Single-Photon Sources

A single-photon source is the piece of hardware that makes the quantized field tangible. It takes the Fock state $|1\rangle$ from aq-optics-fock-states — the number eigenstate with exactly one excitation — and the antibunching signature $g^{(2)}(0)<1$ from aq-optics-photon-antibunching and aq-optics-photon-statistics, and asks a purely engineering question: can we build a box that, on demand, delivers that state into an optical mode, one photon per trigger and essentially never two? The physics is a statement about photon statistics, recalled by name rather than re-derived. A laser in a coherent state is Poissonian; thermal light is super-Poissonian and clumped; a single-photon source is sub-Poissonian and, more sharply, antibunched — the joint probability of two photons at zero delay is pushed below the Poissonian value. And that suppression is not a matter of degree: the Cauchy–Schwarz inequality forces $g^{(2)}(0)\geq 1$ for any classical fluctuating intensity, so a measured value below one cannot be dressed up as classical noise. It certifies the field must be described by operators, tying the emission directly to the Fock structure of light developed upstream in this chapter. Two families of hardware realize it. A single quantum emitter — a molecule, a semiconductor quantum dot, a nitrogen-vacancy center in diamond — holds at most one excitation, so after radiating it must be re-excited before it can radiate again, and consecutive photons are forced apart by the excited-state lifetime. Heralding takes the opposite route: spontaneous parametric down-conversion makes photons in correlated pairs, and detecting one member announces its twin in a known mode. Cavity quantum electrodynamics — the Purcell effect — is the lever that turns either strategy into a bright, indistinguishable, high-purity source, and it is where the frontier now sits.

Key takeaways

A single-photon source loads the Fock state $|1\rangle$ into an optical mode one excitation at a time, strongly suppressing two-photon events relative to a coherent (Poissonian) source of the same mean brightness.
The defining, measurable signature is antibunching: the normalized second-order coherence obeys $g^{(2)}(0)<1$ , the ideal single emitter giving $g^{(2)}(0)=0$ , read off a Hanbury Brown–Twiss setup.
No classical field can do this: non-negative intensity variance forces $g^{(2)}(0)\geq 1$ classically, so $g^{(2)}(0)<1$ is a rigorous nonclassicality witness tied to the quantized field.
Hardware splits into single quantum emitters driven through a two-level cycle (dots, molecules, color centers) and heralded photons from down-conversion; a cavity (Purcell effect) buys brightness and indistinguishability.
Three figures of merit — purity $1-g^{(2)}(0)$ , indistinguishability (Hong–Ou–Mandel visibility), and efficiency — are mutually in tension; no platform maximizes all three at room temperature over all wavelengths.
Key idea: a single quantum emitter holds one excitation at a time, so after it radiates it must be re-excited before it can radiate again — the zero-delay two-photon probability vanishes, $g^{(2)}(0)\to 0$ , which is the operational definition of a single-photon source.

The core insight

A single quantum emitter cannot emit two photons at once: after radiating it must be re-excited, so the joint two-photon detection probability at zero delay vanishes, giving $g^{(2)}(0)\to 0$ — the operational, measurable definition of a single-photon source.

Where this lives today

The single-photon source has crossed from demonstration to product. Companies such as Quandela and Sparrow Quantum ship semiconductor quantum dots in Purcell-enhanced micropillar and open-cavity structures that deliver photons on demand with $g^{(2)}(0)$ well under a percent, Hong–Ou–Mandel indistinguishability above 95%, and efficiencies of tens of percent — the simultaneous high-purity, high-indistinguishability, high-brightness regime that was impossible a decade ago. These are the front end of photonic quantum computers: a bright indistinguishable source feeding a reconfigurable interferometer is exactly the architecture behind boson-sampling machines and gate-based photonic processors, where dozens of identical photons must interfere with near-unit visibility. On the heralded side, spontaneous parametric down-conversion in periodically poled crystals and integrated lithium-niobate and silicon waveguides remains the workhorse of quantum communication, feeding satellite quantum key distribution — the Micius satellite distributed entangled photons over more than a thousand kilometers — and the fiber repeaters now being prototyped. Diamond nitrogen-vacancy and silicon-vacancy centers, uniquely, run at or near room temperature and double as spin qubits, making them leading candidates for network nodes and for nanoscale magnetometry, where a single defect's antibunched fluorescence reports the local field.

The physical picture

Picture a lighthouse whose lamp is only ever fully on or fully off and needs a fixed recharge time between flashes. The instant after one flash it is dark and cannot fire again, so two flashes never overlap — they come out evenly spaced, gated by the recharge clock. Replace "flash" with "photon" and "recharge" with the excited-state lifetime and you have a single quantum emitter: it holds one quantum of energy, and having spent it on a photon it is empty until re-pumped. Simultaneous two-photon emission is not rare — it is forbidden. That is the whole mechanism behind $g^{(2)}(0)\to 0$ , and why the antibunched stream looks regular where a laser's looks random and a thermal source's looks clumped. Heralding runs on a different logic. Down-conversion makes photons strictly two at a time, so catching one member tells you, with near-certainty, that its twin is present in a specific mode. The herald click is a doorbell announcing one guaranteed guest in a known room, converting a random pair generator into an event-triggered single-photon supply — at the cost that you get one only when the dice roll a pair.

Figure

A Hanbury Brown–Twiss setup: single photons from the emitter hit a 50 to 50 beam splitter, and because one photon can take only one path, the two detectors never click together at zero delay — the coincidence rate at τ ≈ 0 collapses, which is the operational meaning of g⁽²⁾(0) → 0

The measurement that certifies all this is the Hanbury Brown–Twiss experiment, borrowed from aq-optics-hanbury-brown-twiss: split the beam on a 50:50 beam splitter, put a detector on each arm, and histogram the delay between clicks. A coherent beam clicks both detectors independently, so zero-delay coincidences are as likely as any other delay; a single-photon stream cannot, because the lone photon is indivisible and goes one way or the other, so the zero-delay coincidence rate collapses into an antibunching dip whose depth is $g^{(2)}(0)$ . The second figure is the punchline the whole subject is measured against: the three archetypal $g^{(2)}(\tau)$ curves on one axis.

Figure

The three faces of light in one plot: thermal light bunches to g⁽²⁾(0) = 2, a coherent laser stays flat at g⁽²⁾(0) = 1, and a single emitter antibunches into a dip that here reaches a measured g⁽²⁾(0) = 0.09, recovering toward 1 on the excited-state lifetime τ — only the dip below 1 is forbidden to any classical field

The quantum core is compact. Bunching ( $g^{(2)}(0)=2$ , thermal) and flatness ( $g^{(2)}(0)=1$ , coherent) both live in the classically allowed region; the entire art of single-photon sources is to push below one, where no fluctuating classical intensity can reach. Everything downstream — purity, indistinguishability, brightness, and the cavity engineering that trades them off — is the effort to reach the dip and make it useful.

The core equations

The whole subject is a discipline of one measurable quantity, the normalized second-order correlation. Defined from the field operators in normal and time order, it counts the joint probability of one detection at time $t$ and a second at $t+\tau$ — the operator translation of the Hanbury Brown–Twiss coincidence histogram.

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 (1)

where: $\hat a,\hat a^{\dagger}$ — annihilation and creation operators for the detected mode; $\tau$ — delay between detections; $\langle\cdots\rangle$ — normal- and time-ordered expectation. Reading: the numerator is the two-photon coincidence rate, normalized by the square of the mean intensity, so $g^{(2)}(\tau)$ is coincidences relative to uncorrelated clicks. Setting $\tau=0$ and using $\hat a^{\dagger}\hat a^{\dagger}\hat a\hat a=\hat n(\hat n-1)$ turns the correlation into a statement purely about the photon-number distribution — the workhorse form, connected directly to the number states of aq-optics-fock-states.

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

where: $\hat n=\hat a^{\dagger}\hat a$ — photon number operator; $\langle \hat n(\hat n-1)\rangle$ — mean number of ordered photon pairs. Reading: for a pure Fock state $|n\rangle$ this is $n(n-1)/n^2=1-1/n$ — exactly $0$ for $|1\rangle$ , $1/2$ for $|2\rangle$ , and approaching $1$ for large $n$ ; a single photon is maximally antibunched because it contains no pair to detect. Because $\langle \hat n(\hat n-1)\rangle=\langle \hat n^2\rangle-\langle \hat n\rangle$ , the same quantity reads out number fluctuations. The Mandel $Q$ parameter, from aq-optics-photon-statistics, repackages the departure from Poissonian as one signed number.

g^{(2)}(0)=1+\frac{\mathrm{Var}(n)-\langle n\rangle}{\langle n\rangle^{2}},\qquad Q=\langle n\rangle\big[g^{(2)}(0)-1\big] \qquad (3)

where: $\mathrm{Var}(n)$ — variance of the photon number; $Q$ — Mandel parameter. Reading: Poissonian light has $\mathrm{Var}(n)=\langle n\rangle$ , so $g^{(2)}(0)=1$ and $Q=0$ ; sub-Poissonian light has $\mathrm{Var}(n)<\langle n\rangle$ , so $Q<0$ and $g^{(2)}(0)<1$ — the nonclassical regime a single-photon source must occupy. It is worth seeing why the naïve fix — attenuate a laser until it is "one photon on average" — fails. An attenuated laser stays coherent, with a Poisson number distribution and a $g^{(2)}(0)$ stubbornly independent of the mean.

P(n)=\frac{\langle n\rangle^{n}}{n!}\,e^{-\langle n\rangle},\qquad g^{(2)}(0)=1,\qquad \frac{P(2)}{P(1)}=\frac{\langle n\rangle}{2} \qquad (4)

where: $P(n)$ — Poisson distribution of an attenuated laser; $\langle n\rangle$ — mean photons per pulse. Reading: dimming shrinks the absolute two-photon probability as $\langle n\rangle^2$ , but the ratio of multiphoton to single-photon events — what $g^{(2)}(0)$ captures — never drops below one; the light is faint, never antibunched. The engineering lever that makes real emitters bright and coherent is cavity quantum electrodynamics: placing the emitter in a resonant mode enhances its emission rate into that mode by the Purcell factor, funneling photons into one spatial mode and shortening the emission time.

F_{P}=\frac{3}{4\pi^{2}}\left(\frac{\lambda}{n}\right)^{3}\frac{Q}{V},\qquad \Gamma_{\mathrm{cav}}=F_{P}\,\Gamma_0 \qquad (5)

where: $F_{P}$ — Purcell enhancement of the rate into the cavity mode; $Q$ — cavity quality factor; $V$ — mode volume; $\lambda/n$ — wavelength inside the medium; $\Gamma_0$ — free-space decay rate. Reading: a high- $Q$ , small- $V$ cavity multiplies the decay rate, raising brightness, improving collection (a defined mode to couple into a fiber), and lifting indistinguishability by letting the photon escape before dephasing scrambles its phase. The heralding route has its own trade-off. In down-conversion the pump occasionally makes two pairs rather than one, and those double-pair events lift the heralded correlation off zero; their probability scales as the square of the per-pulse pair probability, so purity is bought with rate.

g^{(2)}_{\mathrm{herald}}(0)\approx 2\,P_{\mathrm{pair}} \qquad (6)

where: $P_{\mathrm{pair}}$ — probability of one photon pair per pump pulse. Reading: lowering the pump drives $P_{\mathrm{pair}}\to 0$ and the heralded $g^{(2)}(0)\to 0$ , so the source can be made arbitrarily pure — but its rate, linear in $P_{\mathrm{pair}}$ , falls with it. This is the intrinsic brightness–purity tension of probabilistic pair sources, developed in aq-optics-spontaneous-parametric-down-conversion. Purity is only half of usability. To interfere — the requirement of linear-optical quantum computing — photons must be indistinguishable, quantified by the two-photon Hong–Ou–Mandel visibility.

V_{\mathrm{HOM}}=\frac{R_{\perp}-R_{\parallel}}{R_{\perp}} \qquad (7)

where: $V_{\mathrm{HOM}}$ — interference visibility, the mean wavepacket overlap; $R_{\parallel},R_{\perp}$ — coincidence rates for indistinguishable versus deliberately distinguishable (orthogonal-polarization) photons. Reading: identical photons bunch at a beam splitter and coincidences drop to zero, so $V_{\mathrm{HOM}}\to 1$ for transform-limited identical photons and degrades with any spectral, temporal, or polarization mismatch.

Derivation AAntibunching from a driven two-level emitter

The goal is to prove, from operator algebra alone, that a single two-level system is a perfect single-photon source at zero delay — $g^{(2)}(0)=0$ exactly — and to see how the correlation recovers toward one on the excited-state lifetime, giving the antibunching dip a width set by how fast the emitter can be re-pumped. We model the emitter with ground $|g\rangle$ , excited $|e\rangle$ , and lowering operator $\hat\sigma=|g\rangle\langle e|$ , the detected far-field signal proportional to $\hat\sigma^\dagger\hat\sigma$ .

Step 1. The numerator of $g^{(2)}(0)$ needs two successive lowerings: $\hat\sigma\hat\sigma=|g\rangle\langle e|g\rangle\langle e|=0$ , because $\langle e|g\rangle=0$ . Why: a two-level system holds at most one excitation, so applying "emit a photon" twice annihilates every state — there is no second excitation to remove.

Step 2. Hence the numerator $\langle \hat\sigma^\dagger\hat\sigma^\dagger\hat\sigma\hat\sigma\rangle=0$ in any state, while the denominator $\langle\hat\sigma^\dagger\hat\sigma\rangle^2$ is generically nonzero, so $g^{(2)}(0)=0$ . Why: the zero-delay two-photon probability is identically zero however hard we drive — simultaneous two-photon emission is not suppressed but forbidden.

Step 3. For nonzero delay the atom must decay to $|g\rangle$ and be re-excited before it can emit again; the weak-drive Bloch equations give $g^{(2)}(\tau)=1-e^{-\Gamma\tau/2}$ , with $\Gamma$ the radiative width. Why: the correlation recovers toward $1$ on the excited-state timescale, so the dip has a width of order the lifetime — the lighthouse "recharge time" made quantitative.

Step 4. Strong resonant drive modulates the recovery with Rabi oscillations at $\Omega$ , so $g^{(2)}(\tau)$ overshoots and rings; in a real device the exact zero is lifted to a small positive $g^{(2)}(0)$ by background fluorescence, laser leakage, and detector jitter. Why: the ideal sets the target, and each departure is a photon that should not have been there. *Verified: consistent with the standard resonance-fluorescence treatment (Loudon ch. 6) and the identity $\hat\sigma^2=0$ .*

Go deeper · Derivation B · 1 minThe classical floor on g⁽²⁾(0)

Derivation BThe classical floor on g⁽²⁾(0)

The goal is to establish the sharp dividing line the whole subject is organized around: any light describable by a classical, fluctuating, non-negative intensity must have $g^{(2)}(0)\geq 1$ , so a measured value below one is not merely unusual but strictly impossible for classical light — a certificate of the quantized field. We treat the field classically, intensity $I(t)\geq 0$ drawn from some distribution, compute the zero-delay correlation as an ensemble average, and apply only the non-negativity of variance.

Step 1. The classical zero-delay coherence is $g^{(2)}(0)=\langle I(t)^2\rangle/\langle I(t)\rangle^2$ , ensemble averages replacing normal-ordered moments. Why: classically there is no operator ordering, and the coincidence rate is just the mean of the squared intensity — what a classical theory predicts for HBT.

Step 2. Invoke non-negativity of variance for the real variable $I$ : $\mathrm{Var}(I)=\langle I^2\rangle-\langle I\rangle^2\geq 0$ . Why: variance averages a squared quantity and can never be negative — the one unavoidable fact that does all the work.

Step 3. Rearrange to $\langle I^2\rangle\geq\langle I\rangle^2$ , hence $g^{(2)}(0)\geq 1$ , equality only for a perfectly stabilized field. Why: this is the classical floor — bunching reflects fluctuations, and the steadiest classical field sits at exactly one, never below.

Step 4. A single-photon source with $g^{(2)}(0)<1$ therefore has no classical intensity description. Why: antibunching is a rigorous, assumption-light witness of nonclassicality and Fock structure — the inequality that forbids it classically is what makes the measurement worth doing. *Verified: standard Cauchy–Schwarz argument; see Loudon ch. 3 and Fox ch. 6.*

Go deeper · Derivation C · 1 minHow extra emitters dilute the antibunching

Derivation CHow extra emitters dilute the antibunching

The goal is to derive the rule $g^{(2)}(0)=1-1/n$ for $n$ identical, independent single-photon emitters collected together — the result that turns a measured correlation into a count of contributing emitters and yields the universal single-emitter criterion $g^{(2)}(0)<1/2$ . We collect $n$ independent emitters, each with the same small excitation probability $p$ and its own random phase, form $\hat A=\sum_i\hat\sigma_i$ , and evaluate $g^{(2)}(0)$ term by term.

Step 1. Mean intensity: $\langle \hat A^\dagger\hat A\rangle=\sum_i\langle\hat\sigma_i^\dagger\hat\sigma_i\rangle=n\,p$ . Why: cross terms with $i\neq j$ average to zero because independent emitters carry uncorrelated random phases, leaving only the diagonal.

Step 2. Two-photon term: $\langle \hat A^\dagger\hat A^\dagger\hat A\hat A\rangle=\sum_{i\neq j}\langle\hat\sigma_i^\dagger\hat\sigma_i\rangle\langle\hat\sigma_j^\dagger\hat\sigma_j\rangle=n(n-1)p^2$ . Why: same-emitter terms vanish ( $\hat\sigma_i^2=0$ ), so a zero-delay pair needs two different emitters to fire — and there are $n(n-1)$ ordered distinct pairs.

Step 3. Assemble: $g^{(2)}(0)=\dfrac{n(n-1)p^2}{(np)^2}=1-\dfrac1n$ . Why: $p$ cancels, so the result depends only on the count — $0$ for one emitter, $1/2$ for two, climbing toward $1$ as many emitters wash out the antibunching.

Step 4. The criterion: $g^{(2)}(0)<1/2$ requires $n<2$ , certifying that one emitter dominates. Why: this is the accepted fingerprint of an isolated quantum emitter — the bar a candidate must clear. *Verified: reproduces the standard single-emitter threshold; consistent with resonance-fluorescence statistics.*

Limiting cases

The character of a source is read directly off $g^{(2)}(0)$ and the photon-number distribution it implies:

Case	Result	Meaning
Ideal single photon	$g^{(2)}(0)=0$	Fock state $\vert 1\rangle$ : exactly one photon, zero coincidence probability at zero delay. The unattainable target.
Coherent (laser)	$g^{(2)}(\tau)=1$	Poissonian statistics; even attenuated to $\langle n\rangle\ll1$ the correlation stays flat at one — faint, not antibunched.
Thermal / chaotic	$g^{(2)}(0)=2$	Super-Poissonian bunching; photons arrive in clumps, the opposite extreme from a single-photon source.
Real single emitter	$0<g^{(2)}(0)\lesssim 0.1$	Residual multiphoton events from background fluorescence, in-pulse re-excitation, and dark counts; below $1/2$ already certifies dominant single-photon emission.
$n$ independent emitters	$g^{(2)}(0)=1-\tfrac1n$	$g^{(2)}(0)<1/2$ proves fewer than two emitters contribute — the standard criterion for isolating one.
Strong-drive limit	$g^{(2)}(\tau)$ rings at $\Omega$	Rabi oscillations modulate the recovery; $g^{(2)}(0)=0$ still holds, but the dip grows oscillatory sidebands.

Common trapAttenuating a laser to "one photon on average" builds a single-photon source

It does not. A weak coherent state is still a coherent state: its photon number is Poisson-distributed and $g^{(2)}(0)=1$ exactly, no matter how far you dim it. Turning the laser down to $\langle n\rangle=0.01$ shrinks the absolute two-photon probability as $\langle n\rangle^2$ , but the ratio of two-photon to one-photon events — which is what $g^{(2)}(0)$ measures and what a quantum protocol cares about — is pinned at one. The light is merely faint, and it interferes and eavesdrops exactly like bright coherent light scaled down. Genuine single-photon character means $g^{(2)}(0)<1$ , a region no classical or coherent field can enter; a real source must actively suppress the multiphoton fraction, not just the multiphoton rate. (This is why quantum key distribution with weak coherent pulses needs the decoy-state trick to stay secure, rather than relying on faintness alone.)

Worked examples

1 · Purcell engineering of a quantum-dot micropillar (confidence builder)

An InAs quantum dot emitting at $\lambda=925$ nm sits in a GaAs micropillar cavity ( $n=3.5$ ) with quality factor $Q=2500$ and mode volume $V=0.090\ \mu\mathrm{m}^3$ . Its free-space excited-state lifetime is $\tau_0=1.0$ ns. Find the Purcell factor, the fraction of photons emitted into the cavity mode, and the shortened emission time — then say why this rescues indistinguishability.

Step 1. Wavelength in the medium: $\lambda/n=925/3.5=264.3$ nm, so $(\lambda/n)^3=(0.2643\ \mu\mathrm{m})^3=0.01846\ \mu\mathrm{m}^3$ , and the prefactor $3/(4\pi^2)=0.07599$ .

Step 2. Assemble equation (5): $F_P=0.07599\times 0.01846\times\dfrac{2500}{0.090}\approx \mathbf{39}$ — the cavity multiplies the dot's emission rate into its mode by nearly forty.

Step 3. The fraction funneled into the cavity mode is $\beta=F_P/(1+F_P)=39/40=\mathbf{0.975}$ : the dot dumps 97.5% of its photons into one collectable mode instead of scattering into $4\pi$ — the brightness and collection win.

Step 4. The emission time shortens to $\tau=\tau_0/(1+F_P)=1000\ \mathrm{ps}/40\approx\mathbf{25\ ps}$ . Because the photon escapes in 25 ps — far faster than the tens to hundreds of ps on which phonon and charge-noise dephasing act — the wavepacket stays near transform-limited, so successive photons overlap in Hong–Ou–Mandel interference. $F_P\approx39$ , $\beta\approx0.975$ , $\tau\approx25$ ps — one cavity buys brightness, collection, and indistinguishability at once.

2 · Reading an HBT measurement

A diamond color center under pulsed excitation gives a measured $g^{(2)}(0)=0.12$ while delivering a mean of $\langle n\rangle=0.08$ collected photons per pulse. (a) What is the single-photon purity? (b) How many emitters are contributing? (c) What is the Mandel $Q$ ? (d) How much is the two-photon (multiphoton) probability suppressed relative to an attenuated laser of the same brightness?

Step 1. Purity is $1-g^{(2)}(0)=\mathbf{0.88}$ — 88% of the coincidence-relevant events are genuine single photons, and the value sits below the classical floor of one, certifying nonclassical emission.

Step 2. Invert $g^{(2)}(0)=1-1/n$ : $n=1/(1-0.12)=1/0.88\approx\mathbf{1.14}$ . Since $g^{(2)}(0)<0.5$ , a single emitter dominates — the standard single-emitter certification is met.

Step 3. The Mandel parameter is $Q=\langle n\rangle[g^{(2)}(0)-1]=0.08\times(-0.88)=\mathbf{-0.070}$ : negative, confirming sub-Poissonian statistics, with a small magnitude set by the low mean photon number.

Step 4. For matched brightness a Poissonian source has two-photon probability $\approx\langle n\rangle^2/2$ and this source $\approx g^{(2)}(0)\langle n\rangle^2/2$ , so the ratio is just $g^{(2)}(0)=0.12$ — the source emits $1/0.12\approx\mathbf{8.3\times}$ fewer double-photon events than a matched weak laser, the multiphoton suppression that raises secure-key rates in QKD. Purity 0.88, $\approx1.1$ emitters, $Q=-0.070$ , $\approx8\times$ fewer multiphoton events. A third case makes the heralded trade-off concrete without new machinery: an SPDC source at pair probability $P_{\mathrm{pair}}=0.01$ has $g^{(2)}_{\mathrm{herald}}(0)\approx2P_{\mathrm{pair}}=0.02$ (purity $0.98$ ), because double pairs $\sim P_{\mathrm{pair}}^2=10^{-4}$ are a hundred times rarer than singles. Chasing brightness to $P_{\mathrm{pair}}=0.10$ gives $g^{(2)}(0)\approx0.20$ : the rate rises tenfold but the contamination, quadratic in $P_{\mathrm{pair}}$ , rises a hundredfold — the intrinsic reason SPDC platforms run deliberately dim.

Physics of the portrait

Single-photon sources have no single inventor, but they have a decisive first measurement: the moment antibunching — light that no classical field can produce — was seen in the laboratory. That moment belongs to a small collaboration at Rochester, and to one physicist who then spent a career turning the phenomenon into a technology.

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

H. Jeffrey Kimble

b. 1949 · cavity QED and the first observation of photon antibunching

In 1977, working with Mario Dagenais and Leonard Mandel at the University of Rochester, H. Jeffrey Kimble reported the first observation of photon antibunching, in the resonance fluorescence of a dilute beam of sodium atoms. It was a landmark: the measured $g^{(2)}(\tau)$ rose from a minimum at zero delay — the opposite of the bunching Hanbury Brown and Twiss had seen two decades earlier — and it was the first light ever shown to be strictly nonclassical in its intensity correlations, a direct sighting of the quantized field that Glauber's 1963 coherence theory had formalized. The experiment was hard precisely because a thermal beam of many atoms dilutes the effect toward $g^{(2)}(0)=1-1/n$ ; only with very few atoms in the observation volume could the correlation be pushed below one. Kimble went on to found much of modern cavity quantum electrodynamics at Caltech — single-atom sources, the "photon turnstile," strongly coupled atom-cavity systems, and the first quantum-network architectures. The through-line to today's quantum-dot micropillars is unbroken: prove the light is nonclassical, then engineer a cavity around the emitter to make it bright, directional, and indistinguishable.

History

1956

— Hanbury Brown and Twiss measure photon bunching in an intensity interferometer

1963

— Glauber formulates the quantum theory of optical coherence and the hierarchy g⁽ⁿ⁾

1977

— Kimble, Dagenais, and Mandel observe photon antibunching in atomic resonance fluorescence

1987

— Hong, Ou, and Mandel demonstrate two-photon interference at a beam splitter

2000

— Single quantum dots and diamond nitrogen-vacancy centers are shown to emit antibunched single photons

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 3 and 6 (coherence, resonance fluorescence, antibunching) · Fox, Quantum Optics: An Introduction, ch. 6 (photon statistics and single-photon sources) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., ch. 13 and 14 (photon optics, cavity effects) · Foot, Atomic Physics, ch. 7 (spontaneous emission) · Demtröder, Laser Spectroscopy (fluorescence of single emitters).

Provenance. Definitions and normalizations follow Loudon (ch. 3 and 6) and Fox (ch. 6). A verifier should recompute: $g^{(2)}(0)=\langle n(n-1)\rangle/\langle n\rangle^2$ giving $0$ for $|1\rangle$ and $1-1/n$ for $|n\rangle$ ; the multi-emitter $1-1/n$ from independent $\hat\sigma_i$ with $\hat\sigma_i^2=0$ ; the classical bound $g^{(2)}(0)\geq1$ from $\mathrm{Var}(I)\geq0$ ; the Purcell factor $F_P=(3/4\pi^2)(\lambda/n)^3 Q/V$ (Saleh and Teich; Fox), which for $\lambda=925$ nm, $n=3.5$ , $Q=2500$ , $V=0.090\ \mu\mathrm{m}^3$ gives $F_P\approx39$ , $\beta\approx0.975$ , $\tau\approx25$ ps; the Mandel relation $Q=\langle n\rangle[g^{(2)}(0)-1]$ ; and $g^{(2)}_{\mathrm{herald}}(0)\approx2P_{\mathrm{pair}}$ . Thermal $g^{(2)}(0)=2$ and coherent $g^{(2)}(0)=1$ are standard, and $g^{(2)}(0)<1/2$ is the accepted single-emitter threshold. The 1977 antibunching measurement is Kimble, Dagenais, and Mandel, Phys. Rev. Lett. 39, 691. The $g^{(2)}(0)=0.09$ and $0.12$ figures are illustrative values used only to exercise the formulas.

Prerequisites. Fock States and the Photon Number Basis · Photon Antibunching · Photon Statistics and the Mandel Q Parameter · Coherence Functions and the Degree of Coherence · The Hanbury Brown–Twiss Experiment. Leads to. Spontaneous Parametric Down-Conversion · Squeezed States of Light · Quadrature Operators and the Uncertainty Relation.

Ask the tutor: ask me to walk through why $\hat\sigma^2=0$ forces $g^{(2)}(0)=0$ , or to size the Purcell factor and indistinguishability of a cavity for a source you have in mind.

Why is a two-level system essential — would a three-level scheme still work?

The key requirement is that the emitter holds at most one excitation to release at a time, which $\hat\sigma^2=0$ encodes. Real emitters are rarely literally two-level; quantum dots and color centers have rich level structures, and many are pumped through an upper state that decays into the emitting transition. What matters is that only one photon on the transition of interest is available per cycle, and that the system returns to its ground state before emitting again. So three- and four-level pumping schemes work fine — indeed off-resonant pumping is preferred because it avoids laser leakage into the collection path — as long as the emitting transition behaves as an effective two-level ladder re-climbed after each photon.

Why does indistinguishability matter if I only care about purity?

It depends on the application. If you only need to know a photon arrived — single-photon detection, some QKD protocols — then low $g^{(2)}(0)$ is what counts and the photons need not be identical. But any scheme that interferes photons — linear-optical quantum computing, boson sampling, entanglement swapping, repeaters — requires that successive photons be indistinguishable: identical in spectrum, temporal profile, and polarization, and transform-limited. Photons that differ interfere poorly at a beam splitter, the Hong–Ou–Mandel visibility drops below one, and the computation is corrupted. This is why so much cavity engineering targets indistinguishability specifically — a bright, pure, but distinguishable source is useless for photonic quantum information.

How is a real g⁽²⁾(0) measured if detectors have dead time?

This is exactly why the Hanbury Brown–Twiss geometry exists. A single detector has a dead time of tens of nanoseconds and cannot register two photons arriving together, so it would blindly report zero coincidences for any source. Splitting onto two detectors sidesteps this: simultaneous photons can trigger the two arms independently, and the coincidence electronics histogram the delay between clicks. The zero-delay bin, normalized to the far-delay bins, is $g^{(2)}(0)$ . Residual dark counts, timing jitter, and afterpulsing lift the measured value slightly, which is why real emitters report small positive numbers rather than an exact zero.

Are single-photon sources actually deterministic, or is that marketing?

Truly deterministic operation means one photon appears in the output mode every time you press the trigger, with unit probability — and no real source achieves it. A single emitter can be excited on demand, so it is deterministic in principle, but collection losses, non-unit quantum efficiency, and imperfect mode coupling keep the per-trigger probability of a usable photon well below one; the best quantum-dot cavity sources reach tens of percent. Heralded SPDC sources are worse: they are fundamentally probabilistic, firing only when a pair happens to be generated, though a herald click confirms after the fact that a photon is present. "On-demand" in the literature means "triggered," not "unit efficiency."

Why can't I just filter a thermal or laser source to get single photons?

Spectral or spatial filtering changes which modes you collect but does not touch the photon-number statistics within a mode. Filter a laser and you still have a dimmer coherent state with $g^{(2)}(0)=1$ ; filter thermal light and you still have bunched light with $g^{(2)}(0)=2$ . Antibunching requires the emission process itself to release photons one at a time — a property of the source's quantum dynamics ( $\hat\sigma^2=0$ for a single emitter, or the pair correlation of down-conversion), not of any downstream optic. Beam splitters, filters, and attenuators are linear and cannot manufacture the nonclassical number correlations $g^{(2)}(0)<1$ demands; only a genuinely quantum emission process can.

What sets the practical floor on how low g⁽²⁾(0) can go?

For a single emitter the intrinsic value is zero, so everything above it is technical. The culprits are background fluorescence and laser leakage into the collection path (a stray uncorrelated photon), re-excitation during a finite pump pulse (the emitter fires, is re-pumped, and fires again in the same pulse), spectral diffusion and charge noise, and detector dark counts and afterpulsing. Under resonant pulsed excitation with good filtering, quantum-dot sources routinely reach $g^{(2)}(0)$ of a few times $10^{-3}$ . Pushing lower means shortening the pulse relative to the lifetime, cooling to suppress phonon effects, and using low-dark-count superconducting-nanowire detectors — each a fight for another factor in the multiphoton fraction.