Quantum Optics
Photon Antibunching: The g²(0) Dip of a Single Emitter
Point two detectors at a single atom and ask how often they click at the very same instant — the answer is essentially never. This is photon antibunching: a lone quantum emitter cannot emit two photons simultaneously, so its second-order correlation function collapses to a dip at zero delay, g²(0) → 0. First seen in 1977 in the resonance fluorescence of sodium atoms by Kimble, Dagenais and Mandel, it is the cleanest laboratory signature that light is quantized — a feature with no classical wave interpretation.
Concretely, g²(0) < 1 is impossible for any classical field, and g²(0) < 0.5 certifies that fewer than one emitter's worth of photons arrive together. A perfect single emitter gives g²(0) = 0; real quantum dots, molecules and nitrogen-vacancy centers routinely reach g²(0) ≈ 0.01–0.2, the working definition of a "single-photon source."
- RegimeNonclassical light; single quantum emitter
- Key relationg²(0) < 1 (ideal g²(0) = 0)
- First observedKimble, Dagenais & Mandel, 1977 (Na atoms)
- MeasurementHanbury Brown–Twiss intensity correlation
- Characteristic scaleRecovery over the excited-state lifetime ~1–20 ns
- Matters forQuantum key distribution, photonic quantum computing, metrology
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What antibunching is and why it forced light to be quantized
Photon antibunching is the tendency of photons emitted by a single quantum system to arrive spread out in time rather than clumped. Operationally it is defined through the normalized second-order (intensity) correlation function g²(τ) = ⟨:Î(t)Î(t+τ):⟩ / ⟨Î(t)⟩², the probability of detecting a second photon a delay τ after a first, relative to random. Antibunching means g²(τ) rises from a minimum at τ = 0, so g²(0) < g²(τ) — and crucially g²(0) < 1.
This inequality is the punchline. Any classical stochastic field obeys the Cauchy–Schwarz bound g²(0) ≥ 1: intensity fluctuations can only make photons cluster, never anti-cluster. A measured g²(0) < 1 therefore cannot be reproduced by any classical electromagnetic wave, however you shape its noise. As Kimble, Dagenais and Mandel emphasized in 1977, antibunching is comprehensible only if the field is quantized and the atom emits discrete photons one at a time — direct evidence for the photon and for quantum jumps.
The mechanism: an emitter has to re-excite before it can fire again
The physics is disarmingly simple. A single two-level emitter that has just emitted a photon is left in its ground state. It is now incapable of emitting a second photon until it is re-excited — it must absorb energy, climb back to the excited state, and only then decay. This mandatory re-excitation imposes a dead time on the order of the excited-state lifetime, and that dead time is exactly the antibunching dip.
Formally, the emission operator projects the atom to the ground state: the conditional state right after a click has zero excited-state amplitude, so the instantaneous re-emission probability vanishes. Under continuous resonant driving the atom then undergoes Rabi oscillations while relaxing, so g²(τ) recovers toward 1 and can even overshoot — a damped oscillation whose peaks reflect the coherent drive. The single-quantum, projective nature of emission is the conserved bookkeeping: one excitation in, one photon out, never two at once. This is why the effect requires a single emitter (or very few); an ensemble of N independent atoms washes the dip out as ~1/N.
The governing equations, the criterion, and the numbers
For a resonantly driven two-level atom the second-order correlation has the closed form
g²(τ) = 1 − e^(−3Γτ/4)[cos(μτ) + (3Γ/4μ)·sin(μτ)], with μ = √(Ω² − (Γ/4)²),
where Γ is the spontaneous decay rate (Γ = 1/T₁) and Ω the Rabi frequency. At τ = 0 the bracket equals 1, so g²(0) = 0 exactly — perfect antibunching, independent of drive strength. For weak drive (Ω ≪ Γ) the dip recovers monotonically over ~1/Γ; for strong drive (Ω ≫ Γ) g²(τ) oscillates at the Rabi frequency and overshoots 1 before settling.
The practical certification thresholds: g²(0) < 1 proves nonclassicality; g²(0) < 0.5 proves the emission is dominated by a single emitter (two independent emitters bottom out at 0.5). Characteristic recovery times track the lifetime: ~1.6 ns for InAs quantum dots, ~10–25 ns for NV centers in diamond, tens of ns for single molecules. Deviations of the measured floor from zero directly quantify multiphoton contamination.
How it is measured: the Hanbury Brown–Twiss interferometer
You cannot measure g²(0) with one detector, because real single-photon detectors have a dead time of tens of nanoseconds — they are blind to a second photon arriving immediately after the first. The trick, borrowed from Hanbury Brown and Twiss's 1956 stellar intensity interferometer, is to send the light onto a 50:50 beamsplitter and place a detector in each output arm. A start–stop coincidence circuit then histograms the delay τ between a click on detector A and the next on detector B.
For antibunched light this coincidence histogram shows a pronounced dip at τ = 0: if the emitter released only one photon, it went to A or B, never both, so simultaneous clicks are suppressed. The depth of the dip, normalized to the flat coincidence background at long delay, is g²(0). Under pulsed excitation the histogram becomes a comb of peaks spaced by the repetition period, and the vanishing (or suppression) of the central peak relative to its neighbors gives g²(0). This HBT configuration is the universal workhorse for certifying single-photon sources in every platform below.
Where it operates — from atomic beams to solid-state qubits
Antibunching appears in essentially any isolated quantum emitter: single trapped ions and neutral atoms, single dye molecules (Basché, Moerner, Orrit, early 1990s), semiconductor quantum dots (InAs/GaAs, colloidal, perovskite), color centers in diamond (NV and SiV) and in hexagonal boron nitride, and carbon nanotubes. Record purities are extraordinary — epitaxial quantum dots in micropillars and colloidal dots reach g²(0) ≪ 10⁻², while NV centers give g²(0) ≈ 0.05–0.2 at room temperature.
It is important to distinguish antibunching from its opposite, bunching (g²(0) > 1), the effect Hanbury Brown and Twiss originally exploited with thermal starlight (g²(0) = 2 for chaotic light). Antibunching is also distinct from photon blockade — a cavity-QED analog where strong single-photon nonlinearity, not the emitter's dead time, forbids a second photon. And it differs from sub-Poissonian statistics measured on a single detector: antibunching is specifically a statement about the time correlation g²(τ), the hallmark of a genuinely single quantum system.
Why it matters: single-photon sources and open questions
Antibunching is the enabling metric for the deterministic single-photon sources that quantum technology needs. In BB84-style quantum key distribution, multiphoton pulses open the door to photon-number-splitting attacks, so a low g²(0) directly bounds an eavesdropper's information. In linear-optical quantum computing and boson sampling, on-demand indistinguishable single photons with g²(0) → 0 are the raw resource; the field now demands sources that are simultaneously bright, pure (g²(0) < 0.01) and indistinguishable (Hong–Ou–Mandel visibility → 1).
Open frontiers remain. Combining near-perfect purity with high collection efficiency and high indistinguishability in a single device is still hard, especially at room temperature and telecom wavelengths for fiber networks. Cavity Purcell engineering, phonon-sideband suppression and strain tuning are active levers. Meanwhile, higher-order correlations (g³, g⁴) and antibunching in exotic systems — heralded from spontaneous parametric down-conversion, in superconducting-circuit microwave photons, and in interacting-photon fluids — keep extending what "one photon at a time" can mean and measure.
| Light source | g²(0) | Photon statistics | Physical picture |
|---|---|---|---|
| Ideal single emitter | 0 | Antibunched (sub-Poissonian) | One photon at a time; dead time = lifetime |
| Real single-photon source | 0.01–0.2 | Antibunched | Small multiphoton background from imperfections |
| Coherent (laser) | 1 | Poissonian | Independent random emission times |
| Thermal / chaotic light | 2 | Bunched (super-Poissonian) | Photons arrive in clumps (HBT bunching) |
| Two independent emitters | 0.5 | Weakly antibunched | Coincidences from either atom add |
| Classical field (any) | ≥ 1 | Bunched or Poissonian | Cauchy–Schwarz forbids g²(0) < 1 |
Frequently asked questions
Why is g²(0) < 1 impossible for classical light?
For any classical field the intensities are real, nonnegative numbers, and the Cauchy–Schwarz inequality forces ⟨I(t)I(t+τ)⟩ to be maximized at τ = 0, giving g²(0) ≥ g²(τ) and g²(0) ≥ 1. Classical intensity fluctuations can only bunch photons. A measured g²(0) < 1 requires the normal-ordered quantum expectation value to go negative-definite in a way no classical probability distribution allows, so it is a rigorous witness of nonclassical, quantized light.
What is the difference between antibunching and sub-Poissonian statistics?
They are related but not identical. Sub-Poissonian statistics is a statement about the photon-number variance in a fixed time window being below the mean (Mandel Q < 0). Antibunching is specifically about the time correlation g²(τ) rising from a minimum at τ = 0. For stationary light the two usually coincide, but one can construct fields that are antibunched yet not sub-Poissonian (or vice versa), because they probe different aspects of the statistics.
Why does g²(0) = 0.5 matter for counting emitters?
Two independent single emitters each give an antibunching dip, but their coincidences add incoherently: a click can come from atom 1 or atom 2, so the combined g²(0) bottoms out at 0.5, not 0. In general N identical independent emitters give g²(0) = 1 − 1/N. Therefore g²(0) < 0.5 is the standard threshold certifying that a single emitter dominates the emission.
Why do you need two detectors and a beamsplitter (HBT) instead of one?
Real single-photon detectors have a dead time of tens of nanoseconds after firing, during which they are blind. That is longer than the antibunching timescale, so a single detector could never register the suppressed near-simultaneous second photon. Splitting the light onto two detectors lets one detector's click 'start' the clock and the other's 'stop' it, so the τ ≈ 0 region becomes accessible. This is the Hanbury Brown–Twiss configuration.
Who discovered photon antibunching and when?
It was first observed by H. Jeff Kimble, Mario Dagenais and Leonard Mandel at the University of Rochester in 1977, using resonance fluorescence from a dilute atomic beam of sodium excited by a tuned dye laser (Phys. Rev. Lett. 39, 691). Predictions came slightly earlier from Carmichael and Walls, and from Kimble and Mandel, in the mid-1970s. It stands as one of the first unambiguous demonstrations of the quantum nature of the electromagnetic field.
How does antibunching differ from photon blockade?
Both suppress a second photon, but by different mechanisms. Antibunching from a single emitter is due to the emitter's mandatory re-excitation time after it decays to the ground state. Photon blockade is a cavity-QED effect: a strong single-photon nonlinearity (e.g., the anharmonic Jaynes–Cummings ladder) shifts the two-photon resonance out of the drive, so absorbing one photon blocks a second. Blockade produces antibunched output but relies on cavity nonlinearity rather than the bare emitter dead time.