Photon Statistics and the Mandel Q Parameter

Field quantization hands optics a startling reappraisal of what "how bright is this light" even means. Once the single-mode field is quantized — the result carried in from aq-optics-field-quantization and aq-optics-fock-states — its energy comes in discrete photons, and the classical intensity dissolves into a probability distribution over photon number: $P(n)=\langle n|\hat\rho|n\rangle$ , the chance that a detector integrating the mode registers exactly $n$ quanta. This distribution, not a single intensity, is the true observable of photon counting — what a photomultiplier or nanowire actually samples, one click at a time. The remarkable fact is that different sources of identical mean brightness produce visibly different $P(n)$ . An ideal laser gives a Poisson distribution — the statistics of independent, memoryless arrivals. A filtered thermal source gives a broader geometric (Bose–Einstein) distribution whose photons arrive in bunches. And certain engineered emitters give distributions narrower than Poisson, which no classical fluctuating wave can imitate. The width of $P(n)$ relative to its mean is a fingerprint of how the light was made. Leonard Mandel's 1979 $Q$ parameter compresses that comparison into one dimensionless number, anchored on the Poisson fact that variance equals mean. $Q=0$ marks coherent light, the classical–quantum boundary; $Q>0$ marks classically explicable bunched light; $Q<0$ marks genuinely nonclassical sub-Poissonian light with no positive $P$ -function. Photon statistics is thus the gateway to $g^{(2)}$ , antibunching, and single-photon sources — the vocabulary separating classical from quantum light.

Key takeaways

A light mode is described quantum-mechanically by its photon-number distribution $P(n)=\langle n|\hat\rho|n\rangle$ — the probability of finding exactly $n$ photons — not by a single intensity.
The Mandel parameter $Q=(\langle(\Delta\hat n)^2\rangle-\langle\hat n\rangle)/\langle\hat n\rangle$ measures how far the number variance departs from the Poissonian value (variance = mean).
Coherent light gives $Q=0$ ; thermal light gives $Q=\bar n>0$ (super-Poissonian, bunched); nonclassical light can reach $-1\le Q<0$ (sub-Poissonian).
Sub-Poissonian statistics ( $Q<0$ ) has no classical description — it forces the Glauber–Sudarshan $P$ -function negative or singular, the definition of nonclassicality.
$Q$ links to the intensity correlation through $Q=\langle\hat n\rangle[g^{(2)}(0)-1]$ , tying photon counting to Hanbury Brown–Twiss measurements.
Key idea: compare the photon-number variance to its mean — a Poisson distribution has them equal, so $Q$ is zero for coherent light, positive for classically bunched light, and negative only for genuinely nonclassical sub-Poissonian light.

The core insight

Compare the photon-number variance to its mean: for a Poisson distribution they are equal, so $Q=(\langle(\Delta\hat n)^2\rangle-\langle\hat n\rangle)/\langle\hat n\rangle$ is zero for coherent light, positive for bunched (classical) light, and negative only for genuinely nonclassical sub-Poissonian light.

Where this lives today

Photon statistics is now a routine benchmark. Every laboratory building a single-photon source — a quantum dot, a nitrogen-vacancy centre in diamond, a trapped ion — reports a Hanbury Brown–Twiss measurement of $g^{(2)}(0)$ as its headline figure of merit, with the best deterministic emitters pushed below $0.01$ ; a value under one-half is the threshold for calling a source "single-photon." The same statistics underwrite quantum key distribution, where a source that occasionally emits two photons opens a photon-number-splitting attack, making sub-Poissonian output a security requirement. On the super-Poissonian side, the bunching Hanbury Brown and Twiss exploited to measure stellar diameters is reborn in intensity interferometry across Cherenkov-telescope arrays. And amplitude-squeezed light engineered to sit at $Q<0$ now circulates in gravitational-wave detectors, where injecting sub-shot-noise light into the dark port has directly improved LIGO and Virgo's sensitivity beyond the standard quantum limit. In every case the idea is Mandel's: count the photons, compare variance to mean, and let the sign of the excess tell you what light you hold.

The physical picture

Picture photons landing on a detector like raindrops, and ask: in a fixed window, how much does the count wobble from window to window? If every photon arrives independently — no memory, no correlation — the count follows Poisson statistics, variance equal to mean. That is exactly what an ideal laser delivers: coherent light is "as random as independence allows," not orderly but uncorrelated — on the fence between classical and quantum. Thermal light is clumpy. The chaotic superposition of many uncorrelated emitters makes the field's intensity fluctuate, and where it spikes, photons arrive in a burst — inflating the variance above the mean, so the light is bunched, $Q>0$ . Sub-Poissonian light is the opposite: anti-clumpy. If the emitter is regularized — a single atom that must be re-excited before radiating again — successive photons space more evenly than chance dictates and the variance drops below the mean. That regime alone is off-limits to any classical wave, because a fluctuating classical intensity can only add noise.

Figure

Three photon streams carrying the same mean count of 10 detections per window — regular sub-Poissonian arrivals with Q negative, independent Poissonian arrivals with Q = 0, and clumped super-Poissonian arrivals with Q positive; only the regular stream has no classical wave description

The same story reads off the distribution's shape. Pin $P(n)$ to a single spike and you have a Fock state, the most sub-Poissonian light possible ( $Q=-1$ ); spread it into a Poisson bell and you have coherent light ( $Q=0$ ); broaden it into a monotonically decaying geometric curve and you have thermal light ( $Q=\bar n$ ). The second figure overlays the two classical extremes at identical mean.

Figure

Photon-number distributions P(n) at equal mean ⟨n⟩ = 4 — the Poisson distribution of a coherent state (blue, Q = 0) peaks near the mean, while the thermal Bose–Einstein distribution (amber, Q = 4, g⁽²⁾(0) = 2) decays monotonically and is broader; a single-photon Fock state would be one spike at n = 1 with Q = −1

Both curves have $\langle\hat n\rangle=4$ , yet the thermal one peaks at $n=0$ while the Poisson one humps near its mean — the difference is entirely in the width, which is what $Q$ reports.

The core equations

For number-counting purposes a single field mode is captured by the diagonal of its density operator in the Fock basis — the photon-number distribution, the raw material for all that follows.

P(n)=\langle n|\hat\rho|n\rangle \qquad (1)

where: $P(n)$ — probability of finding exactly $n$ photons; $\hat\rho$ — density operator of the field state; $|n\rangle$ — Fock eigenstate, $\hat n|n\rangle=n|n\rangle$ . Physically, $P(n)$ is the histogram a photon counter builds over many windows. From the distribution follow its first two moments — the mean (brightness) and the variance (spread of the counts), all $Q$ needs.

\langle\hat n\rangle=\sum_{n}nP(n),\qquad \langle(\Delta\hat n)^2\rangle=\sum_n(n-\langle\hat n\rangle)^2P(n) \qquad (2)

where: $\hat n=\hat a^\dagger\hat a$ — photon-number operator; $\langle\hat n\rangle$ — mean photon number; $\langle(\Delta\hat n)^2\rangle$ — variance. The variance is the whole game: its size relative to the mean is what distinguishes the sources. The Mandel parameter subtracts the Poissonian benchmark and normalizes by the mean, so pure independent statistics land at zero and everything else reads as fractional excess or deficit noise.

Q=\frac{\langle(\Delta\hat n)^2\rangle-\langle\hat n\rangle}{\langle\hat n\rangle} \qquad (3)

where: $Q$ — normalized excess variance; $Q=0$ Poissonian, $Q>0$ super-Poissonian (bunched), $Q<0$ sub-Poissonian (nonclassical). Because a Poisson distribution has variance equal to mean, the numerator vanishes for coherent light, so $Q$ is a pure meter of departure from independence. Rewriting the same variance with the operators in normal order exposes a bridge to what a Hanbury Brown–Twiss experiment measures — the second-order coherence at zero delay.

Q=\langle\hat n\rangle\big[g^{(2)}(0)-1\big] \qquad (4)

where: $g^{(2)}(0)=\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle/\langle\hat a^\dagger\hat a\rangle^2$ — normalized zero-delay intensity correlation. For a single stationary mode $Q$ and $g^{(2)}(0)$ carry the same verdict: $Q<0\iff g^{(2)}(0)<1$ , since $\langle\hat n\rangle>0$ . Two distributions anchor the classical extremes. The coherent state — the model of an ideal laser far above threshold — has a Poisson distribution set by its amplitude.

P(n)=e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!} \qquad (5)

where: $\alpha$ — coherent-state amplitude, mean $\langle\hat n\rangle=|\alpha|^2$ . The discrete Poisson law with $\lambda=|\alpha|^2$ ; equal mean and variance are the origin of $Q=0$ . Chaotic (thermal) light instead has the geometric Bose–Einstein distribution, falling monotonically in $n$ .

P(n)=\frac{\bar n^{\,n}}{(1+\bar n)^{n+1}}=\frac{1}{1+\bar n}\Big(\frac{\bar n}{1+\bar n}\Big)^{n} \qquad (6)

where: $\bar n$ — mean photon number, $1/(e^{\hbar\omega/k_BT}-1)$ for a single mode at temperature $T$ . Decaying from $n=0$ , thermal light is broad and bunched, with $Q=\bar n$ . Finally, the parameter is bounded below but not above, the floor reached only by a definite-photon-number state.

-1\le Q<\infty \qquad (7)

where: $Q=-1$ is the lower bound, attained solely by a Fock eigenstate whose zero variance drives the numerator to $-\langle\hat n\rangle$ . The unbounded ceiling is thermal bunching, growing without limit with brightness.

Derivation ACoherent light is Poissonian so Q vanishes

The goal is to show from scratch that the coherent state a laser prepares carries Poisson photon statistics, and that Poisson statistics force the Mandel parameter to exactly zero — fixing $Q=0$ as the reference against which every other source is graded, and justifying the slogan that coherent light is "as random as independence allows."

Step 1. Expand the coherent state in the number basis, $|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n}(\alpha^n/\sqrt{n!})|n\rangle$ , the normalized eigenstate of the annihilation operator, $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ . Why: the number distribution is read straight off the overlap with $|n\rangle$ .

Step 2. Project onto $|n\rangle$ and square: $P(n)=|\langle n|\alpha\rangle|^2=e^{-|\alpha|^2}|\alpha|^{2n}/n!$ , a Poisson distribution with parameter $\lambda=|\alpha|^2$ . Why: identifying the Poisson law lets us import its equal mean and variance rather than re-deriving them.

Step 4. For the second moment, $\hat n^2=\hat a^\dagger\hat a^\dagger\hat a\hat a+\hat a^\dagger\hat a$ gives $\langle\hat n^2\rangle=|\alpha|^4+|\alpha|^2$ , so $\langle(\Delta\hat n)^2\rangle=|\alpha|^2$ . Why: variance equals mean — the Poisson signature — from the algebra, not quoted.

Step 5. Insert into the definition: $Q=(|\alpha|^2-|\alpha|^2)/|\alpha|^2=0$ , equivalently $g^{(2)}(0)=1+Q/\langle\hat n\rangle=1$ . Why: this pins coherent light to the classical–quantum boundary, the anchor for every excursion. *Verified: standard Poisson result (Loudon ch. 5, Saleh and Teich ch. 12); $\langle\hat n\rangle=\langle(\Delta\hat n)^2\rangle=|\alpha|^2$ gives $Q=0$ .*

Go deeper · Derivation B · 1 minThermal light is super-Poissonian with Q equal to the mean

Derivation BThermal light is super-Poissonian with Q equal to the mean

The goal is to work out the photon statistics of chaotic light from its equilibrium density operator and expose the extra term that makes it noisier than a laser of equal brightness. The payoff is a pair of famous results — variance $\bar n+\bar n^2$ and a second-order coherence of exactly two — that certify the photon bunching Hanbury Brown and Twiss first detected.

Step 1. Write the thermal density operator, diagonal in number states, $\hat\rho_{\rm th}=\sum_n P(n)|n\rangle\langle n|$ with $P(n)=\bar n^{\,n}/(1+\bar n)^{n+1}$ , from the Boltzmann weight $e^{-n\hbar\omega/k_BT}$ and $\bar n=(e^{\hbar\omega/k_BT}-1)^{-1}$ . Why: chaotic light has no phase coherence — a classical mixture over number states.

Step 2. Set $x=\bar n/(1+\bar n)$ so $P(n)=(1-x)x^n$ , a geometric distribution. The mean is $\langle\hat n\rangle=x/(1-x)=\bar n$ , the Planck occupation. Why: the geometric series makes both moments elementary sums.

Step 3. The second moment is $\langle\hat n^2\rangle=x(1+x)/(1-x)^2=\bar n(1+2\bar n)=\bar n+2\bar n^2$ , so $\langle(\Delta\hat n)^2\rangle=\bar n+2\bar n^2-\bar n^2=\bar n+\bar n^2$ . Why: the variance splits into shot noise $\bar n$ plus an extra $\bar n^2$ of intensity noise — the face of bunching.

Step 4. Form the parameter: $Q=(\bar n+\bar n^2-\bar n)/\bar n=\bar n>0$ , growing without bound as the source brightens. Why: chaotic light is always super-Poissonian.

Step 5. Convert: $g^{(2)}(0)=1+Q/\bar n=2$ , the same value for every brightness. Why: the brightness cancels, leaving the universal factor-of-two bunching. *Verified: Bose–Einstein statistics and $g^{(2)}(0)=2$ (Loudon ch. 3, 6; Mandel and Wolf ch. 12); $\bar n=4$ gives variance $20$ , $Q=4$ , $g^{(2)}(0)=2$ .*

Go deeper · Derivation C · 1 minWhy classical light can never reach Q below zero

Derivation CWhy classical light can never reach Q below zero

The goal is to prove the most important structural fact about the Mandel parameter: any state a classical stochastic wave could produce has $Q\ge0$ , so a measured $Q<0$ is a rigorous certificate of nonclassicality. The argument turns the operator average into an integral over field amplitudes, then uses the fact that a variance is never negative.

Step 1. Represent the state in Glauber–Sudarshan form, $\hat\rho=\int P(\alpha)|\alpha\rangle\langle\alpha|\,\mathrm{d}^2\alpha$ , where classical light is exactly the case that $P(\alpha)$ is a non-negative, non-singular probability density. Why: the question of $Q$ 's sign becomes a question about the sign of $P(\alpha)$ .

Step 2. Because coherent states are eigenstates of $\hat a$ , normal-ordered averages become $c$ -number integrals: $\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle=\int P(\alpha)|\alpha|^4\,\mathrm{d}^2\alpha$ and $\langle\hat n\rangle=\int P(\alpha)|\alpha|^2\,\mathrm{d}^2\alpha$ . Why: normal ordering lets us treat $|\alpha|^2$ as a classical intensity distributed by $P(\alpha)$ .

Step 3. Then $\langle\hat n\rangle Q=\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle-\langle\hat n\rangle^2$ equals $\int P|\alpha|^4\,\mathrm{d}^2\alpha-(\int P|\alpha|^2\,\mathrm{d}^2\alpha)^2$ , precisely the variance of the classical intensity $I=|\alpha|^2$ under $P(\alpha)$ . Why: recognizing the combination as a variance is the whole trick.

Step 4. A variance over a genuine density satisfies $\mathrm{Var}_P(|\alpha|^2)\ge0$ , so $\langle\hat n\rangle Q\ge0$ , and since $\langle\hat n\rangle>0$ , $Q\ge0$ . Why: a classical intensity can only add noise to the counts, never remove it.

Step 5. By contraposition, observing $Q<0$ (equivalently $g^{(2)}(0)<1$ ) is impossible for any non-negative $P(\alpha)$ : it forces $P(\alpha)$ negative or more singular than a delta — the definition of nonclassical light. Why: a negative $Q$ is experimental proof of quantumness, not bookkeeping. *Verified: standard nonclassicality criterion (Loudon ch. 5; Mandel and Wolf ch. 12); $\langle\hat n\rangle Q=\mathrm{Var}_P(|\alpha|^2)\ge0$ for classical $P$ .*

Limiting cases

The sign and magnitude of $Q$ sort every source into one of four regimes.

Case	Result	Meaning
Fock state $\vert n_0\rangle$	$P(n)=\delta_{n,n_0},\ Q=-1$	Definite number, zero variance: maximally sub-Poissonian. $\vert 1\rangle$ is the canonical nonclassical source.
Coherent state $\vert\alpha\rangle$	$P(n)$ Poisson, $Q=0$	Variance equals mean $\|\alpha\|^2$ : the classical–quantum boundary, the ideal laser.
Thermal (chaotic)	$P(n)$ Bose–Einstein, $Q=\bar n$	Variance $\bar n+\bar n^2$ exceeds mean: bunched, $g^{(2)}(0)=2$ for any brightness. A lamp or blackbody.
Sub-Poissonian nonclassical	$-1\le Q<0$	Variance below mean, $g^{(2)}(0)<1$ : no positive $P$ -function. Single emitters, squeezed light, regulated pumping.

Common trapSub-Poissonian statistics and photon antibunching are not the same thing

It is tempting to treat $Q<0$ , $g^{(2)}(0)<1$ , and "antibunching" as three names for one effect. For a single stationary mode the first two are strictly equivalent, because $Q=\langle\hat n\rangle[g^{(2)}(0)-1]$ ties their signs together. But antibunching proper is about the time dependence — that $g^{(2)}(\tau)>g^{(2)}(0)$ , counts growing more likely as the delay opens up — which is logically distinct from sub-Poissonian counts in a single window. In multimode or non-stationary fields one can occur without the other. Keep the number statistics ( $Q$ ) and the temporal correlation ( $g^{(2)}(\tau)$ versus $g^{(2)}(0)$ ) as separate diagnostics.

Worked examples

1 · Counting an attenuated laser (confidence builder)

A single-mode laser is attenuated to a mean count of $\langle\hat n\rangle=|\alpha|^2=4$ photons per window. Find the probability of zero, one, and two photons, the variance, and $Q$ .

Step 1. The laser is coherent, so $P(n)$ is Poisson with $\lambda=4$ (equation 5): $P(n)=e^{-4}\,4^{\,n}/n!$ . With $e^{-4}=0.01832$ : $P(0)=\mathbf{0.0183}$ , $P(1)=4e^{-4}=\mathbf{0.0733}$ , $P(2)=8e^{-4}=\mathbf{0.1465}$ . Even at mean four, roughly one window in fifty-five records nothing.

Step 2. For a Poisson distribution the variance equals the mean, $\langle(\Delta\hat n)^2\rangle=\mathbf{4}$ : pure shot noise, standard deviation $2$ photons.

Step 3. So $Q=(4-4)/4=\mathbf{0}$ and $g^{(2)}(0)=1+Q/\langle\hat n\rangle=\mathbf{1}$ : exactly on the classical–quantum boundary, no bunching, no antibunching. P(0) = 1.83%, variance = 4, Q = 0, g⁽²⁾(0) = 1.

2 · Thermal light of the same brightness

A single mode of filtered thermal light has mean $\bar n=2$ . Compare its variance, $Q$ , and zero-count probability against a coherent beam of the same mean.

Step 1. From the thermal derivation, $\langle(\Delta\hat n)^2\rangle=\bar n+\bar n^2=2+4=\mathbf{6}$ — three times the coherent variance of $2$ , the extra $\bar n^2=4$ being intensity noise the laser lacks. Hence $Q=\bar n=\mathbf{2}$ and $g^{(2)}(0)=\mathbf{2}$ , the universal factor-of-two bunching.

Step 2. The zero-count probability is $P(0)=1/(1+\bar n)=1/3=\mathbf{0.333}$ , versus $e^{-2}=\mathbf{0.135}$ for the coherent beam — chaotic light is more than twice as likely to deliver nothing, because it hoards photons into bursts. Thermal: variance 6, Q = 2, g⁽²⁾(0) = 2, P(0) = 33.3% vs the laser's 13.5%.

3 · How sub-Poissonian is a lossy single-photon source?

A gated single-photon source is meant to emit $|1\rangle$ , but losses mean it delivers one photon with probability $0.70$ and vacuum with probability $0.30$ per window (never two). Find $\langle\hat n\rangle$ , the variance, $Q$ , and $g^{(2)}(0)$ , and explain why the two measures disagree.

Step 1. Only $n=0,1$ occur, so $\langle\hat n\rangle=\mathbf{0.70}$ , and since $n^2=n$ here, $\langle\hat n^2\rangle=0.70$ too. The variance is $\langle(\Delta\hat n)^2\rangle=0.70-0.49=\mathbf{0.21}$ , below the mean.

Step 2. The Mandel parameter is $Q=(0.21-0.70)/0.70=\mathbf{-0.70}$ : negative, so nonclassical, but short of the ideal $Q=-1$ of pure $|1\rangle$ — the vacuum admixture drags it toward zero.

Step 3. The coherence is $g^{(2)}(0)=(\langle\hat n^2\rangle-\langle\hat n\rangle)/\langle\hat n\rangle^2=(0.70-0.70)/0.49=\mathbf{0}$ : perfect antibunching survives the loss, because $g^{(2)}(0)$ counts only two-photon events and there are none. Q = −0.70 (loss-degraded) yet g⁽²⁾(0) = 0 (loss-independent): a source can look ideal to a Hanbury Brown–Twiss test while its Mandel parameter is nowhere near −1.

Physics of the portrait

The Mandel parameter is inseparable from the man who built the experimental language of quantum light around it.

Origins · 1 minLeonard Mandel and how this was discovered

Leonard Mandel

1927–2001 · coherence theory and quantum optics

Leonard Mandel escaped Nazi Germany as a child, trained in London, and spent the heart of his career at the University of Rochester, where he became the quiet architect of experimental quantum optics. After Glauber and Sudarshan built the coherent-state theory of optical coherence, Mandel asked what its predictions would look like on a photon counter, and made the counter the arbiter of what is and is not classical light. In 1977, with Kimble and Dagenais, he reported photon antibunching in single-atom resonance fluorescence — the first light ever shown to have $g^{(2)}(0)<1$ . Two years later, in a three-page Optics Letters note, he introduced the parameter that bears his name to grade how far a counting distribution departs from Poisson, and in 1983 he and Short reported the first directly measured sub-Poissonian light. His textbook with Emil Wolf, Optical Coherence and Quantum Optics, remains the standard reference. Mandel's instinct throughout was that the deepest questions about light are settled not by philosophy but by carefully counting clicks — and the $Q$ parameter is that instinct distilled into one number.

History

1956

— Hanbury Brown and Twiss detect photon bunching in thermal light with an intensity interferometer

1963

— Glauber and Sudarshan formulate the coherent-state and P-representation theory of optical coherence

1977

— Kimble, Dagenais and Mandel observe photon antibunching in single-atom resonance fluorescence

1979

— Mandel introduces the Q parameter to grade departures from Poissonian photon statistics

1983

— Short and Mandel report the first directly measured sub-Poissonian light

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 5–6 · Foot, Atomic Physics, ch. 9 · Saleh and Teich, Fundamentals of Photonics, ch. 12 · Mandel and Wolf, Optical Coherence and Quantum Optics, ch. 12 · Gerry and Knight, Introductory Quantum Optics, ch. 5–6.

Provenance. The distribution $P(n)=\langle n|\hat\rho|n\rangle$ , the classical bound, and the normal-ordering identities follow Loudon (3rd ed., ch. 5–6) and Mandel and Wolf ch. 12; the $Q$ parameter is Mandel, Optics Letters 4, 205 (1979). A verifier should recompute: (1) the Poisson moments giving $Q=0$ for $|\alpha\rangle$ ; (2) the Bose–Einstein moments giving variance $\bar n+\bar n^2$ , $Q=\bar n$ , $g^{(2)}(0)=2$ ; (3) the rearrangement $Q=\langle\hat n\rangle[g^{(2)}(0)-1]$ ; (4) the $P$ -function positivity giving $Q\ge0$ classically. The anchors (thermal $g^{(2)}(0)=2$ , Fock $Q=-1$ ) are standard textbook results.

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon-Number Basis · Coherent States of Light. Leads to. Photon Antibunching · The Hanbury Brown–Twiss Experiment · Single-Photon Sources · Squeezed States of Light.

Ask the tutor: Ask me to walk through why an attenuated laser and a filtered lamp of the very same brightness give completely different photon-counting histograms, and where the line of nonclassicality falls between them.

Is P(n) something a real detector measures, or just a theoretical object?

It is directly measurable, with one caveat. A photon-number-resolving detector — a transition-edge sensor, a nanowire array, or a photomultiplier in the counting regime — repeatedly integrates the same mode over identical windows and builds a histogram of clicks per window, and that histogram is an estimate of $P(n)$ . The caveat is efficiency: any loss reshuffles the statistics (pulling $Q$ toward zero, as worked example 3 shows), so the measured distribution is the true one only for a lossless detector. In practice one corrects for known loss or reports the loss-robust $g^{(2)}(0)$ , unchanged by linear loss for two-level-emitter light.

Why is variance-equals-mean the natural dividing line rather than, say, zero variance?

Because independence, not order, is the meaningful reference. If photons arrive completely independently — no attraction, no repulsion, no memory — the count is Poisson-distributed, variance exactly equal to mean: the statistics of "pure chance" that an ideal laser produces. Anything noisier (variance above mean) means the photons are correlated and clump; anything quieter (variance below mean) means they anticorrelate and space out. Zero variance is just the extreme quiet end, reached only by a Fock state. So variance-equals-mean is the honest zero, separating correlated from anticorrelated, classical from not.

How can thermal light be "bunched" when its most likely outcome is zero photons?

These two facts are the same fact. The Bose–Einstein distribution decays monotonically from $n=0$ , so the single most probable count is zero — but it also has a long tail, and its variance $\bar n+\bar n^2$ far exceeds Poisson of the same mean. Bunching means photons are not spread evenly in time: the field's intensity fluctuates, and photons pile into the moments when it is high, leaving the rest empty. "Most likely zero" and "occasionally a big burst" are the two halves of bunching.

What exactly is the difference between Q and g-squared-of-zero?

For a single stationary mode they carry identical information, linked by $Q=\langle\hat n\rangle[g^{(2)}(0)-1]$ , so both flip sign at the classical boundary. The difference is robustness: $Q$ is an absolute excess-noise measure depending on the mean, so loss drags it toward zero, while $g^{(2)}(0)$ is a ratio of intensity correlations, untouched by linear loss for light built only from zero- and one-photon components. That is why experiments quote $g^{(2)}(0)$ when grading a single-photon source, and reserve $Q$ for the intrinsic photon-number squeezing of a bright beam.

Why does the classical bound use the Glauber–Sudarshan P-function specifically?

Because the $P$ -function is the representation in which "classical" has a clean definition. Any state is $\hat\rho=\int P(\alpha)|\alpha\rangle\langle\alpha|\,\mathrm{d}^2\alpha$ , a weighted mixture of coherent states — the most classical states the theory allows. If $P(\alpha)$ is a legitimate probability density (non-negative, no worse than a delta), the state is exactly what a classical stochastic wave of random amplitude would produce, every normal-ordered average becomes a statistical average over intensities, and the numerator of $Q$ is a genuine variance that cannot be negative. A sub-Poissonian state defeats this by forcing $P(\alpha)$ negative or more singular than a delta — a "quasi-probability" no classical ensemble can realize.

Does antibunching from a single atom violate any conservation law or classical intuition?

No conservation law — it violates only the classical assumption that light is a fluctuating wave. A single two-level atom holds just one quantum of excitation, so after emitting a photon it must be re-excited before emitting again; two photons cannot leave back-to-back. That dead time spaces the photons out, giving $g^{(2)}(0)=0$ and $Q<0$ . A classical wave has no such mechanism: its intensity can be large at two instants arbitrarily close together, so classically the zero-delay correlation can only be at least as large as at any other delay. The single atom breaks that inequality — the first light ever proven to have no classical description.