24.1 Provenance and structure of a phase-scan dataset
24.1.1 Inspecting a seeded sixteen-phase binomial interference scan
Throughout Chapters 1–23, we paired apparatus schematics with explicit probabilistic models and separated what a dataset can establish from what it leaves open. In this capstone investigation, you will audit a complete phase-scan dataset from raw detector counts to parameter estimates, uncertainty intervals, and residual diagnostics. The baseline dataset included in the interactive workbench is a reproducible synthetic reference scan generated with fixed random seed from the binomial interference model
across \(M = 16\) uniformly spaced phases \(\phi_m = 2\pi m / M\) (\(m = 0, 1, \dots, M-1\)) with \(N_m = 100\) trials per phase setting and true visibility \(V_{\mathrm{true}} = 0.82\).
24.1.2 Preserving raw counts, trial denominators, and generator metadata
Every scientifically auditable dataset—whether simulated for estimator validation or recorded from a laboratory interferometer—must preserve three layers of provenance: (1) whether the counts are synthetic or physical (and, if synthetic, the exact generator model, parameters, and pseudorandom seed), (2) the raw integer counts \(k_m\) alongside the trial denominators \(N_m\) before any background subtraction or postselection, and (3) the calibration of the independent control variable \(\phi_m\). Storing only pre-normalized ratios \(\hat{p}_m = k_m/N_m\) without \(N_m\) discards the information required to compute binomial variances and likelihood ratios.
24.1.3 Criteria for substituting a published laboratory dataset
You can substitute a real optical, neutron, or qubit Ramsey phase scan directly into the same analysis pipeline provided it supplies at least \(M \ge 8\) phase settings spanning a full \(2\pi\) period (or a calibrated non-uniform phase vector \(\{\phi_m\}\)), raw integer counts \((k_m, N_m)\) per setting, and documentation of dark-count rates and dead-time corrections.
Visual animation · Section 24.1
Provenance and structure of a phase-scan dataset
Interactive visual model for 24.1 Provenance and structure of a phase-scan dataset (Inspecting a seeded sixteen-phase binomial interference scan · Preserving raw counts, trial denominators, and generator metadata · Criteria for substituting a published laboratory dataset). Use Animate or Step to follow the physical state evolution and quantitative predictions.
24.2 Fourier-moment visibility estimation and plug-in variance
24.2.1 Unbiased cosine-moment estimator over a uniform phase grid
For a uniform phase grid \(\phi_m = 2\pi m / M\) (\(M \ge 3\)), the orthogonality identities
Similarly, the baseline mean is estimated unbiasedly by \(\hat{p}_0 = \frac{1}{M}\sum_{m=0}^{M-1}\hat{p}_m\) (expected value \(1/2\)), and any uncalibrated phase offset \(\delta\phi\) in \(p(\phi) = \frac{1}{2}[1 + V\cos(\phi - \delta\phi)]\) is captured by the quadrature sine moment \(\hat{V}_s = \frac{4}{M}\sum_{m=0}^{M-1}\hat{p}_m\sin\phi_m\), with total phase-invariant visibility \(\hat{V}_{\mathrm{tot}} = \sqrt{\hat{V}_c^2 + \hat{V}_s^2}\).
24.2.2 Binomial variance propagation and boundary breakdown near unit visibility
Because counts at distinct phase settings \(m\) are statistically independent with binomial variance \(\mathrm{Var}(\hat{p}_m) = p_m(1 - p_m)/N_m\), the exact variance of the linear estimator \(\hat{V}_c\) is
When \(V\) is close to \(1\) and \(N_m\) is small, settings near \(\phi_m = 0\) or \(\pi\) often yield \(k_m = N_m\) or \(k_m = 0\), making the plug-in factor \(\hat{p}_m(1 - \hat{p}_m) = 0\) underestimate the variance at the boundary and occasionally allowing \(\hat{V}_c > 1\). Adding a Jeffreys/Wilson continuity pseudocount (\(\tilde{p}_m = (k_m + 0.5)/(N_m + 1)\)) in the variance formula or computing a binomial profile-likelihood interval prevents zero-variance collapse at the boundaries.
24.2.3 Diagnosing unmodeled phase offsets, background, and detector imbalance
Once \(\hat{p}_0\), \(\hat{V}_c\), and \(\hat{V}_s\) are estimated, inspect the standardized Pearson residuals
Three diagnostic checks immediately catch systematic model failures: (1) if \(|\hat{p}_0 - 0.50|\) exceeds \(2\,\mathrm{SE}(\hat{p}_0)\), detector efficiency imbalance or dark background is shifting the mean; (2) if \(|\hat{V}_s|\) exceeds \(2\,\mathrm{SE}(\hat{V}_s)\), the zero-phase reference \(\phi = 0\) is shifted by \(\widehat{\delta\phi} = \arctan2(\hat{V}_s, \hat{V}_c)\); and (3) if \(\chi^2\) far exceeds the degrees of freedom \(\nu = M - 3\) (or if \(r_m\) shows a \(2\phi_m\) second-harmonic pattern), phase-calibration nonlinearity or multi-pass reflections are distorting the sinusoid.
Visual animation · Section 24.2
Fourier-moment visibility estimation and plug-in variance
Interactive visual model for 24.2 Fourier-moment visibility estimation and plug-in variance (Unbiased cosine-moment estimator over a uniform phase grid · Binomial variance propagation and boundary breakdown near unit visibility · Diagnosing unmodeled phase offsets, background, and detector imbalance). Use Animate or Step to follow the physical state evolution and quantitative predictions.
24.3 Writing a reproducible experimental audit report
24.3.1 Essential components: apparatus model, estimator, residuals, and code
A complete experimental audit report contains five self-contained sections:
Data provenance and apparatus schematic: State clearly whether the dataset is synthetic (with generator equation and seed) or physical (with source, analyzer, and detector specifications), and tabulate all raw triples \((\phi_m, k_m, N_m)\).
Explicit statistical model and estimators: Write down the forward probability model \(p(\phi_m)\) and the exact formulas for \(\hat{p}_0, \hat{V}_c, \hat{V}_s\) and their standard errors.
Point estimates and confidence intervals: Report \(\hat{V}_c \pm \widehat{\mathrm{SE}}(\hat{V}_c)\) alongside \(\hat{p}_0\) and \(\hat{V}_s\).
Residual and goodness-of-fit diagnostics: Plot \(\hat{p}_m\) against \(\hat{p}_{\mathrm{fit}}(\phi_m)\), plot the standardized residuals \(r_m\), and report \(\chi^2/\nu\).
Scope and limitations of the physical claim: State precisely which alternative hypotheses the data rule out and which remain open.
24.3.2 Bounding the claim: why a cosine fit does not certify nonclassicality
A high-visibility sinusoidal fringe \(p(\phi) = \frac{1}{2}(1 + V\cos\phi)\) in a single-port or two-port phase scan certifies first-order phase coherence \(V = |g^{(1)}|\), but by itself it does not distinguish single-particle quantum self-interference from a classical continuous-wave laser or radio field, which produces the exact same \(\cos\phi\) intensity curve. As established across Chapters 6, 16, and 17, certifying nonclassicality requires either heralded single-quantum anti-bunching (\(g^{(2)}(0) < 1\)), two-photon Hong–Ou–Mandel visibility \(V_{\mathrm{HOM}} > 0.50\), or a multi-setting Bell/contextuality inequality violation (\(|S| > 2\)).
24.3.3 Verification checklist for synthetic and physical interference audits
Before finalizing your report, verify that: (a) the estimator recovers \(V_{\mathrm{true}}\) within expected binomial scatter across multiple random seeds, (b) every symbol and unit is defined in MathJax notation, and (c) no interpretive or nonclassical claim exceeds what the recorded count table actually tests.
Visual animation · Section 24.3
Writing a reproducible experimental audit report
Interactive visual model for 24.3 Writing a reproducible experimental audit report (Essential components: apparatus model, estimator, residuals, and code · Bounding the claim: why a cosine fit does not certify nonclassicality · Verification checklist for synthetic and physical interference audits). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the Apparatus
Inspect the 16-phase binomial scan \((\phi_m, k_m, N_m)\), adjust the true visibility \(V_{\mathrm{true}}\), shot count \(N_m\), or phase offset \(\delta\phi\), and compare the Fourier cosine estimator \(\hat{V}_c \pm \widehat{\mathrm{SE}}(\hat{V}_c)\) against the quadrature sine residual \(\hat{V}_s\) and the standardized Pearson residuals \(r_m\).
Interactive model · Synthetic observations
Audit a synthetic phase scan
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
Can you turn counts into a reproducible, appropriately limited conclusion? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
Worked Check
Problem: Consider a compact \(M = 4\) phase scan at \(\phi_m \in \{0, \pi/2, \pi, 3\pi/2\}\) with \(N_m = 100\) trials per setting, yielding counts \(k_0 = 90\), \(k_1 = 52\), \(k_2 = 10\), and \(k_3 = 48\). Compute the baseline mean \(\hat{p}_0\), the cosine visibility estimate \(\hat{V}_c\), the quadrature sine estimate \(\hat{V}_s\), and the plug-in standard error \(\widehat{\mathrm{SE}}(\hat{V}_c)\).
Solution:
1. Convert counts to empirical probabilities \(\hat{p}_m = k_m/100\): \(\hat{p}_0 = 0.90\), \(\hat{p}_1 = 0.52\), \(\hat{p}_2 = 0.10\), and \(\hat{p}_3 = 0.48\). The baseline mean is
Thus \(\hat{V}_c = 0.800 \pm 0.042\), with baseline \(\hat{p}_0^{\mathrm{mean}} = 0.500\) and negligible phase offset (\(\hat{V}_s = 0.040 \pm 0.071\)).
Summary
Auditing an experimental quantum claim requires verifying every step from raw timestamped coincidence tables to background subtraction, efficiency normalization, and propagation of statistical uncertainties.
True visibility must be bounded with confidence intervals after accounting for accidental coincidences \(R_A R_B \tau_c\), dark counts, and postselection fractions.
A rigorous scientific conclusion explicitly states which alternative models the data rule out, which instrumental loopholes remain open, and how the analysis can be reproduced.
Practice Problems
Problem 24.1: Effect of an uncalibrated phase shift \(\delta\phi\) on \(\hat{V}_c\) and \(\hat{V}_s\)
Suppose the true probability curve has phase offset \(\delta\phi = 30^\circ\) (\(\pi/6\text{ rad}\)) and true visibility \(V = 0.80\), so \(p(\phi_m) = \frac{1}{2}[1 + V\cos(\phi_m - \delta\phi)]\). What are the expected values \(\mathbb{E}[\hat{V}_c]\) and \(\mathbb{E}[\hat{V}_s]\) on a uniform \(M = 16\) grid, and how are \(V\) and \(\delta\phi\) recovered from them?
Hint
Expand \(\cos(\phi_m - \delta\phi) = \cos\delta\phi\cos\phi_m + \sin\delta\phi\sin\phi_m\) and apply the discrete orthogonality identities.
Solution
By orthogonality, \(\mathbb{E}[\hat{V}_c] = V\cos\delta\phi = 0.80\cos 30^\circ \approx 0.6928\) and \(\mathbb{E}[\hat{V}_s] = V\sin\delta\phi = 0.80\sin 30^\circ = 0.4000\). Measuring \(\hat{V}_c\) alone would underestimate \(V\) as \(0.693\), whereas combining both moments recovers \(\sqrt{\hat{V}_c^2 + \hat{V}_s^2} = 0.800\) and \(\delta\phi = \arctan2(0.4000, 0.6928) = 30^\circ\).
Problem 24.2: Scaling of visibility uncertainty with \(M\) and \(N\)
For low visibility \(V \ll 1\) where \(p_m(1-p_m) \approx 1/4\) at every setting, show that \(\mathrm{Var}(\hat{V}_c) \approx 2/(MN)\), where \(MN\) is the total number of shots across the entire \(M\)-setting scan. How many total shots \(MN\) are required to reach a standard error \(\mathrm{SE}(\hat{V}_c) = 0.010\)?
Hint
Substitute \(p_m(1-p_m) = 1/4\) and \(\sum_{m=0}^{M-1}\cos^2\phi_m = M/2\) into \(\mathrm{Var}(\hat{V}_c)\).
Solution
Substituting \(p_m(1-p_m) \approx 1/4\) gives \(\mathrm{Var}(\hat{V}_c) \approx \frac{16}{M^2}\frac{1}{4N}\left(\frac{M}{2}\right) = \frac{2}{MN}\). Setting \(\sqrt{2/(MN)} = 0.010\) gives \(MN = 2 / 10^{-4} = 20{,}000\) total shots (for example, \(M = 16\) settings with \(N = 1250\) shots per setting).
Problem 24.3: Distinguishing background offset from reduced coherence
An interferometer with perfectly coherent paths (\(V_0 = 1.00\)) suffers from an unpolarized stray-light background that adds a constant click probability \(b = 0.10\) to one detector, changing the normalized detection probability to \(p(\phi) = 0.55 + 0.45\cos\phi\). What values of \(\hat{p}_0^{\mathrm{mean}}\) and \(\hat{V}_c\) will the Fourier audit report, and how does \(\hat{p}_0^{\mathrm{mean}}\) alert the experimenter that the \(0.90\) visibility is caused by background asymmetry rather than pure dephasing?
Hint
Compare \(\hat{p}_0^{\mathrm{mean}}\) for symmetric pure dephasing (\(0.50\)) with \(0.55\).
Solution
The audit reports \(\hat{V}_c = 2(0.45) = 0.90\) and \(\hat{p}_0^{\mathrm{mean}} = 0.55\). Pure dephasing leaves the mean probability strictly at \(0.50\); observing \(\hat{p}_0^{\mathrm{mean}} = 0.55\) far above \(0.50 + 2\,\mathrm{SE}(\hat{p}_0^{\mathrm{mean}})\) diagnoses an asymmetric background or detector efficiency mismatch rather than phase decoherence.
Key Takeaways
A reproducible experimental audit preserves raw counts \(k_m\), trial denominators \(N_m\), phase settings \(\phi_m\), and explicit provenance metadata (synthetic generator seed vs.\ physical calibration).
On a uniform \(M\)-point phase grid, \(\hat{V}_c = \frac{4}{M}\sum_{m=0}^{M-1}\hat{p}_m\cos\phi_m\) is an exact unbiased estimator of fringe visibility with plug-in variance \(\frac{16}{M^2}\sum_{m=0}^{M-1}[\hat{p}_m(1-\hat{p}_m)/N_m]\cos^2\phi_m\).
Checking the baseline mean \(\hat{p}_0\), the quadrature sine moment \(\hat{V}_s\), and the standardized Pearson residuals \(r_m\) separates genuine coherence decay from phase offsets, detector imbalance, and phase-scan nonlinearity.
First-order cosine interference measures coherence \(V = |g^{(1)}|\); certifying uniquely quantum behavior requires pairing fringe visibility with anti-bunching (\(g^{(2)}(0) < 1\)), two-photon HOM interference, or Bell/contextuality tests.