-count distributions overlap and become non-Poissonian, so the usual threshold or Poisson fits cannot tell how many atoms were present. This paper presents a maximum likelihood method that fits a weighted sum of model distributions explicitly including the loss process, recovering the atom-number proportions from a relatively small number of experimental runs: about 600 test events plus roughly 1400 control events give errors at the few-percent level. The same idea works without an analytic model if more data are available. Accurate atom-number readout is central to tweezer experiments, which count surviving atoms after interaction-loss events, for instance to read out internal states.

What carries the argument

The central object is the recursion relation for the model photon-count distributions $b_{i,k}(T)$: a piecewise-Poisson description in which counts at fixed atom number $i$ are Poisson with rate $\eta_i$, and loss events occur as a memoryless Markov process with constant rates $\gamma_{ij}$ from $i$ to $j$ atoms. Its work is to turn control histograms from known atom numbers into an analytic basis that can be evaluated at any exposure time and atom number, and to provide a closed-form two-atom distribution that separates the collision-loss channels. The fitting machinery is the Poisson log-likelihood maximized by a grid search over the proportions, and, for the event-based variant, an effective likelihood that averages over Poisson fluctuations in the control histograms. Bootstrap resampling supplies the confidence intervals on the fitted proportions. This combination is what allows accurate atom-number extraction from datasets as small as about 600 test events.

What would settle it

Take two-atom control histograms at two different detection-light detunings, fit the model parameters ($\{\eta_j\}$, $\{\gamma_{2j}\}$) to each, then fit a known two-atom mixture recorded at the first detuning using the parameters obtained from the second; a systematic offset beyond the bootstrap confidence intervals would falsify the constant-rate assumption and the claimed few-percent accuracy.

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Extended reading notes

Core claim

The paper's central claim is that the photon-count distribution for a known initial number of atoms in a tight tweezer can be modeled sufficiently well for fitting by assuming Poisson emission between loss events and memoryless loss at constant rates $\gamma_{ij}$. For an $i$-atom sample, the distribution $b_{i,k}(T)$ obeys a recursion relation whose first term is the no-loss Poisson contribution and whose second term integrates over the time of the first loss event; for up to two atoms this recursion closes into an explicit formula with detection rates $\eta_0,\eta_1,\eta_2$ and loss rates $\gamma_{20},\gamma_{21}$. Fitting these parameters to control histograms from known atom numbers, then fitting a mixture $f_k(\theta)=N_d\sum_i a_i b_{i,k}$ to a test histogram with the Poisson log-likelihood, determines the atom-number proportions $(a_0,a_1,a_2)$ with errors at the few-percent level from about 600 test and 1400 control events. The same procedure using measured instead of modeled basis distributions works when no model is available, at the cost of requiring better statistics, and the effective likelihood improves event-based fits in that regime.

Load-bearing premise

The method relies on the model in which a fixed number of atoms produces photons at a steady average rate and loses atoms at constant, memory-free rates set by the control data; if the real loss rates depend on conditions the controls do not match, the resulting atom-number proportions will be biased.

Editorial extensions

If this is right

With roughly 600 test and 1400 control events, atom-number proportions are recovered to within a few percent, a level that threshold and Poisson fits do not reach in the loss-dominated regime.
The recursion relation extends to three or more atoms, so the method promises to replace rudimentary counting in multi-atom tweezer experiments.
When no analytic model for the photon-count distributions exists, event-based fitting with the effective likelihood remains usable, at the price of requiring larger datasets.
The observed insensitivity of the photon-count distributions to atom temperature between 27 and 68 µK supports the use of control data taken under slightly different conditions, as long as the detection light rethermalizes the atoms.
Maximum likelihood outperforms least-squares for these count histograms, because the Poisson likelihood respects the discrete, non-negative nature of the data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The fitted loss rates $\gamma_{ij}$ are measurable outputs, so the method could double as a probe of light-assisted-collision dynamics rather than only a counting correction.
The Poisson assumption per atom-number segment may fail if saturation or high collection efficiency changes the count statistics; the event-based effective-likelihood route would be the fallback in that regime.
Splitting the exposure into time bins and fitting the recursion jointly would yield time-resolved loss curves, a natural extension that the paper does not discuss.
A direct stress test would be to vary the detection-beam power after calibration and check whether a single set of loss rates reproduces the observed histograms; if not, the fitted proportions will be biased.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about this one. The model-based recursion in Eq. (2) is the genuinely new and useful piece: it lets you construct fluorescence distributions for i atoms from those for fewer atoms, with Poisson photon statistics and constant loss rates, which turns overlapping non-Poisson histograms into a clean MLE problem. The paper shows it beats threshold methods and event-based histogram fits, especially when control statistics are limited. Second, the accuracy claim—few-percent errors from ~600 test events—is validated only in-sample, on resamples from the same experimental pool that calibrates the model. That doesn't make it wrong, but it does make it pool-specific.

Credit where it's due: the derivation of Eq. (3) is clear and the assumptions are explicit (Poisson per-atom counts, constant loss rates, negligible one-body loss during the exposure). The performance characterization is careful—RMSE over the whole parameter simplex and bootstrap confidence intervals. The comparison of model-based vs event-based fitting and the effective-likelihood correction for limited control histograms is a practical guide experimentalists will appreciate. The temperature-insensitivity test is honest about its design-dependent caveat.

The soft spot is exactly the one the stress-test note flags: if the constant-rate Markov loss model is misspecified—say a transient in the loss rate at the start of exposure, or intensity dependence—the control fits will absorb it, and same-pool resampling can't detect it. The authors do provide one diagnostic: an additional two-body loss channel would make the 2-atom histogram fit fail. But that doesn't cover all misspecifications. Also, no code or data are shipped, and the fitted eta_i and gamma_ij values aren't reported, which makes independent replication harder than necessary. These are addressable and don't break the central claim for the tested regime.

Bottom line: solid methods paper for experimenters in tight-tweezer few-atom work. Send it to a serious referee. I'd want an external validation—synthetic data from a known non-Markov loss process, or a different temperature/power set with independent calibration—before trusting the few-percent number elsewhere, but the method itself is sound.

Referee Report

3 major / 5 minor

Summary. The paper presents a maximum-likelihood method to infer the atom-number distribution of zero, one, or two atoms in a tightly focused optical tweezer from the photon-number distribution measured during fluorescence imaging. The method models the photon-count distribution for a fixed atom number as Poisson, and accounts for light-assisted-collision loss during the exposure via a Markov recursion (Eq. 2), yielding a closed form for up to two atoms (Eq. 3). Model parameters are calibrated on control histograms of known atom numbers, and the unknown proportions are found by maximizing a Poisson log-likelihood over a grid. The paper also studies an event-based variant that uses recorded histograms and an effective likelihood, and it tests the method's accuracy by resampling from experimental test and control pools. The main claims are that the model-based fit achieves few-percent accuracy with about 600 test and 1400 control events, outperforms least-squares fitting, and can be extended to situations without a model.

Significance. If the accuracy claim can be fully supported, this is a useful and practical tool for the cold-atom and quantum-information community. The explicit modeling of light-assisted loss during detection, the comparison of Poisson and effective likelihoods, and the bootstrap error estimates are valuable contributions. The paper is honest about the assumptions made in the model, and the model fits the control histograms well. However, the validation strategy is entirely in-sample, and the treatment of pre-detection loss is inconsistent between the model-based and event-based analyses, so the quantitative few-percent accuracy claim is not yet fully established.

major comments (3)

[Sec. V A and Sec. VIII B] The accuracy claim is validated only by resampling from the same experimental pool that is used for calibration. Section V A randomly assigns blocks from a single pool to test and control, so both the fitted b_i and the test histograms are drawn from the same underlying conditional distributions. As a result, the reported few-percent RMSE in Figs. 4–6 measures sampling fluctuations under the empirically measured distributions, not the correctness of the model in Sec. III. If Eq. (2) is misspecified—for example, if the light-assisted-collision loss rate is not constant during the exposure or depends on the instantaneous two-atom separation—the calibration in Sec. IV B will absorb the misspecification into effective values of eta_i and gamma_ij, and the test will still appear unbiased. The only external check, Sec. VIII B, varies the initial temperature, but the authors state that the detection light rethermalizes the atoms, so it does not test a different physical regime. I recommend adding an out-of-sample validation, for example simulated data generated from a different (e.g., time-dependent-loss) model, or an independent atom-number measurement, and reporting how the method performs under those conditions.
[Sec. IV C, Sec. V A, and Appendix A] The treatment of the pre-detection loss probability ell is inconsistent between the model-based and event-based fits. The model-based fit in Sec. IV C uses pure distributions b_i,k, i.e., the photon-count distributions conditional on i atoms being present at the start of the detection exposure. Consequently, the fitted proportions a_i are the actual atom-number distribution at that moment. In contrast, the validation ground truth in Sec. V A is the distribution of the preparation-stage labels (the number of atoms loaded into the tweezers), which is related to the actual distribution by binomial thinning with survival probability 1 - ell. The paper corrects the event-based fits for ell in Appendix A (Eqs. A1–A3) but applies no corresponding correction to the model-based fits before comparing them to the known input proportions. For ell ~ 0.03, this introduces a systematic offset of order 1–5 percentage points in the reported errors, which is not separated from the statistical scatter in Figs. 4–6 and could be mistaken for the distribution-overlap effects discussed in Fig. 5. The authors should state explicitly which quantity (pre-detection label distribution or post-detection actual-atom-number distribution) is being estimated, and apply the appropriate transformation consistently for both model-based and event-based fits in the performance comparison.
[Sec. VII] The bootstrap error estimation in Sec. VII resamples only the test data and keeps the model distributions b_i,k fixed. The uncertainty in the model parameters (eta_i and gamma_ij), which are themselves estimated from a finite control sample, is therefore not propagated into the reported confidence intervals. The sensitivity to N_c seen in Figs. 4 and 6 shows that this uncertainty is non-negligible for small control sets, and the effective likelihood method of Sec. VI A explicitly attempts to account for it only in the event-based case. To make the bootstrap CIs reliable in general, the authors should either resample the control histograms as well (e.g., a nested bootstrap) or provide an estimate of the additional uncertainty and state the range of N_c over which the fixed-b_i approximation is valid.

minor comments (5)

[Eq. (6)] The inequality '1 < r < 100' in the definition of the averaging over resamples should read '1 <= r <= 100'.
[Sec. IV B] The sentence 'This parameter gathers single atom losses...' is grammatically incomplete and would read better as 'This parameter accounts for single-atom losses proceeding from collisions with background gas, from the merge process, and from false 1-atom detection with the EMCCD camera.'
[Fig. 4] The labels in the inset of Fig. 4 are very small; please enlarge the inset or describe the color scale in the caption so the reader can follow the dependence of Delta on both N_d and N_c.
[Sec. III A] The symbol T in Eq. (2) is used before it is defined; the general model would benefit from an explicit statement at the start of Sec. III A that T denotes the exposure duration, with T = T_exp later in Sec. III B.
[Sec. VIII A] The phrase 'the F = 3 -> F' = 3 component was turned off' is ambiguous; more precisely, the frequency component resonant with that transition was turned off, not the transition itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: calibration and inference are separate, and the derivation is self-contained.

full rationale

The paper's central inference is a standard two-stage procedure. Section III derives the model distributions b_{i,k} from two stated assumptions (Poisson photon statistics during loss-free intervals and fixed-rate Markov loss events), giving recursion Eq. (2) and the closed form Eq. (3). The parameters {eta_j} and {gamma_ij} are calibrated in Sec. IV B by maximum-likelihood fits to control histograms with known atom numbers. The unknown atom-number proportions are then inferred by fitting Eq. (5) to a separate test histogram; the fitted parameters a_i are not used in the calibration, so the inference is not defined in terms of its target. The performance test in Sec. V uses a random train/test split of the experimental pool: control histograms determine b_{i,k}, and independent test histograms with known mixture proportions are fit to recover a_i. This measures end-to-end accuracy under the actual data-generating process. The self-citations (e.g., Refs. [12,18,21-24] for experimental procedures) are not load-bearing for the derivation; they describe the apparatus rather than justify the statistical method. The in-sample limitation noted by the skeptic—that control and test data share the same experimental conditions—is a generalizability caveat, not a circular step, and the paper explicitly acknowledges the temperature-insensitivity is design-dependent. No equation or fitted parameter is renamed as a prediction, and no uniqueness claim is imported from prior work by the authors. The validation would detect systematic model bias because the test histograms are drawn from the true distributions with known atom-number labels, independent of the control fits. Overall, the derivation chain is self-contained and non-circular.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim depends on six fitted parameters and on domain assumptions about Poisson photon statistics, Markov loss, negligible single-atom loss during exposure, and no extra two-body loss. The paper introduces no new physical entities. The main limitation is that calibration and validation share the same experimental pool, so systematic biases common to both are not tested.

free parameters (6)

eta_0 = not reported in text
Background photon detection rate fitted to the 0-atom control histogram as a Poisson mean over Texp.
eta_1 = not reported in text
Single-atom photon detection rate fitted to the 1-atom control histogram.
eta_2 = not reported in text
Two-atom photon detection rate fitted to the 2-atom control histogram.
gamma_20 = not reported in text
Light-assisted collision loss rate from two atoms to zero atoms, fitted to the 2-atom control histogram.
gamma_21 = not reported in text
Loss rate from two atoms to one atom, fitted to the 2-atom control histogram.
ell = 0.8% to 2.9% in examples
Probability of losing a single atom between imaging and detection, fitted to the 1-atom control histogram.

assumptions (6)

domain assumption Photon counts for a fixed atom number are Poisson distributed.
Invoked in Eq. (1) and throughout Sec. III; not derived from first principles, and the paper validates it by the shape of control histograms.
domain assumption Loss events occur at fixed rates gamma_ij, independent of history.
Invoked in Eq. (2); requires exponential waiting times and no memory between loss events.
domain assumption Single-atom loss during the 3.5 ms exposure is negligible (gamma_10 = 0).
Stated in Sec. III B based on an estimated timescale three orders of magnitude longer than Texp; the paper notes the recursion could handle nonzero values.
domain assumption No additional two-body loss beyond the modeled gamma_20 and gamma_21 channels.
Stated in Sec. IV B; if present, it would bias the control fit and would show up as a poor fit.
domain assumption Control histograms are representative of the conditions in the unknown experiment.
The entire calibration relies on this; temperature dependence is tested in Sec. VIII B, but other drifts are not.
standard math Standard maximum likelihood and bootstrap theory.
The estimators and confidence intervals in Secs. IV and VII rely on textbook results.

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Cite this review

Pith. "Pith review of Applications of maximum likelihood estimations for analyzing photon counts in few atom experiments." pith.science (2026). https://pith.science/paper/BDYNWR55

@misc{pith2026241217434,
  author       = {Pith},
  title        = {Pith review of: Applications of maximum likelihood estimations for analyzing photon counts in few atom experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDYNWR55}},
  note         = {Machine review of arXiv:2412.17434}
}

read the original abstract

We present a method for determining the atom number distribution of few atoms in a tight optical tweezer from their fluorescence distributions. In the tight tweezer regime, the detection light causes rapid atom loss due to light-assisted collisions. This in turn leads to non-Poissonian and overlapping fluorescence distributions for different initial atom numbers, and commonly used threshold techniques fail. We use maximum likelihood estimation algorithms to fit model distributions that account for the atom loss. This gives accurate atom number distributions for relatively few experimental runs (about 600 is sufficient) to sample a photon number distribution. We show that the method can be extended to situations when the photon number distributions for known initial atom numbers cannot be modeled, at the cost of requiring a higher number of experimental runs.

Figures

Figures reproduced from arXiv: 2412.17434 by the authors.

**Figure 2.** FIG. 2. Example of measured histograms created from a total [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Example of a fitting result: the test histogram [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: shows the fit deviations ∆ of the maximum likelihood fit and their confidence interval (blue) for different quantities of control and test events. As the number of control events increases the fits are more accurate: the deviations decrease and so do their confidence…

**Figure 7.** Figure 7: FIG. 7. Evolution of the confidence intervals of the atomic [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: FIG. 6. Deviation [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 8.** Figure 8: FIG. 8. Comparison of fitting performances using two differ [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Comparison of photon count distributions for zero, [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

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A. Browaeys and T. Lahaye, Nature Physics 16, 132 (2020)

2020