Common pitfalls

Chapter opening

Words such as “single photon,” “entanglement,” “superresolution,” “non-Poisson,” and “new physics” can make a problem sound larger than it is. The data have to be more specific. How does the event table become a joint probability? How is the correlation peak calibrated? How is the false-alarm rate computed? Under which assumptions can the model parameters still be estimated? The basic quantum-optical criteria, astronomical implementations of intensity interferometry, SPADE information limits, astrophysical masers and lasers, and statistical tests in astronomy are discussed in Glauber [1963] [Abeysekara et al., 2020, Brown and Twiss, 1957, Cash, 1979, Elitzur, 1992, Goodman, 1985, Hanbury Brown, 1956, Hanbury Brown, 1974, Kimble et al., 1977, Mandel and Wolf, 1965, Mandel and Wolf, 1995, Scargle et al., 2013, Sudarshan, 1963, Tsang et al., 2016].

Seeing photons is not the same as doing quantum astronomy

A single-photon detector is only the entrance to the data. If the analysis simply sums the event table of Eq. (1) in Chapter Mathematical and Physical Foundations into a mean count rate \(R=N_\gamma/T\), the result is still an ordinary light curve. The new information in quantum astronomy comes from the joint distribution of events. The simplest independent case, the second-order correlation, and pair counts were introduced in Eqs. (44) and (46) of Chapter Why quantum astronomy is needed. A useful test is whether the event labels enter the model. \(t\) is arrival time, in s or ns. \(d\) is a detector or telescope identifier. \(\nu\) is a frequency channel, in Hz or as a channel number. \(p\) is a polarization label. If all of these labels are discarded before analysis, the fact that the detector counts single photons is not enough to claim a new quantum observable.

A common mistake is to equate low photon number with quantum behavior. X-ray and gamma-ray astronomy have long used Poisson counts and Cash likelihoods, but many of those problems are still estimates of mean intensity or spectrum [Cash, 1979, Gehrels, 1986]. Conversely, radio and millimeter astronomy often operate at high occupation number, close to the classical-wave limit, yet still discuss coherence functions, phase, polarization, and quantum noise. The criterion is whether the model needs multi-photon joint probabilities or optical-field modes to be expressed.

An event table also does not generate new physics automatically. The Poisson count model was given in Eq. (9) of Chapter Mathematical and Physical Foundations, and the binned version in Eq. (118) of Chapter Data analysis for event tables. \(\mu_k=R_k\Delta t_k\) is the expected count in bin \(k\), and is dimensionless. If the source varies, \(\mu_k\) varies with time. If the effective exposure \(E_k\) changes because of clouds, dead time, or the selection function, the expectation should be written as \(R_kE_k\Delta t_k\). Comparing all \(N_k\) with one constant \(\mu\) and then announcing “non-Poisson” usually mistakes ordinary variability, bad weather, or exposure variation for a statistical anomaly. Bayesian Blocks, period searches, and likelihood-ratio tests can all handle these problems, but each has its own conditions of validity and trial factors [Gregory and Loredo, 1992, Protassov et al., 2002, Scargle, 1982, Scargle et al., 2013].

\(g^{(2)}=1\) is not a sufficient criterion

\(g^{(2)}(0)=1\) is often misstated as “the light must be a classical laser,” or in the opposite direction as “there is no quantum information.” A coherent state has a Poisson photon-number distribution and \(g^{(2)}(0)=1\), but \(g^{(2)}(0)=1\) is not a unique fingerprint of a coherent state. Multimode thermal light approaches 1 when the number of modes \(M\) is large, and finite time response dilutes the bunching peak further. The basic contrast-dilution form was given in Eq. (18) of Chapter Mathematical and Physical Foundations and Eq. (83) of Chapter Photon statistics and coherence functions. \(\tau_c\) is coherence time, in s. \(\Delta t_{\rm eff}\) is the effective detector-response or correlation-bin width, in s. \(M_{\rm eff}\) includes spatial, frequency, polarization, and temporal modes. A visible \(10\,{\rm nm}\) filter near \(500\,{\rm nm}\) gives \(\tau_c\sim10^{-14}\,{\rm s}\), while electronic bins are often \(10^{-9}\)–\(10^{-8}\,{\rm s}\). Even for \(M_{\rm eff}=1\), the contrast is only \(10^{-6}\)–\(10^{-5}\). The stellar-bunching experiment of Guerin, the AquEYE/Iqueye proposals, and VERITAS/MAGIC intensity interferometry all treat this dilution as a central quantity [Abeysekara et al., 2020, Abe et al., 2024, Acharyya et al., 2024, Guerin et al., 2017, Zampieri et al., 2016, Zampieri et al., 2021].

Figure 124 Dilution of g⁽²⁾ contrast by time bin, spectral bandwidth, and mode number. The left panel shows how Δt/τ_c and \(M_{\rm eff}\) suppress the bunching peak of single-mode thermal light; the dashed line marks a 10⁻⁵-level systematic floor. The right panel converts filter bandwidth into coherence time. With 1 ns sampling, broadband visible-light bunching can easily fall below the calibratable range.

Antibunching is a stronger diagnostic than \(g^{(2)}=1\), because \(g^{(2)}(0)<1\) cannot be produced by a classical random intensity. The resonance-fluorescence experiment of Kimble, Dagenais, and Mandel is the classic example [Kimble et al., 1977]. Seeing antibunching in astronomical sources is extremely difficult. The source normally contains many independent emitters, propagation paths, and detector-response effects, so the sub-Poisson signature of one quantum emitter is averaged over space, frequency, and time. A report that says “\(g^{(2)}\simeq1\), therefore it is a laser,” or “\(g^{(2)}\simeq1\), therefore there is no quantum information,” has not given enough evidence.

Intensity interferometry still needs calibration

Intensity interferometry is insensitive to atmospheric piston phase, but the correlation amplitude is still a calibrated quantity. Transparency, scintillation, mirror reflectivity, filter profile, photomultiplier gain, SPAD dead time, electronic crosstalk, and background light all enter the measured amplitude. A practical data model is

(357)\[\hat c_b(\tau)= \beta_b\,|V_b|^2\,h_b(\tau) +c_{{\rm inst},b}(\tau) +c_{{\rm sel},b}(\tau) +\epsilon_b(\tau).\]

\(\hat c_b=\hat g_b^{(2)}-1\) is the normalized correlation excess on baseline \(b\). \(\beta_b\) is the zero-baseline contrast or instrument-dilution factor. \(|V_b|^2\) is the squared astronomical visibility. \(h_b(\tau)\) is the normalized peak shape. \(c_{\rm inst}\) contains electronics and detector terms. \(c_{\rm sel}\) contains selection-function, weather, and background processing terms. \(\epsilon\) is random error. All terms are dimensionless correlation amplitudes. VERITAS explicitly models background light and zero-baseline correlation when fitting stellar angular diameters, while the MAGIC real-time correlator is designed around continuous recording, low crosstalk, and fast mode switching [Abeysekara et al., 2020, Abe et al., 2024, Acharyya et al., 2024].

Figure 125 How calibration errors can imitate correlation signals. In the left panel, an electronic artifact peak is more than an order of magnitude larger than the 5 × 10⁻⁵ stellar bunching peak. If template subtraction leaves a narrow residual, it contaminates ĝ⁽²⁾(0). The right panel shows that integration time reduces only the statistical error as T^−1/2; once a 10⁻⁵–3 × 10⁻⁵ systematic floor is present, more integration does not automatically improve the conclusion.

Null tests should cover time, space, channel, and integration behavior. A time-shift test offsets one event stream by much more than the instrument response and astronomical correlation width; the astronomical peak should disappear, while slow drifts and background structure remain. An off-source or off-band test moves to a position without starlight or to a spectral region without the target line; the zero-delay peak should not remain. Channel permutation cross-correlates detector, polarization, or spectral channels that should not be physically correlated. If a peak remains, crosstalk is the first suspect. Block convergence divides the data into time segments; the mean correlation amplitude should be stable, and the uncertainty should fall approximately as \(T^{-1/2}\) until a systematic floor appears. Guerin et al. showed that uncorrected TDC artifacts can saturate the SNR. The AquEYE/Iqueye literature also lists the effects of SPAD dead time, fiber delay, and environmental control on systematic error [Cova et al., 1996, Guerin et al., 2017, Jansweijer et al., 2013, Wahl et al., 2020, Zampieri et al., 2016].

Missing phase does not mean imaging is impossible

Another pitfall is the claim that “intensity interferometry has no phase, so it can only measure diameters and cannot image.” The Fourier definition of complex visibility was given in Eq. (5) of Chapter Mathematical and Physical Foundations and Eq. (92) of Chapter Spatial coherence and intensity interferometry. Intensity interferometry directly measures \(|V(u,v)|^2\), so it does lose the phase of the first-order complex visibility. But \(|V|^2\) still contains a large amount of structural information. Uniform disks, limb darkening, binaries, oblate rotating stars, and simple disks can be fit directly in \(|V|^2\). The main difficulties are degeneracies and mirror ambiguity. A mirror brightness distribution \(I(-\boldsymbol\theta)\) has visibility \(V^\ast(\boldsymbol u)\), so \(|V|^2\) is exactly the same; this phase-loss problem was shown in Fig. Figure 30 of Chapter Spatial coherence and intensity interferometry. Separating mirror solutions, asymmetric structure, and complex morphology requires multibaseline coverage, physical priors, regularized phase retrieval, or higher-order information such as third-order correlations [Dravins et al., 2012, Foellmi, 2009, Karl et al., 2022, Malvimat et al., 2014, Monnier, 2003].

Figure 126 Mirror degeneracy when only |V|² is measured. The left panel shows an unequal-brightness binary and its mirror image. The right panel shows that the two have identical |V|², while the visibility phase changes sign. Intensity interferometry can constrain structure, but multibaseline data, model priors, or higher-order correlations are needed to recover the missing directional information.

Third-order intensity correlations can, in ideal circumstances, provide information analogous to closure phase, but their signals are weaker than second-order correlations. If second-order correlation errors are already near the systematic floor, third-order correlations are limited by the same photon rate and calibration problems. Large arrays and extremely bright targets can explore this direction; first-generation science cases usually do better by starting with low-parameter \(|V|^2\) models and then adding prior-assisted imaging gradually.

Quantum superresolution cannot break diffraction arbitrarily

SPADE and related mode measurements correct a statistical misunderstanding: the Rayleigh criterion is not the information limit for every parameter estimation problem. For two weak, equal-brightness, incoherent point sources, direct imaging loses separation Fisher information when \(s/\sigma\ll1\), while ideal mode sorting retains finite Fisher information [Nair and Tsang, 2016, Tham et al., 2017, Tsang, 2015, Tsang et al., 2016]. This does not mean that an arbitrary complex image can be reconstructed from a finite number of photons. The Cramer–Rao bound and the quantum Fisher information for a Gaussian PSF were given in Eqs. (138) and (141) of Chapter Quantum estimation, the Rayleigh limit, and sub-resolution information. In those equations, \(s\) is the angular separation to be estimated, in radians, mas, or in units of the PSF width \(\sigma\). \(N_\gamma\) is the number of detected photons that enter the correct mode analysis. \(F_s\) is the Fisher information per photon, with units of angle to the power \(-2\). Mode measurement improves \(F_s\); it does not replace \(N_\gamma\), and it does not remove background, aberration, mode crosstalk, or centroid error.

Figure 127 Photon budget and systematic floor for quantum superresolution. The left panel shows the statistical-error scaling at s = 0.2σ for direct imaging and ideal mode sorting. Mode sorting greatly reduces the error, but a 0.003σ–0.01σ mode or centroid-error floor stops further improvement. The right panel uses a toy model to show how the photon requirement rises rapidly when higher moments or modes are recovered; a complex image cannot be reconstructed arbitrarily from a small photon sample.

In astronomy one must also distinguish the source’s own coherence from the first-order spatial coherence after propagation. Partial source coherence changes the mode probabilities, and the literature on whether the Rayleigh curse reappears turns on this condition [Ang et al., 2017, Larson and Saleh, 2018, Paúr et al., 2018, Tsang, 2017, Tsang and Nair, 2019]. It is therefore not valid to transfer the two-incoherent-Gaussian-source result directly to galaxies, accretion disks, jets, or strong-lensing arcs. Given a source model, PSF, background, and measurement basis, mode measurement can increase the information efficiency for some parameters. Outside those conditions, superresolution is only a slogan.

Astrophysical lasers, masers, and non-Poisson statistics

It is also wrong to assume that an astrophysical laser or maser must have \(g^{(2)}=1\). Stimulated emission can produce high brightness temperature, narrow lines, and strong polarization, but the astrophysical environment usually has no stable laboratory-style cavity. Unsaturated masers, saturated masers, velocity-coherent paths, pump fluctuations, multiple blended spots, and continuum background all reshape the photon statistics [Dulk, 1985, Elitzur, 1982, Elitzur, 1992]. In the Weigelt blobs of Eta Carinae, Fe II and O I natural-laser candidates require evidence from Ly\(\alpha\) pumping, population inversion, line width, spatial location, and continuum subtraction. Brown–Twiss–Townes heterodyne correlation has been proposed as a way to measure the width of narrow Fe II lines [Dravins and Germanà, 2008, Johansson and Letokhov, 2004, Johansson and Letokhov, 2005, Johansson and Letokhov, 2005].

If the observed light is a sum of independent components, \(I=\sum_a I_a\), and different components do not have intensity correlations with one another, the zero-delay correlation excess is approximately

(358)\[g_{\rm mix}^{(2)}(0)-1 = \sum_a f_a^2\left[g_a^{(2)}(0)-1\right], \qquad f_a=\frac{\langle I_a\rangle}{\sum_b\langle I_b\rangle}.\]

\(f_a\) is the dimensionless flux fraction. A thermal component that provides only \(20\%\) of the total flux contributes only \(0.04\) to the contrast even if its own \(g^{(2)}(0)-1=1\). If there are also \(M=100\) modes and \(\tau_c/\Delta t=10^{-5}\), the observed contrast falls to \(4\times10^{-9}\). Conversely, an unsaturated stimulated line with pump-intensity noise may show super-Poisson statistics. Such a result must first separate line, continuum, and instrument response before it is attributed to new physics.

Figure 128 Mixed statistics in astrophysical maser and laser candidates. The left panel shows how thermal-continuum bunching is diluted by the square of the narrow-line flux fraction; if the narrow line itself carries unsaturated intensity noise, a separate super-Poisson branch appears. The right panel shows how line width and electronic time bin dilute correlation contrast. MHz–GHz line widths require ns or faster timing to retain a large peak.

Non-Poisson statistics also have ordinary sources. A randomly varying source can be described as a Cox process: conditional on the instantaneous rate \(\Lambda\), the counts are Poisson, but \(\Lambda\) itself varies with time or environment. Then

(359)\[F_{\rm Fano} \equiv \frac{{\rm Var}(N)}{\mathbb E(N)} = 1+\frac{{\rm Var}(\Lambda)}{\mathbb E(\Lambda)}\]

is greater than 1 in the appropriate time bin. Dead time suppresses short intervals and can make \(F_{\rm Fano}<1\) or produce negative correlation at short delay. Afterpulsing creates positive correlation after the detector’s characteristic time. The afterpulsing probabilities of \(10^{-4}\)–\(10^{-2}\) discussed in Chapter Detectors, clocks, and event tables are already enough to contaminate many short-delay searches. Before declaring “non-Poisson,” one should plot Fano factor versus bin width, short-delay auto- and cross-correlations, dark-field data, off-source data, and time-shift backgrounds.

Figure 129 Ordinary sources of non-Poisson statistics. The left panel uses the Fano factor to distinguish steady Poisson counts, slow source variability, dead time, and afterpulsing. A slowly varying source becomes super-Poisson in long bins; dead time produces under-dispersion. The right panel shows that afterpulsing and dead time have opposite signatures in the short-delay correlation function. These instrument features should be quantified first using dark fields and cross channels.

False alarms and the boundary of new physics

A single low-probability event is another common false discovery route. If the tail probability for one test is \(p_0\), and the number of independent trials is \(N_{\rm trial}\), the probability of at least one occurrence is

(360)\[p_{\rm any}=1-(1-p_0)^{N_{\rm trial}} \simeq N_{\rm trial}p_0 \quad (N_{\rm trial}p_0\ll1).\]

\(N_{\rm trial}\) can come from time bins, frequency channels, baselines, targets, model grids, and windows selected after looking at the data. If one night searches \(10^6\) candidate delay, frequency, and target combinations, a single-trial event with \(p_0=10^{-5}\) is almost guaranteed to occur. Multiple testing is not bookkeeping. It decides whether a peak is an astronomical signal, an instrument anomaly, or a natural tail of the search space.

Figure 130 Poisson tails and multiple testing. In the left panel, a single bin with μ = 8 occasionally produces a high-count tail such as N ≥ 17. The right panel shows how the same single-trial p₀ becomes a high global false-alarm probability when many trials are searched. Event-table searches, period searches, cross-frequency correlations, and transient filters should report \(N_{\rm trial}\) or an equivalent global significance.

“Quantum astronomy” is also not the same as “quantum gravity.” Most of this book studies field statistics, mode measurements, coherence functions, instruments, and information limits. Black holes, the CMB, axions, cosmic birefringence, or dark-matter lensing enter the same program when they can be written as observable time, frequency, polarization, or spatial correlations, not because every topic requires Planck-scale physics. A checkable anomaly has to separate the data sources:

(361)\[d_{\rm obs}=d_{\rm sky}(\boldsymbol\theta) +d_{\rm inst}(\boldsymbol\phi) +d_{\rm sel}(\boldsymbol\psi) +n,\]

where \(d_{\rm obs}\) is the data vector, such as a delay histogram, \(|V|^2\), polarization correlation, or event time; \(d_{\rm sky}\) is the astrophysical model; \(d_{\rm inst}\) is the instrument model; \(d_{\rm sel}\) is the selection function; and \(n\) is noise. If these four terms have not been separated, an anomalous peak, polarization rotation, or non-Poisson count should not be assigned directly to new physics.

The final page of a research plan should return to the added observable. Is it mean intensity, a second-order correlation, a cross-frequency covariance, a polarization correlation, or a mode count? If the target is a second-order correlation, list how the event table becomes a delay histogram, how the time-shift background is defined, and how the zero-delay peak is calibrated. If the target is polarization or mode counting, list the channel definition, response matrix, background term, and covariance. Each quantity needs units, typical range, and validity conditions, such as thermal light, Gaussian PSF, weak light, low-parameter source model, or stable instrument response. At least two null tests should be chosen from time shift, off source, off band, dark field, wrong polarization, and injection recovery. Global significance and the systematic floor have to enter the conclusion together.