Photon Antibunching

Every light source answers the same question differently: given that a photon just clicked, how likely is the next to click behind it? The tools for asking it — the quantized field, the photon-number (Fock) states, and the normally ordered correlation functions — are inherited from this chapter's prerequisites; antibunching is what happens when they meet a source that will not emit two photons at once. A laser answers "just as likely as ever": arrivals are Poissonian and $g^{(2)}(\tau)=1$ for every delay. Thermal light answers "more likely": intensity fluctuations make photons travel in clumps, so $g^{(2)}(0)=2$ and detections bunch. Photon antibunching is the third, strictly non-classical answer, $g^{(2)}(0)<1$ : photons actively avoid one another and arrive more evenly spaced than chance allows. The origin is transparent for a single quantum emitter — an atom, ion, molecule, quantum dot, or colour centre. The instant it radiates it is left in its ground state and cannot emit again until re-excited, a recovery of order the excited-state lifetime; that enforced dead time is a real gap in the photon stream, and the probability of a second detection at zero delay collapses to zero, giving $g^{(2)}(0)=0$ for a perfect single emitter. No classical field can reproduce this — a positive intensity distribution forces $g^{(2)}(0)\ge 1$ — so the dip below unity is not a small correction but a qualitative signature: direct evidence that the electromagnetic field is granular, delivered one indivisible quantum at a time. It is the cleanest temporal proof that light is quantized.

Key takeaways

Photon antibunching is $g^{(2)}(0)<1$ : detected photons are more evenly spaced than the random Poissonian arrivals of laser light.
It is genuinely non-classical — any classical field obeys both $g^{(2)}(0)\ge 1$ and $g^{(2)}(0)\ge g^{(2)}(\tau)$ , so a dip below unity violates a hard classical inequality.
The canonical source is a single quantum emitter: after radiating one photon it must be re-excited before emitting again, forcing a dead time and $g^{(2)}(0)=0$ for an ideal single photon.
The value classifies the light: $0$ (single photon), $\tfrac12$ (two-photon or two-emitter), $1$ (laser), $2$ (thermal); below $0.5$ is the standard single-emitter certificate.
It is measured with the Hanbury Brown–Twiss setup: a beamsplitter, two detectors, and a coincidence histogram whose zero-delay value is $g^{(2)}(0)$ .
Key idea: antibunching is the statement $g^{(2)}(0)<1$ — normally-ordered intensity correlations at zero delay fall below the coherent-state value, which is impossible for any classical intensity distribution and is the temporal hallmark of quantized, single-quantum emission.

Where this lives today

The antibunching dip is the acceptance test for the entire single-photon-source industry. A quantum emitter — a nitrogen-vacancy centre in diamond, an InGaAs quantum dot, a single molecule, or a trapped ion — is only called a single-photon source once an HBT measurement returns $g^{(2)}(0)$ well below $0.5$ , and the best deterministic quantum-dot sources now report the $10^{-2}$ range on demand into a waveguide. The same number gates technology: in quantum key distribution multi-photon pulses open the door to photon-number-splitting attacks, so $g^{(2)}(0)\ll1$ directly raises the secure key rate and reach; in photonic quantum computing, arrays of indistinguishable single photons feed boson-sampling and fusion architectures, each qualified by its $g^{(2)}(0)$ . Antibunching is also a measurement tool: because $N$ equal emitters give $g^{(2)}(0)=1-1/N$ , the dip depth counts molecules or defects in a diffraction-limited spot, and heralded down-conversion sources use the conditional $g^{(2)}(0)$ to certify near-single-photon states. From starlight interferometry to the quantum internet, the same shallow dip at $\tau=0$ proves the light in front of you is granular.

The physical picture

Picture the emitter as a bucket that holds at most one photon. Pour it out — a detection — and the bucket is empty; it must refill before another photon appears. That mandatory refill creates a gap after every click, so two clicks essentially never land at the same instant and the correlation at $\tau=0$ collapses. Contrast a laser, whose steady intensity makes arrivals fall at random like raindrops (Poissonian, $g^{(2)}(0)=1$ ), and thermal light, whose intensity spikes cluster photons into bursts (bunched, $g^{(2)}(0)=2$ ). Antibunched light is smoother than random — metered out one quantum at a time, a regularity no positive-intensity classical source can manufacture, as the first figure shows.

Figure

Three photon streams on a common time axis — thermal light clusters into bursts with g²(0) ≈ 2, laser light arrives at random with g²(0) = 1, and a single emitter meters photons out one at a time with a minimum gap set by the excited-state lifetime, g²(0) → 0

The measured quantity is really a conditional probability: given a photon now, how likely is another after delay $\tau$ ? For a single emitter that is the probability it has been re-excited, starting at zero and climbing back as the excited state recovers — hence the dip that fills in over the lifetime. Weak driving gives a monotonic rise to $g^{(2)}=1$ ; strong driving makes the re-pumped atom cycle coherently, so the conditional probability overshoots and rings at the Rabi frequency before damping to $1$ , as the second figure shows for all three sources.

Figure

The correlation g²(τ) versus delay — thermal light starts bunched at g²(0) = 2 and decays to 1, a laser stays flat at g²(0) = 1, and a strongly driven single emitter dips to g²(0) = 0 then rings with damped Rabi oscillations at Ω′ = 3Γ before settling to 1; the dashed line at 0.5 is the single-emitter threshold

The core equations

The second-order correlation is the whole subject in one ratio: it compares the joint probability of detecting a photon at $t$ and another at $t+\tau$ against the product of the two independent probabilities. Normalized this way an uncorrelated source sits at exactly $1$ ; departures measure bunching or antibunching. The operators are in normal order because that is what a photon-absorbing detector does.

g^{(2)}(\tau)=\frac{\langle a^{\dagger}(t)\,a^{\dagger}(t+\tau)\,a(t+\tau)\,a(t)\rangle}{\langle a^{\dagger}(t)\,a(t)\rangle\,\langle a^{\dagger}(t+\tau)\,a(t+\tau)\rangle} \qquad (1)

where: $a,\,a^{\dagger}$ — annihilation and creation operators of the detected mode; $\tau$ — delay between detections; $\langle\cdot\rangle$ — quantum expectation. The numerator is the rate of ordered photon pairs separated by $\tau$ ; the denominator normalizes by what an uncorrelated stream of the same brightness gives, so $g^{(2)}=1$ means "no memory". Setting $\tau=0$ in a single mode collapses the correlator onto the number operator $n=a^{\dagger}a$ , turning arrival times into photon-number statistics.

g^{(2)}(0)=\frac{\langle a^{\dagger}a^{\dagger}aa\rangle}{\langle a^{\dagger}a\rangle^{2}}=\frac{\langle n(n-1)\rangle}{\langle n\rangle^{2}} \qquad (2)

where: $n=a^{\dagger}a$ — photon-number operator; $\langle n(n-1)\rangle$ — the normally ordered second moment, the expected number of ordered photon pairs (using $[a,a^{\dagger}]=1$ , $a^{\dagger}a^{\dagger}aa=n(n-1)$ ). A state that never holds two photons at once has zero numerator. For a Fock state $|n\rangle$ the photon number is sharp, so equation (2) gives a clean discrete fingerprint of the granularity of light.

g^{(2)}(0)=\frac{n(n-1)}{n^{2}}=1-\frac{1}{n}\quad\text{for the Fock state }|n\rangle \qquad (3)

where: $n$ — the fixed photon number. This is $0$ for $n=1$ , exactly $\tfrac12$ for $n=2$ , $\tfrac23$ for $n=3$ , creeping toward $1$ as $n\to\infty$ — large Fock states look nearly Poissonian, yet every one is formally antibunched. The zero-delay correlation ties to photon-number noise through the Mandel $Q$ parameter, linking it to the single-time variance $(\Delta n)^2$ — why antibunched sources are so often sub-Poissonian.

g^{(2)}(0)=1+\frac{(\Delta n)^{2}-\langle n\rangle}{\langle n\rangle^{2}}=1+\frac{Q}{\langle n\rangle},\qquad Q=\frac{(\Delta n)^{2}-\langle n\rangle}{\langle n\rangle} \qquad (4)

where: $(\Delta n)^2$ — photon-number variance; $Q$ — Mandel parameter. A coherent state has $Q=0$ ; sub-Poissonian light has $Q<0$ and drags $g^{(2)}(0)$ below unity. Antibunching is decisive because it breaks a bound no classical field can escape. Treating a classical intensity as a non-negative random variable, the variance inequality forces two constraints at once.

g^{(2)}(0)\ge 1,\qquad g^{(2)}(0)\ge g^{(2)}(\tau) \qquad (5)

where: $g^{(2)}(\tau)$ — correlation at finite delay. The first says classical light can never dip below unity; the second says its correlation can never have a minimum at $\tau=0$ . Antibunching violates both — possible only if the Glauber $P$ -function fails to be a probability density, the definition of a non-classical state. For the archetypal source, a single two-level atom driven near resonance, the optical Bloch equations give the full delay dependence: it starts at zero and, under strong driving, rings at the reduced Rabi frequency before damping to unity.

g^{(2)}(\tau)=1-e^{-3\Gamma\tau/4}\left[\cos(\Omega'\tau)+\frac{3\Gamma}{4\Omega'}\sin(\Omega'\tau)\right],\qquad \Omega'=\sqrt{\Omega^{2}-\left(\tfrac{\Gamma}{4}\right)^{2}} \qquad (6)

where: $\Gamma$ — excited-state decay rate (Einstein $A$ coefficient); $\Omega$ — Rabi frequency; $\Omega'$ — reduced Rabi frequency. At $\tau=0$ the bracket equals $1$ and $g^{(2)}(0)=0$ exactly. Weak drive gives a monotonic rise; strong drive a real oscillation — the emitter's Rabi cycling imprinted on the photon statistics.

Derivation AAntibunching of a photon-number state

The goal is to prove the central discrete result, that a Fock state $|n\rangle$ has $g^{(2)}(0)=n(n-1)/n^2$ and in particular that a single photon $|1\rangle$ gives exactly zero, by evaluating the normally ordered correlator directly in the number basis. It shows that the granularity of light — an integer count of indivisible quanta — is what forces the correlation to vanish, with no dynamics involved, purely operator algebra on a definite-photon state.

Step 1. Insert the number state into equation (2): $g^{(2)}(0)=\langle n|a^{\dagger}a^{\dagger}aa|n\rangle/\langle n|a^{\dagger}a|n\rangle^{2}$ . Why: a Fock state is an eigenstate of $n$ , so every expectation reduces to counting, and the normally ordered form is what a coincidence detector measures.

Step 2. Apply the two annihilation operators via $a|m\rangle=\sqrt{m}\,|m-1\rangle$ : $aa|n\rangle=\sqrt{n}\sqrt{n-1}\,|n-2\rangle$ . Why: two lowerings drop the state two rungs, and the product of ladder coefficients $\sqrt{n(n-1)}$ is the amplitude for an ordered photon pair.

Step 3. The numerator is the squared norm, $\langle n|a^{\dagger}a^{\dagger}aa|n\rangle=n(n-1)$ ; the denominator is $\langle n|a^{\dagger}a|n\rangle^2=n^2$ . Why: the numerator counts ordered pairs, so for $n=1$ there are none and it vanishes — the whole phenomenon in one line.

Step 4. Assemble: $g^{(2)}(0)=n(n-1)/n^2=1-1/n$ , so $|1\rangle\to0$ , $|2\rangle\to\tfrac12$ , $|3\rangle\to\tfrac23$ , $|n\rangle\to1$ . Why: the single-photon zero is the perfect-antibunching benchmark, and the slow climb explains why large Fock states look nearly Poissonian. *Verified: reproduces Loudon ch. 6 and Saleh and Teich ch. 12; the check $aa|1\rangle=0$ makes the numerator vanish so $g^{(2)}(0)=0$ .*

Go deeper · Derivation B · 1 minThe classical bound and its quantum violation

Derivation BThe classical bound and its quantum violation

The aim is to establish that every classical field obeys $g^{(2)}(0)\ge 1$ and then to show single-emitter light sits below that floor — what elevates antibunching from a curiosity to a proof of quantization. We treat the intensity as a positive random variable, use the non-negativity of variance to bound the correlation, then confront the bound with the Fock result, so the mere existence of $g^{(2)}(0)<1$ certifies no positive intensity distribution can describe the light.

Step 1. Classically the correlation is an intensity average, $g^{(2)}(0)=\langle I^2\rangle/\langle I\rangle^2$ with $I\ge 0$ and $P(I)\ge 0$ a genuine density. Why: the classical field is a definite (if noisy) function of time, so correlations are ordinary moments of a real positive quantity.

Step 2. The variance is never negative, $(\Delta I)^2=\langle I^2\rangle-\langle I\rangle^2\ge 0$ , so $\langle I^2\rangle\ge\langle I\rangle^2$ . Why: variance cannot be negative is the entire content of the classical bound.

Step 3. Dividing gives $g^{(2)}(0)=1+(\Delta I)^2/\langle I\rangle^2\ge 1$ : no classical field dips below unity, thermal fluctuations pushing it to $2$ . Why: this fixes the classical floor at $1$ , so any value beneath it is off the classical map.

Step 4. The Fock state $|1\rangle$ gives $g^{(2)}(0)=0<1$ , which no non-negative $P(I)$ can reproduce — the Glauber $P$ -representation must go negative or singular. Why: a $P$ -function that is not a density signals non-classical light, making antibunching a model-free witness of quantization. *Verified: standard non-classicality argument (Loudon ch. 5–6); $g^{(2)}(0)<1$ demands a $P$ -function that is not a valid probability density.*

Go deeper · Derivation C · 1 minWhy a single two-level atom dips to exactly zero

Derivation CWhy a single two-level atom dips to exactly zero

The purpose is to trace the perfect antibunching of a single atom to a one-line identity, $\sigma^2=0$ , and to connect the recovery of $g^{(2)}(\tau)$ to the atom's re-excitation, so both the zero at $\tau=0$ and the return to unity have a mechanical cause. This is the engine behind every single-photon source: a system with one excitation cannot be lowered twice, so it cannot emit two photons at once, and its radiated field inherits that impossibility.

Step 1. In the far field the radiated amplitude is proportional to the atomic lowering operator, $a\propto\sigma$ , so the photon correlator maps onto an atomic one, $g^{(2)}(\tau)=\langle\sigma^{\dagger}(0)\sigma^{\dagger}(\tau)\sigma(\tau)\sigma(0)\rangle/\langle\sigma^{\dagger}\sigma\rangle^2$ . Why: the detector sees emission from the atomic dipole, so the light's statistics are those of $\sigma$ .

Step 2. A two-level atom satisfies $\sigma^2=0$ , since $\sigma|e\rangle=|g\rangle$ and $\sigma|g\rangle=0$ . Why: with one quantum of excitation available, two lowerings in a row annihilate any state — the algebra of "at most one photon at a time".

Step 3. At $\tau=0$ the numerator contains $\sigma\sigma=0$ , so $g^{(2)}(0)=0$ identically, independent of drive, detuning, or temperature. Why: single-atom antibunching is a kinematic fact of the two-level structure, guaranteed by the operator algebra alone.

Step 4. For $\tau>0$ the correlation recovers as the conditional excited-state population rebuilds: weak drive gives a monotonic rise $\sim1-e^{-\Gamma\tau/2}$ , strong drive the damped Rabi ringing of equation (6). Why: after emission the atom is in its ground state, so $g^{(2)}(\tau)$ is the probability it has been re-excited by delay $\tau$ — the photon statistics image the atom's dynamics. *Verified: matches Foot ch. 9 and the Kimble–Dagenais–Mandel 1977 measurement; the identity $\sigma^2=0$ for the Pauli lowering operator forces the zero-delay numerator to vanish.*

Limiting cases

The single number $g^{(2)}(0)$ sorts every optical source into a physical category, with sharp boundaries.

Case	Result	Meaning
Single-photon Fock state	$g^{(2)}(0)=\left.\frac{n(n-1)}{n^2}\right\vert_{n=1}=0$	Perfect antibunching; zero chance of two simultaneous detections.
Two-photon or two-emitter	$g^{(2)}(0)=\frac{2\cdot 1}{2^2}=\tfrac12$	Still non-classical, but at the threshold that separates one emitter from two.
Coherent (laser) light	$g^{(2)}(\tau)=1$	Poissonian; the exact classical–quantum boundary, neither bunched nor antibunched.
Thermal (chaotic) light	$g^{(2)}(0)=1+\vert g^{(1)}(0)\vert^2=2$	Bunched; photons clump, the opposite extreme from antibunching.
$N$ equal independent emitters	$g^{(2)}(0)=1-\tfrac1N$	The dip depth counts emitters in the spot: $N=1\to0$ , $N=2\to\tfrac12$ , $N\to\infty\to1$ .
Classical bound	$g^{(2)}(0)\ge 1,\ \ g^{(2)}(0)\ge g^{(2)}(\tau)$	Every classical field obeys both; antibunching violates each of them.

Worked examples

1 · Certifying a single-photon source from a pulsed HBT histogram (confidence builder)

An InGaAs quantum dot is excited by a pulsed laser into an HBT setup. The coincidence histogram shows peaks at each multiple of the laser period; the side peaks (uncorrelated pulses) average $A_{\text{side}}=840$ counts, the central peak at zero delay $A_0=52$ counts. Is this a single-photon source, and how much does it suppress multi-photon emission relative to a laser?

Step 1. In pulsed operation the normalized zero-delay correlation is the ratio of central to average side-peak area (each side peak being two photons from independent pulses, the $g^{(2)}=1$ reference): $g^{(2)}(0)=A_0/A_{\text{side}}=52/840=\mathbf{0.062}$ .

Step 2. A value below $0.5$ rules out two independent emitters and below $0.1$ marks a high-purity source, so at $0.062$ this dot is comfortably single-photon.

Step 3. An attenuated-laser pulse of the same brightness has $g^{(2)}(0)=1$ , so the ratio $1/0.062\approx 16$ means the dot emits multi-photon events about 16 times less often — the property that beats photon-number-splitting attacks. g²(0) = 0.062, suppressing two-photon events ≈ 16× below a laser.

2 · Timescales of the antibunching dip and Rabi ringing

A nitrogen-vacancy centre has lifetime $T_1=12\ \mathrm{ns}$ , so $\Gamma=1/T_1$ . Estimate (a) the weak-drive recovery time of the dip, and (b) at strong drive $\Omega=5\Gamma$ , the Rabi period in $g^{(2)}(\tau)$ and the first overshoot height.

Step 1. The decay rate is $\Gamma=1/T_1=1/(12\times10^{-9}\ \mathrm{s})=\mathbf{8.3\times10^{7}\ s^{-1}}$ . Under weak driving the dip recovers monotonically, $g^{(2)}(\tau)$ climbing from $0$ toward $1$ over roughly $2T_1\approx\mathbf{24\ ns}$ (the $e^{-\Gamma\tau/2}$ limit).

Step 2. At strong drive $\Omega'=\sqrt{\Omega^2-(\Gamma/4)^2}=\Gamma\sqrt{25-0.0625}=4.99\,\Gamma\approx5\Gamma$ , giving period $T_{\text{R}}=2\pi/\Omega'=2\pi T_1/5=\mathbf{15\ ns}$ .

Step 3. The first overshoot is at $\Omega'\tau=\pi$ , i.e. $\tau=\pi/\Omega'\approx\mathbf{7.5\ ns}$ , where the bracket in equation (6) equals $-1$ so $g^{(2)}(\tau)=1+e^{-3\Gamma\tau/4}$ . With $\Gamma\tau=0.63$ this gives $g^{(2)}\approx1+0.62=\mathbf{1.62}$ : the stream briefly looks bunched as coherent Rabi cycling re-synchronizes emission before decay damps it. Rabi period ≈ 15 ns, first overshoot g²(7.5 ns) ≈ 1.6.

3 · Counting emitters in a diffraction-limited spot

A confocal microscope focuses on a cluster of identical molecules and a CW HBT measurement returns $g^{(2)}(0)=0.63$ . Assuming equally bright independent emitters, how many are in the spot? What would $0.45$ imply?

Step 1. For $N$ equal independent single-photon emitters the dip depth is $g^{(2)}(0)=1-1/N$ , so $N=1/(1-0.63)=1/0.37=\mathbf{2.7}$ : a non-integer count meaning unequal brightness or background, but pinning the answer to about three molecules.

Step 2. For $g^{(2)}(0)=0.45$ the same formula gives $N=1/0.55=\mathbf{1.8}$ , essentially two emitters — and being below $0.5$ it excludes a clean single molecule. Only $g^{(2)}(0)<0.5$ guarantees a dominant single emitter. N ≈ 2.7 (about three); 0.45 means about two, and only below 0.5 is a single emitter certified.

Physics of the portrait

Photon antibunching is where quantum optics stopped being a formal possibility and became a measured fact — and the person who first caught a single atom breaking the classical bound gave the field its image.

Origins · 1 minH. Jeff Kimble and how this was discovered

H. Jeff Kimble

b. 1949 · quantum optics and cavity QED

H. Jeff Kimble made the measurement that turned photon antibunching from prediction into observation. In 1977, at the University of Rochester with Mario Dagenais and Leonard Mandel, he trained a Hanbury Brown–Twiss interferometer on the resonance fluorescence of a very dilute sodium beam — dilute enough that, for stretches of time, a single atom sat in the laser focus. The coincidence histogram showed a dip at zero delay, $g^{(2)}(0)<g^{(2)}(\tau)$ : a direct violation of a classical inequality and the first observation of light no classical field can produce. The experiment was extraordinarily demanding — short transit times, low count rates, a shallow dip pulled out of noise — but it stood as the cleanest early proof that the electromagnetic field must be quantized. Kimble went on to become one of the architects of cavity quantum electrodynamics and the quantum-network programme at Caltech. The antibunching dip he first resolved is now a daily laboratory diagnostic worldwide.

History

1956

— Hanbury Brown and Twiss measure intensity correlations of starlight, opening the study of photon coincidences

1963

— Glauber formulates the quantum coherence functions and the P-representation, defining g² operationally

1976

— Carmichael and Walls predict antibunching in the resonance fluorescence of a two-level atom

1977

— Kimble, Dagenais and Mandel observe antibunching in the fluorescence of single sodium atoms

2000

— single molecules, quantum dots and colour centres reach g²(0) well below 0.5 at room temperature

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 3, 5, 6 (correlation functions, non-classicality, resonance fluorescence) · Foot, Atomic Physics, ch. 9 (single-atom fluorescence) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., ch. 12–13 (photon statistics) · Walls and Milburn, Quantum Optics, ch. 2–3 · Fox, Quantum Optics: An Introduction, ch. 5–6 · Kimble, Dagenais and Mandel, Phys. Rev. Lett. 39, 691 (1977).

Provenance. The operator definitions and the Fock, thermal, and coherent values are from Loudon (ch. 3, 5, 6) and Saleh and Teich (ch. 12). The single-atom result $g^{(2)}(0)=0$ and the resonance-fluorescence delay dependence follow Foot (ch. 9) and Loudon (ch. 8); the strong-driving formula is quoted with the standard $3\Gamma/4$ damping of the optical Bloch equations, $\Omega'=\sqrt{\Omega^2-(\Gamma/4)^2}$ . The classical inequalities are standard (Loudon ch. 5). A verifier should recompute: $aa|n\rangle=\sqrt{n(n-1)}\,|n-2\rangle$ and $g^{(2)}(0)=n(n-1)/n^2$ giving $0,\tfrac12,\tfrac23\to1$ ; $\sigma^2=0$ ; $\langle I^2\rangle\ge\langle I\rangle^2$ ; $g^{(2)}(0)=1-1/N$ ; and the worked numbers $52/840=0.062$ , $\Omega'=4.99\Gamma$ at $\Omega=5\Gamma$ , and $N=1/(1-0.63)=2.7$ . Experimental benchmark values (best sources $g^{(2)}(0)\sim10^{-2}$ ) are order-of-magnitude, not cited as exact.

Prerequisites. Coherence Functions and the Degree of Coherence · Photon Statistics and the Mandel Q Parameter · Fock States and the Photon Number Basis · Coherent States · Quantization of the Electromagnetic Field. Leads to. Hanbury Brown–Twiss Experiment · Single-Photon Sources · Spontaneous Parametric Down-Conversion · Squeezed States of Light.

Ask the tutor: Ask me to walk through why $\sigma^2=0$ forces the antibunching dip to exactly zero, or to turn a raw HBT coincidence histogram into a background-corrected $g^{(2)}(0)$ .

What is the difference between antibunching and sub-Poissonian statistics?

Antibunching is $g^{(2)}(0)<1$ , a two-time property of how detections are ordered in time — a second photon is unlikely right after the first. Sub-Poissonian statistics is $Q<0$ , a single-time property of the photon-number variance being below its mean. For an ideal single mode the two are tied by $g^{(2)}(0)=1+Q/\langle n\rangle$ , so one guarantees the other. But in multimode, pulsed, or time-resolved measurements they come apart — a source can be antibunched on short timescales while slow classical intensity noise makes it super-Poissonian overall. Cousins, not synonyms.

Why does a single two-level atom give exactly g²(0) equal to zero, not just a small number?

Because the zero is enforced by algebra, not tuning. A two-level atom has only one excitation to give, so its lowering operator satisfies $\sigma^2=0$ — you cannot de-excite it twice. The zero-delay correlator contains exactly $\sigma\sigma$ , which annihilates any state, so the numerator is identically zero regardless of drive, detuning, or temperature: the atom cannot emit two photons at once, but must radiate, fall to the ground state, and be re-excited first. Real emitters read slightly above zero only from background, dark counts, and any residual second emitter — never because the single-atom physics allows it.

Why can't any classical wave, however cleverly modulated, produce antibunching?

Because classically $g^{(2)}(0)=\langle I^2\rangle/\langle I\rangle^2$ , an average of squared intensity over a positive distribution $P(I)$ . Since the variance is never negative, $\langle I^2\rangle\ge\langle I\rangle^2$ and $g^{(2)}(0)\ge1$ always. To go below one you would need a $P(I)$ that goes negative, which is not a valid probability distribution — so $g^{(2)}(0)<1$ is a direct, assumption-free witness that the light has no classical description.

Why is 0.5 the threshold for calling something a single-photon source?

It comes from counting emitters. For $N$ identical independent single-photon emitters, $g^{(2)}(0)=1-1/N$ : one gives $0$ , two give $\tfrac12$ . So $0.5$ is exactly the value from two equally bright emitters. Only a value strictly below $0.5$ cannot be explained by two independent emitters, which is why crossing that line is the minimum bar for claiming a single quantum system dominates; high-quality sources push far below, to $0.1$ or $0.01$ .

Why do we need two detectors and a beamsplitter instead of one fast detector?

Because every real single-photon detector has a dead time — after a click it is blind for tens of nanoseconds, exactly the timescale on which antibunching lives, so a single detector cannot register two photons close enough together to probe $g^{(2)}(0)$ . The Hanbury Brown–Twiss trick splits the beam onto two detectors with independent dead times and histograms their coincidences versus delay; now two near-simultaneous photons can be caught, one per detector, and the depth of the zero-delay dip gives $g^{(2)}(0)$ directly — which is why HBT is the universal geometry.

Why does g²(τ) rise above 1 under strong driving — isn't that bunching?

It is a transient overshoot, not true bunching, from coherent Rabi cycling. Under strong resonant driving the atom does not passively wait to be re-excited; it oscillates coherently between ground and excited states, so the conditional probability of a second emission starts at zero, swings past its steady value, dips, and rings — tracking $\cos(\Omega'\tau)$ before decay at rate $3\Gamma/4$ damps everything to $g^{(2)}=1$ . The excursions above unity mean a second photon is briefly more likely at those delays, but the defining signature — $g^{(2)}(0)=0$ and the minimum at $\tau=0$ — is untouched: the light is still antibunched, the ringing being the emitter's internal clock printed on its photons.