Hanbury Brown–Twiss Experiment

Two detectors, one beam splitter, and a stopwatch — that is the whole of the Hanbury Brown–Twiss experiment, and yet it split optics into a before and an after. Every interferometer built before it, from Young's slits to Michelson's stellar rig, measured the amplitude of light: it overlapped two field paths and read the phase through fringe visibility, the first-order coherence $g^{(1)}$ . HBT threw the phase away and asked a different question — when one detector clicks, is the other more or less likely to click at the same instant? That correlation of intensities, the second-order coherence $g^{(2)}(\tau)$ , carries information no amplitude measurement can reach. The payoff is a single number, $g^{(2)}(0)$ , that sorts light into three families. Laser light gives exactly $1$ : the detectors are independent beyond shot noise. Thermal light — starlight, a discharge lamp, the sun — gives $2$ : its photons arrive in clumps, so coincidences pile up at zero delay. And a single atom, quantum dot, or colour centre gives less than $1$ , bottoming out at $0$ for a perfect single-photon emitter that cannot send a photon to both detectors at once. That last regime is the prize, because $g^{(2)}(0) \ge 1$ is a theorem for every classical field — it follows from nothing but the non-negativity of a probability distribution — so a dip below unity is model-independent proof that light is made of photons. This article draws its machinery from the quantum theory of the field — the annihilation operators of field quantization, the number statistics of photon statistics, the Poissonian signature of coherent states — and shows how a 50:50 beam splitter physically realises the normally ordered moment $\langle \hat a^\dagger \hat a^\dagger \hat a \hat a\rangle$ those theories say we measure.

Key takeaways

The HBT experiment splits a beam onto two detectors and correlates their intensities (photon coincidences) versus a delay $\tau$ , measuring the second-order coherence $g^{(2)}(\tau)$ — not the field amplitude of classical interferometry.
The zero-delay value classifies light: $g^{(2)}(0)=1$ for coherent laser light, $2$ for single-mode thermal (bunched) light, and $<1$ for antibunched nonclassical light, reaching $0$ for a single photon.
A measured $g^{(2)}(0) < 1$ is impossible for any classical field — it follows from $\langle I^2\rangle \ge \langle I\rangle^2$ — so a dip below unity is a rigorous signature of the quantum nature of light.
A 50:50 beam splitter plus two detectors turns the coincidence rate into the normally ordered moment $\langle \hat a^\dagger \hat a^\dagger \hat a \hat a\rangle$ , sidestepping the deadtime that blinds a single detector to a simultaneous pair.
Invented in the 1950s to measure stellar angular diameters from intensity correlations (robust against atmospheric phase noise), the same box now certifies single-photon sources.
Key idea: HBT measures the normalised intensity correlation $g^{(2)}(\tau)$ ; its zero-delay value separates coherent ( $=1$ ), bunched thermal ( $>1$ ), and antibunched nonclassical ( $<1$ ) light, and $g^{(2)}(0)<1$ is forbidden to every classical field.

The core insight

The Hanbury Brown–Twiss experiment measures the normalised second-order (intensity) correlation $g^{(2)}(\tau)$ ; its zero-delay value $g^{(2)}(0)$ separates coherent ( $=1$ ), bunched thermal ( $>1$ ), and nonclassical antibunched ( $<1$ ) light, with $g^{(2)}(0)<1$ being impossible for any classical field.

Where this lives today

The HBT box is now standard laboratory furniture. Any group building a single-photon source — quantum dots, nitrogen-vacancy and silicon-vacancy centres in diamond, single molecules, defect emitters in 2D materials — certifies it by wiring two single-photon detectors to a fast time-tagger and reading $g^{(2)}(0)$ off the coincidence histogram. A central peak below $0.5$ is the entry ticket for quantum key distribution and photonic quantum computing, where a stray second photon leaks information or corrupts a gate; the best sources report $g^{(2)}(0)\sim10^{-3}$ . The conditional version — $g^{(2)}(0)$ on the signal arm of a down-conversion pair, gated by the idler — is the figure of merit for heralded single photons. Intensity interferometry has also been reborn. Arrays of imaging atmospheric Cherenkov telescopes built for gamma-ray astronomy — VERITAS, MAGIC — double as photon-correlation interferometers: in 2019–2020 they correlated starlight between telescopes hundreds of metres apart and resolved hot-star diameters to sub-milliarcsecond scale. The idea also crossed into matter waves, where correlating atom-detector clicks revealed bunching of thermal bosonic clouds and antibunching of fermionic atoms — Bose and Fermi statistics read directly off an HBT-style measurement.

The physical picture

Start with the apparatus, because its geometry is the whole trick. A beam hits a 50:50 splitter; each output feeds its own photon counter; a fast timer histograms the start–stop delay $\tau$ over billions of events to give $g^{(2)}(\tau)$ directly — the coincidence rate, normalised so uncorrelated clicks sit at $1$ .

Figure

The HBT layout: a beam splitter sends light to detectors D1 and D2 and a timer builds the coincidence histogram versus delay τ. Splitting a pair to opposite detectors lets a genuine simultaneous pair register as one start and one stop, beating the deadtime of a single detector.

Now the three families. Thermal light is a sum of many independent radiators whose fields beat into brief intensity spikes — photon clumps — that hit both detectors at once, so coincidences pile up at $\tau=0$ : bunching, doubling the single-mode rate to $g^{(2)}(0)=2$ . Coherent laser light has a steady amplitude with only shot noise, so the intensities are independent and the histogram is flat at $1$ . A single-photon emitter is the opposite: it holds exactly one photon, routed to one detector or the other, never both, so coincidences vanish and $g^{(2)}(0)=0$ — antibunching — climbing back to $1$ as it re-excites over its lifetime.

Figure

The signature curves g²(τ): thermal light bunches to g²(0) = 2.0 and decays to 1 over the coherence time; coherent light is flat at 1.0; a single-photon emitter dips to 0.0 and recovers to 1 over its lifetime. The dashed line at 1 is the classical floor that nothing classical crosses.

The spatial version made HBT famous. Intensity fluctuations at two separated detectors stay correlated only within the source's coherence area; move them apart and the correlation falls, its width encoding the angular size $\theta \sim \lambda/d$ . Because it reads only the modulus of the spatial coherence, intensity interferometry ignores the atmospheric phase turbulence that cripples amplitude interferometry — which is how HBT sized a star from the ground with two searchlight mirrors.

The core equations

The fundamental observable correlates the field intensity at two times, built from normally ordered field operators. The colons of normal ordering are not decoration — the two-detector geometry physically imposes them, because each detection absorbs a photon and so applies an annihilation operator.

g^{(2)}(\tau)=\frac{\langle \hat E^{(-)}(t)\,\hat E^{(-)}(t+\tau)\,\hat E^{(+)}(t+\tau)\,\hat E^{(+)}(t)\rangle}{\langle \hat E^{(-)}(t)\hat E^{(+)}(t)\rangle\,\langle \hat E^{(-)}(t+\tau)\hat E^{(+)}(t+\tau)\rangle} \qquad (1)

where: $\hat E^{(+)}$ and $\hat E^{(-)}=(\hat E^{(+)})^\dagger$ — positive- and negative-frequency field operators; $\tau$ — delay between the two detections; $\langle\cdots\rangle$ — expectation, operators normally and time ordered. Physical reading: the numerator is the joint two-photon detection rate; dividing by the two single rates makes $g^{(2)}=1$ the benchmark of independence. For a single mode the fields collapse to the mode operators $\hat a, \hat a^\dagger$ . The expression then compresses to a ratio of photon-number moments — the form used in almost every calculation below.

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

where: $\hat a, \hat a^\dagger$ — annihilation and creation operators of the detected mode. Physical reading: the numerator counts ordered photon pairs in the mode, the denominator the squared mean count, so $g^{(2)}$ is a pure intensity-normalised number. At zero delay the single-mode form becomes a statement about photon-number statistics alone. Using $\hat a^\dagger \hat a^\dagger \hat a \hat a = \hat n(\hat n-1)$ and the definition of the variance $(\Delta n)^2 = \langle \hat n^2\rangle - \langle \hat n\rangle^2$ , the correlation reads off the shape of the number distribution.

g^{(2)}(0)=\frac{\langle \hat n(\hat n-1)\rangle}{\langle \hat n\rangle^{2}}=1+\frac{(\Delta n)^{2}-\langle \hat n\rangle}{\langle \hat n\rangle^{2}} \qquad (3)

where: $\hat n=\hat a^\dagger \hat a$ — number operator; $(\Delta n)^2$ — its variance. Physical reading: Poissonian statistics $(\Delta n)^2=\langle \hat n\rangle$ give exactly $1$ ; super-Poissonian (thermal) fluctuations push above $1$ ; sub-Poissonian pull below. The same combination defines the Mandel parameter, the standard scalar measure of how far the statistics depart from Poissonian. It rescales $g^{(2)}(0)-1$ by the mean count, vanishing exactly for coherent light.

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

where: $Q_M$ — Mandel parameter. Physical reading: $Q_M=0$ Poissonian (coherent), $Q_M>0$ super-Poissonian (bunched), $Q_M<0$ sub-Poissonian — a strictly nonclassical regime. For chaotic (thermal) light the full delay dependence is fixed by the first-order coherence through the Siegert relation, which follows from Gaussian moment factorisation. It ties the intensity correlation HBT measures to the amplitude correlation an interferometer measures.

g^{(2)}(\tau)=1+\big|g^{(1)}(\tau)\big|^{2} \qquad (5)

where: $g^{(1)}(\tau)$ — normalised first-order (field) coherence, $|g^{(1)}(0)|=1$ . Physical reading: $\tau=0$ gives $g^{(2)}(0)=2$ ; beyond the coherence time $|g^{(1)}|\to 0$ and $g^{(2)}\to 1$ , so the bunching peak has the width of the field coherence. The classical bound is the theorem that gives HBT its power. For any real, non-negative instantaneous intensity $I$ drawn from any probability distribution, the variance cannot be negative.

g^{(2)}_{\text{classical}}(0)=\frac{\langle I^{2}\rangle}{\langle I\rangle^{2}}=1+\frac{(\Delta I)^2}{\langle I\rangle^2}\ge 1 \qquad (6)

where: $I\ge 0$ — classical (c-number) intensity. Physical reading: no classical field, however exotic its fluctuations, can give $g^{(2)}(0)<1$ ; a measured value below unity is model-independent proof of nonclassical light. Fock states break the bound cleanly — a pure number state $|n\rangle$ has zero variance in $\hat n$ , giving $g^{(2)}(0)=1-1/n$ : that is $0$ for a single photon, $0.5$ for $|2\rangle$ , and $\to1$ from below as $n\to\infty$ .

Derivation AWhy a beam splitter turns coincidences into a normally ordered moment

The goal is to prove the HBT geometry is not an approximation to $g^{(2)}(0)$ but an exact realisation of it: a lossless 50:50 splitter with signal in one port and vacuum in the other gives a coincidence rate $\langle \hat n_c \hat n_d\rangle=\tfrac14\langle \hat a^\dagger \hat a^\dagger \hat a \hat a\rangle$ , so after normalisation the two-detector correlation equals the input mode's $g^{(2)}(0)$ . This licenses every measurement below — the box literally computes the normally ordered second moment the field theory names as the observable.

Step 1. Write the mode transformation for a symmetric lossless splitter: $\hat c=\tfrac{1}{\sqrt2}(\hat a+i\hat b)$ and $\hat d=\tfrac{1}{\sqrt2}(i\hat a+\hat b)$ , with $\hat a$ the input signal and $\hat b$ the open port. Why: the reflection factor $i$ enforces unitarity and is what makes the cross terms cancel below.

Step 2. Put vacuum in the open port, state $|\psi\rangle_a|0\rangle_b$ , and form the coincidence operator $\hat c^\dagger \hat d^\dagger \hat d \hat c$ . Every term carrying a $\hat b$ or $\hat b^\dagger$ acts on the vacuum and, normally ordered, contributes nothing. Why: an open port fed by vacuum can neither supply nor absorb a photon, so only pure-signal terms survive.

Step 3. Collecting the survivors gives $\langle \hat c^\dagger \hat d^\dagger \hat d \hat c\rangle=\tfrac14\langle \hat a^\dagger \hat a^\dagger \hat a \hat a\rangle$ ; the $\tfrac14$ is geometry, since each photon reaches a given detector with probability $\tfrac12$ . Why: the coincidence rate is proportional to the input normally ordered moment, so the constant cancels on normalisation.

Step 4. Divide by $\langle \hat n_c\rangle\langle \hat n_d\rangle=\tfrac14\langle \hat a^\dagger\hat a\rangle^2$ : the quarters cancel and $\langle \hat n_c\hat n_d\rangle/(\langle\hat n_c\rangle\langle\hat n_d\rangle)=g^{(2)}(0)$ . Why: the two-detector correlation is exactly the input $g^{(2)}(0)$ — the deadtime that blinds a single detector to a simultaneous pair is defeated because the pair is split first. For a single input photon $|1\rangle_a|0\rangle_b$ , $\hat a\hat a$ annihilates the state, so the coincidence expectation is zero and $g^{(2)}(0)=0$ — one photon can never fire both detectors. Consistent with Loudon, Ch. 6.

Go deeper · Derivation B · 1 minSingle-mode thermal light gives exactly g²(0) = 2

Derivation BSingle-mode thermal light gives exactly g²(0) = 2

The goal is to compute the zero-delay coherence of a single-mode thermal state from first principles and show it equals $2$ independently of brightness. This value is the calibration point of the subject: the largest bunching a single-mode classical field can produce, and the fixed target every thermal-light HBT run is checked against.

Step 1. A single-mode thermal state has the Bose–Einstein distribution $P(n)=\bar n^{\,n}/(1+\bar n)^{\,n+1}$ ; its second factorial moment is $\langle \hat n(\hat n-1)\rangle = 2\bar n^{2}$ , and its variance is $(\Delta n)^2=\bar n+\bar n^2$ . Why: the factor $2$ is the combinatorial fingerprint of bosons crowding one mode — the same $2$ that becomes the bunching peak.

Step 2. Insert into equation (3): $g^{(2)}(0)= 2\bar n^2/\bar n^2 = 2$ , with $\bar n$ cancelling — equivalently $1+\bar n^2/\bar n^2=2$ from the variance, super-Poissonian with $Q_M=\bar n>0$ . Why: bunching depends on the shape of the distribution, not its scale — a dim and a bright thermal source bunch identically, which is why HBT works on a faint star. The delay dependence follows from Siegert (5): $g^{(2)}(\tau)=1+|g^{(1)}(\tau)|^2$ gives peak $2$ at $\tau=0$ falling to $1$ beyond the coherence time. Matches Loudon and Foot; multimode averaging softens the peak to $g^{(2)}(0)=1+1/M$ .

Go deeper · Derivation C · 1 minSingle-atom antibunching: g²(0) = 0 and the recovery of coincidences

Derivation CSingle-atom antibunching: g²(0) = 0 and the recovery of coincidences

The goal is to explain why a single two-level emitter drives the intensity correlation below the classical floor — the decisive nonclassical result — and how $g^{(2)}(\tau)$ climbs back to $1$ over the atomic lifetime. The emitted field is proportional to the lowering operator $\hat\sigma$ , so the light inherits the algebra of a two-level system, and that algebra forbids two simultaneous photons.

Step 1. For resonance fluorescence the correlation is atomic: $g^{(2)}(\tau)=\langle\hat\sigma^\dagger(t)\hat\sigma^\dagger(t+\tau)\hat\sigma(t+\tau)\hat\sigma(t)\rangle/\langle\hat\sigma^\dagger\hat\sigma\rangle^2$ , since the detected field is $\propto\hat\sigma$ . Why: a two-level source has a spin-½ ladder operator for a dipole, not a harmonic mode — the entire reason its light differs from a laser or a lamp.

Step 2. At zero delay, $\hat\sigma\hat\sigma=0$ (you cannot lower twice from a two-rung ladder), so the numerator vanishes and $g^{(2)}(0)=0$ . Why: after one emission the atom is in its ground state and empty of a second photon — $\hat\sigma^2=0$ is bookkeeping for "one atom, one photon at a time".

Step 3. The recovery: $g^{(2)}(\tau)$ is the excited-state population conditioned on emission at $\tau=0$ , normalised to steady state. In the weak-drive limit the optical Bloch equations give $g^{(2)}(\tau)=\big(1-e^{-\Gamma\tau/2}\big)^{2}$ , rising monotonically from $0$ to $1$ over the lifetime $1/\Gamma$ . Why: the dip refills as the atom re-excites, its width set by the natural linewidth $\Gamma$ — a direct read-out of the excited-state lifetime. Under strong driving the recovery is non-monotonic — the atom Rabi-flops, so $g^{(2)}(\tau)$ overshoots $1$ and oscillates at the Rabi frequency before damping, with the $\tau=0$ antibunching surviving regardless. Reproduces the Kimble–Dagenais–Mandel (1977) result; because $g^{(2)}(0)=0<1$ violates the classical bound (6), single-atom fluorescence is rigorously nonclassical light.

Limiting cases

Case	Result	Meaning
Coherent (laser)	$g^{(2)}(\tau)=1$ for all $\tau$	Poissonian; detectors independent beyond shot noise — the reference state.
Single-mode thermal / chaotic	$g^{(2)}(0)=2$	Maximal classical bunching; peak width equal to the coherence time.
Multimode thermal ( $M$ modes)	$g^{(2)}(0)=1+1/M$	Mode averaging washes out bunching; broadband thermal light approaches $1$ .
Fock state $\lvert n\rangle$	$g^{(2)}(0)=1-\tfrac{1}{n}$	Sub-Poissonian, antibunched; $n=1$ gives $0$ , a perfect single photon.
Chaotic light, long delay	$g^{(2)}(\tau\gg\tau_c)\to 1$	Correlations decay over the coherence time; detectors become independent.
Classical bound	$g^{(2)}(0)\ge 1$ , $g^{(2)}(0)\ge g^{(2)}(\tau)$	Every classical field obeys both; a sub-unity dip or a rising central dip is nonclassical.

Common trapA measured g²(0) above zero means the source is not really a single-photon emitter

Not so — a real antibunched source almost always reports $g^{(2)}(0)>0$ for reasons unrelated to its purity. Timing jitter smears the infinitely narrow dip into the surrounding $\tau\ne0$ correlations, so finite resolution alone lifts the minimum; background and dark counts add a Poissonian floor; and genuine multi-photon emission raises it further. The accepted threshold is $g^{(2)}(0)<0.5$ , which already guarantees more single photons than pairs, with values near $0.1$ marking a high-purity emitter. Separately, do not conflate the two coherences: laser light is fully first-order coherent yet has $g^{(2)}=1$ — amplitude and intensity coherence are independent axes.

Worked examples

1 · Certifying a single-photon source (the modern flagship)

A pulsed quantum-dot source is fed into an HBT box. The coincidence histogram shows side peaks (uncorrelated pulses one laser period apart) averaging $A_{\text{side}}=500$ counts and a central peak at $\tau=0$ of $A_0=55$ counts. Is this a single-photon source?

Step 1. For a pulsed source the normalised zero-delay correlation is the central-peak area over the average side-peak area, since side peaks come from different pulses and define the $g^{(2)}=1$ baseline: $g^{(2)}(0)=A_0/A_{\text{side}}$ .

Step 2. Insert: $g^{(2)}(0)=55/500=\mathbf{0.11}$ , far below the certification threshold of $0.5$ — a good single-photon emitter.

Step 3. The residual: a perfect single photon gives $0$ , so $0.11$ signals a few percent of pulses carrying an unwanted second photon (plus background and jitter), with two-photon probability relative to single of about $g^{(2)}(0)/2\approx 0.05$ . Certified single-photon: g²(0) = 0.11, well below 0.5, with a few-percent multi-photon contamination.

2 · Thermal light: bunching, variance, and Mandel Q

A single-mode thermal (chaotic) source has mean photon number $\bar n=1.0$ per coherence cell. Find $g^{(2)}(0)$ , the photon-number variance, and the Mandel parameter, and contrast with a coherent beam of the same mean.

Step 1. Bunching value: for any single-mode thermal state $g^{(2)}(0)=2$ regardless of $\bar n$ (equation 3 with $\langle\hat n(\hat n-1)\rangle=2\bar n^2$ ), so $g^{(2)}(0)=\mathbf{2.0}$ .

Step 2. Variance: a thermal mode has $(\Delta n)^2=\bar n+\bar n^2=1.0+1.0=\mathbf{2.0}$ , twice the Poissonian variance $\bar n=1$ — the excess $\bar n^2=1$ is the bunching. The Mandel parameter is $Q_M=\bar n[g^{(2)}(0)-1]=1.0\times(2-1)=\mathbf{+1.0}$ , super-Poissonian.

Step 3. Contrast: a coherent beam of the same $\bar n=1$ is Poissonian, $(\Delta n)^2=1$ , $g^{(2)}(0)=1$ , $Q_M=0$ . HBT separates them instantly even when both are equally first-order coherent — the thermal histogram peaks to $2$ at $\tau=0$ while the laser histogram is flat. Thermal: g²(0) = 2.0, variance 2.0, Q = +1.0; coherent of equal brightness: g²(0) = 1.0, variance 1.0, Q = 0.

3 · Sirius: a stellar diameter from intensity correlations

The 1956 HBT stellar interferometer worked near $\lambda=443\ \mathrm{nm}$ . Treating Sirius A as a uniform disk of angular diameter $\theta=6.9\ \mathrm{mas}$ , estimate the baseline at which the intensity correlation first nulls.

Step 1. Convert: $\theta = 6.9\times10^{-3}\times4.848\times10^{-6}=3.35\times10^{-8}\ \mathrm{rad}$ .

Step 2. For a uniform disk the correlation (modulus-squared of the spatial coherence) first vanishes at $d_1=1.22\,\lambda/\theta=1.22\times(443\times10^{-9})/(3.35\times10^{-8})=\mathbf{16\ m}$ . The follow-up Narrabri interferometer used baselines to $188\ \mathrm{m}$ ; smaller stars push the null out since $d_1\propto1/\theta$ . Because HBT correlates intensities, it is immune to the atmospheric phase jitter that would scramble a $16\ \mathrm{m}$ amplitude interferometer. The correlation first nulls near 16 m; its width reads the star's angular size.

Physics of the portrait

The HBT effect is where astronomy, radio engineering, and quantum field theory collided. Its inventor came to optics from radar, and the controversy he provoked forced the first rigorous definition of what "coherence" means for a quantum field.

Origins · 1 minRobert Hanbury Brown and how this was discovered

Robert Hanbury Brown

1916–2002 · radio astronomer and pioneer of intensity interferometry

Robert Hanbury Brown trained as a radar engineer, working through the Second World War on airborne interception — a background that taught him to think about fluctuations in received power rather than phase. At Jodrell Bank, with the mathematician Richard Q. Twiss, he asked whether the intensity fluctuations at two separated radio antennas would be correlated; in 1952 they showed they were, and used it to size the radio galaxies Cygnus A and Cassiopeia A. The bold leap was into the optical band, where the "signal" is a stream of photons: their 1956 tabletop experiment correlated the photocurrents from a split mercury-lamp beam and found a genuine positive correlation — photons arriving together more often than chance. The claim was fiercely disputed, since a billiard-ball picture says photons arrive independently, and several physicists called it an artefact. Hanbury Brown and Twiss answered with a semiclassical calculation and, decisively, a working stellar instrument: the same year they measured the diameter of Sirius, and by the mid-1960s the Narrabri interferometer had catalogued dozens of hot-star diameters. The resolution — that bosonic bunching is real, and that a quantum theory of coherence must track intensity as well as amplitude — came with Roy Glauber's 1963 theory of $g^{(n)}$ , work that shared the 2005 Nobel Prize. A stubbornly practical question thus opened the field of quantum optics.

History

1952

— Hanbury Brown, Jennison and Das Gupta measure radio-source sizes by correlating intensity at two antennas

1956

— Hanbury Brown and Twiss correlate photons in two split beams and measure the angular diameter of Sirius

1963

— Glauber's quantum theory of optical coherence defines g⁽ⁿ⁾ and the role of normal ordering

1966

— the Narrabri Stellar Intensity Interferometer measures the diameters of dozens of hot stars

1977

— Kimble, Dagenais and Mandel observe antibunching, g²(0) less than 1, in single-atom resonance fluorescence

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 3 and 6 · Fox, Quantum Optics: An Introduction, Ch. 5–6 · Foot, Atomic Physics, Ch. 9 · Saleh and Teich, Fundamentals of Photonics, 2nd ed., Ch. 12–13 · Gerry and Knight, Introductory Quantum Optics, Ch. 5–6 · Hanbury Brown and Twiss, Nature 177, 27 (1956); 178, 1046 (1956).

Provenance. Formulas cross-checked against Loudon (Ch. 3 and 6), Foot (Ch. 9), Fox (Ch. 5–6), and Saleh and Teich (Ch. 12–13). A verifier should confirm: the beam-splitter coincidence $=\tfrac14\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle$ (vacuum in the open port); thermal $g^{(2)}(0)=2\bar n^2/\bar n^2=2$ with variance $\bar n+\bar n^2$ , $Q_M=\bar n$ ; Fock $g^{(2)}(0)=1-1/n$ giving $0,0.5,2/3$ for $n=1,2,3$ ; Siegert $g^{(2)}(\tau)=1+|g^{(1)}(\tau)|^2$ ; the classical bound $g^{(2)}(0)\ge1$ from $(\Delta I)^2\ge0$ ; the pulsed ratio $55/500=0.11$ ; and the Sirius null $d_1=1.22\lambda/\theta\approx16\ \mathrm{m}$ at $\lambda=443\ \mathrm{nm}$ , $\theta=6.9\ \mathrm{mas}$ . Historical claims (HBT 1952 and 1956; Glauber 1963; Kimble–Dagenais–Mandel 1977) are standard and citable.

Prerequisites. Coherence Functions and the Degree of Coherence · Photon Statistics and the Mandel Q Parameter · Quantization of the Electromagnetic Field · Coherent States. Leads to. Photon Antibunching · Single-Photon Sources · Spontaneous Parametric Down-Conversion · Phase-Space Quasiprobability Distributions.

Ask the tutor: Ask me to walk through the beam-splitter derivation operator by operator, or to work out $g^{(2)}(\tau)$ for a driven atom showing Rabi oscillations in the recovery.

Why does the HBT setup measure a normally ordered moment when the detectors just count light?

Because photodetection is absorption. Each click removes a photon, applying $\hat a$ ; a coincidence removes two, applying $\hat a\hat a$ . The measured rate is the squared magnitude of that amplitude, $\langle(\hat a\hat a)^\dagger(\hat a\hat a)\rangle=\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle$ , with all annihilation operators right of all creation operators — that is normal ordering. It falls out of absorptive detection, not imposed by hand, which is why vacuum in the open port contributes nothing and single-photon input gives an exact zero.

If photons are independent particles, how can thermal light make them arrive in clumps?

They are not independent — that was the surprise of 1956. Thermal light is a superposition of many radiators with random relative phases that beat into moments of high and low total intensity. High-intensity moments are simply times when more photons are present, so detections cluster there — a correlation through the shared classical intensity, amplified by Bose statistics favouring multiple bosons in one mode. The clumping is classical in origin (a positive intensity variance), which is why thermal bunching, unlike antibunching, needs no quantum theory.

What is the difference between first-order and second-order coherence, in one line each?

First-order coherence $g^{(1)}(\tau)$ is an amplitude correlation governing whether two field paths interfere — what fringe visibility and a Michelson interferometer measure. Second-order coherence $g^{(2)}(\tau)$ is an intensity correlation governing whether two detectors click together — what HBT measures. The two are independent: a laser is perfectly first-order coherent yet has $g^{(2)}=1$ , while thermal light can be first-order coherent within its coherence time yet has $g^{(2)}(0)=2$ . Only $g^{(2)}$ can dip below $1$ , and only that dip proves the light nonclassical.

Why is g²(0) < 1 impossible classically, and what makes a single atom escape the bound?

For a classical field the intensity $I$ is a real non-negative number with some distribution, so $g^{(2)}(0)=1+(\Delta I)^2/\langle I\rangle^2\ge1$ , since a variance cannot be negative — full stop, for every classical field. A single two-level atom escapes because its light has no positive intensity distribution: it holds one quantum, so after one photon leaves there is nothing to detect until it re-excites, and the $\tau=0$ coincidence rate is exactly zero. That $0<1$ is the cleanest proof that the field is quantised.

What does a value like g²(0) = 0.3 tell me about a source, in practice?

It certifies the source as nonclassical and predominantly single-photon, but not perfect. Anything below $1$ is classically forbidden, so $0.3$ is unambiguously antibunched and passes the $0.5$ threshold. The residual above $0$ mixes three effects to disentangle before quoting a purity: genuine multi-photon emission, background and dark counts, and finite timing resolution smearing the dip. Better optics and detectors typically pull a $0.3$ toward $0.05$ or lower; the best emitters reach $g^{(2)}(0)\sim10^{-3}$ .

How did HBT measure a star's diameter without phase-stable optics?

By correlating intensities rather than amplitudes. Intensity fluctuations at two separated detectors stay correlated only within the source's coherence area; as the baseline $d$ widens the correlation falls, first nulling at $d_1\approx1.22\lambda/\theta$ for a uniform disk, so the falloff reads off the diameter $\theta$ . Crucially the observable depends on the modulus of the spatial coherence, so a random atmospheric phase shift on each beam cancels — which is why it works through turbulence that would destroy a Michelson interferometer of the same baseline.