Coherence Functions and the Degree of Coherence

The Hanbury Brown–Twiss layout of that 1956 experiment — a beam splitter, two detectors, a coincidence counter — is now the standard bench for measuring $g^{(2)}(\tau)$, and it certifies every serious single-photon source. Quantum dots, nitrogen-vacancy centres in diamond, and defects in two-dimensional materials are graded by their measured $g^{(2)}(0)$, read out with superconducting-nanowire detectors whose picosecond timing resolves the antibunching dip directly. These sources feed quantum key distribution, where a low $g^{(2)}(0)$ closes the multiphoton loophole an eavesdropper exploits, and photonic quantum computers, where indistinguishable single photons are the raw material. The idea has travelled far beyond photons. Intensity interferometry, invented to size stars without phase stability, has been revived on arrays of Cherenkov telescopes to resolve stars at microarcsecond scales. And ultracold-atom experiments measure the atomic analogue of $g^{(2)}$ and see the contrast quantum statistics demands — bunching for bosons, antibunching for a degenerate Fermi gas, the Pauli principle written into a coincidence histogram. In every case the same correlation function, born to describe fringes and clumped starlight, is doing the work.

Start with the older, first-order question. Split light in a Michelson interferometer and recombine it with a delay $\tau$. If the field keeps a steady phase across that delay, the beams add and subtract in step and bright-dark-bright fringes appear; if the phase wanders faster than $\tau$, the crest of one arm lands on the trough of the other as often as not and the fringes wash out to flat grey. The complex degree of coherence $g^{(1)}(\tau)$ is the bookkeeping of that phase memory, and its magnitude is the fringe visibility. The delay at which the memory fades is the coherence time $\tau_c$, and $c\tau_c$ is the coherence length. Now the newer, second-order question, which no interferometer answers. Put two photon detectors on the outputs of a beam splitter and ask: given a click on A at time $t$, is a click on B at $t+\tau$ more, less, or equally likely than chance? That normalized conditional rate is $g^{(2)}(\tau)$. Thermal light says "more likely" — its photons arrive in clumps, so a first click flags a bright moment and a second follows: bunching, $g^{(2)}(0)=2$. Laser light says "equally likely," $g^{(2)}=1$. A single-photon emitter says "impossible at zero delay" — one photon cannot fire both detectors at once, so $g^{(2)}(0)=0$, antibunching.

The crux is that these two axes do not talk to each other. First-order coherence is fixed by the spectrum — through Wiener–Khinchin, $g^{(1)}(\tau)$ is the Fourier transform of the lineshape — while second-order coherence is fixed by the photon statistics. Filter a thermal lamp until its spectrum is as narrow as a laser's, matching $g^{(1)}$ exactly, and you will not have removed one iota of its bunching. Quantum-mechanically, "perfectly coherent" means every correlation function factorizes, $\langle\hat a^{\dagger}\cdots\hat a\rangle=\prod\langle\hat a^{\dagger}\rangle\langle\hat a\rangle$, which only coherent states achieve; every departure is a failure of that factorization, and below $1$ no classical field can imitate it. The next figure plots the three canonical answers on one axis.

First-order coherence is built from the field autocorrelation. Under stationarity — statistics unchanged by shifting the time origin — every correlation depends only on the delay $\tau$, so we form the mutual coherence function and normalize by the mean intensity, which fixes $g^{(1)}(0)=1$ and makes the modulus a number between $0$ and $1$.

where: $E(t)$ — complex analytic signal of the scalar field at a fixed point; $\langle\cdot\rangle$ — time (ensemble) average, stationarity assumed; $\Gamma^{(1)}$ — mutual coherence function; $g^{(1)}(\tau)$ — complex degree of first-order coherence, $g^{(1)}(0)=1$. Physically it records how well the field's phase at $t+\tau$ is predicted from its phase at $t$; a slow decay means a long coherence time. Second-order coherence compares the intensity now with the intensity a delay $\tau$ later. It is a fourth-order field moment, normalized so an uncorrelated stream gives exactly $1$ — the quantity a two-detector experiment measures.

where: $I(t)\propto|E(t)|^2$ — instantaneous intensity; $g^{(2)}(\tau)$ — normalized intensity correlation, dimensionless. Above $1$, bright moments coincide (bunching); below $1$ they avoid one another (antibunching); at $1$, intensity now says nothing about later. The link between the abstract $g^{(1)}$ and a laboratory reading is fringe visibility: when a field interferes with a delayed copy of itself, the depth of the fringes is governed entirely by the modulus of the degree of coherence.

where: $\mathcal{V}$ — Michelson fringe visibility at path delay $\tau$; $I_{\max},I_{\min}$ — maximum and minimum fringe intensities. Because $0\le\mathcal V\le1$, this is why $|g^{(1)}|$ never exceeds $1$: full fringes ($\mathcal V=1$) mean perfect first-order coherence, a flat field ($\mathcal V=0$) means none. First-order coherence carries the spectrum and nothing else. The Wiener–Khinchin theorem states that $g^{(1)}(\tau)$ and the power spectral density are a Fourier-transform pair, so measuring coherence versus delay reconstructs the lineshape.

where: $S(\omega)$ — power spectral density; $g^{(1)}(\tau)$ — first-order coherence, the Fourier conjugate of the spectrum. A broad spectrum gives a fast-decaying $g^{(1)}$, a narrow one a slow decay: the coherence time is roughly the inverse bandwidth, $\tau_c\sim1/\Delta\nu$. For chaotic (thermal) light alone, the two orders are tied by the Siegert relation. It is not a general law — it holds precisely because thermal light has complex-Gaussian statistics — but where it holds it is exact and it drives intensity interferometry.

where: $g^{(2)}(\tau)$ — intensity correlation of chaotic light. At zero delay $|g^{(1)}(0)|=1$ so $g^{(2)}(0)=2$; at large delay $g^{(1)}\to0$ so $g^{(2)}\to1$. The bunching decays from $2$ to $1$ over exactly the coherence time set by the spectrum. To describe nonclassical light the correlation functions are promoted to quantum expectation values of the mode operators, written in normal order — every creation operator to the left of every annihilation operator. This is not a matter of taste; it is forced by the physics of detection, which absorbs photons.

where: $\hat a,\hat a^{\dagger}$ — annihilation/creation operators of the detected mode; the numerator is normally ordered. Because detection clicks by absorbing a photon, the coincidence rate is $\langle\hat a^{\dagger}\hat a^{\dagger}\hat a\hat a\rangle$; normal ordering discards the vacuum term, so an empty mode gives zero counts. For a single mode at zero delay the numerator collapses to a photon-number expression, so $g^{(2)}(0)$ becomes a direct reporter of the number variance — the equation that connects coherence to photon statistics and exposes the classical boundary.

where: $\hat n=\hat a^{\dagger}\hat a$ — number operator; $\langle(\Delta\hat n)^2\rangle$ — number variance. Poissonian ($\langle(\Delta\hat n)^2\rangle=\langle\hat n\rangle$) gives $g^{(2)}(0)=1$; super-Poissonian gives bunching above $1$; sub-Poissonian ($\langle(\Delta\hat n)^2\rangle<\langle\hat n\rangle$) drives it below $1$ — the exclusively quantum regime. Any classical field obeys two inequalities, and their violation is the operational definition of nonclassical light. Both follow from the fact that a classical intensity is real and non-negative.

where: $g^{(2)}(0)$ — zero-delay intensity correlation. Classical correlations are strongest at zero delay, so a measured $g^{(2)}(0)<1$, or a correlation that grows away from zero delay, cannot come from any classical field.

The value of $g^{(2)}(0)$ classifies the source at a glance; the standard cases are:

1 · Coherence length of a filtered spectral lamp, and its bunching (confidence builder)

A low-pressure mercury lamp is filtered to a Lorentzian lineshape of full width $\Delta\nu=1.0\ \mathrm{GHz}$ around the green $546.1\ \mathrm{nm}$ line. Find its coherence time and length, the fringe visibility at a Michelson delay equal to one coherence time, and the second-order coherence there.

Step 1. For a Lorentzian spectrum the Wiener–Khinchin partner is a two-sided exponential, $|g^{(1)}(\tau)|=e^{-\pi\Delta\nu|\tau|}$, so the $1/e$ coherence time is $\tau_c=1/(\pi\Delta\nu)=1/(\pi\times1.0\times10^{9})=\mathbf{0.318\ ns}$. The narrower the line, the longer the phase memory.

Step 2. The coherence length is the distance light travels in that time, $\ell_c=c\,\tau_c=(2.998\times10^{8})(3.18\times10^{-10})=\mathbf{9.5\ cm}$ — the largest Michelson path imbalance the fringes survive.

Step 3. At delay $\tau=\tau_c$ the modulus has fallen to $|g^{(1)}(\tau_c)|=e^{-1}=0.368$, and by equation (3) that is the fringe visibility, $\mathcal V=\mathbf{0.37}$ — fringes washed to about a third of full contrast.

Step 4. Because the lamp is chaotic, Siegert gives the second-order coherence at that delay: $g^{(2)}(\tau_c)=1+|g^{(1)}(\tau_c)|^{2}=1+e^{-2}=\mathbf{1.135}$. The bunching, which starts at $g^{(2)}(0)=2$, has decayed most of the way to $1$ over one coherence time. τ_c = 0.318 ns, ℓ_c = 9.5 cm, 𝒱 = 0.37, g²(τ_c) = 1.135 — one bandwidth fixes all four.

2 · Certifying a single-photon source

A quantum-dot emitter is characterized in a Hanbury Brown–Twiss setup. Compare the ideal Fock-state coincidence values, explain why simply attenuating a laser can never mimic them, and interpret a measured $g^{(2)}(0)=0.08$.

Step 1. From $g^{(2)}(0)=1-1/n$, the ideal number states give $|1\rangle\to0$, $|2\rangle\to0.5$, $|5\rangle\to0.8$, $|10\rangle\to0.9$. A perfect single-photon source reads exactly $\mathbf{0}$; even a two-photon state already sits at $0.5$, on the classical boundary. Antibunching weakens fast with photon number.

Step 2. Attenuating a bright laser does not help. A coherent state has $g^{(2)}(0)=1$ regardless of $|\alpha|$, so dimming it to one photon per pulse still gives $g^{(2)}(0)=1$ — the weak beam is mostly vacuum with an occasional Poissonian burst, including two-photon events. Weakness is not antibunching; a faint laser is still classical.

Step 3. The certification threshold is $g^{(2)}(0)<0.5$, which rules out any classical mixture faking single photons by emitting two-photon states half the time. A measurement of $g^{(2)}(0)=0.08$ is comfortably below it.

Step 4. As a purity, $g^{(2)}(0)\approx0.08$ means two-photon emission is suppressed to roughly $8\%$ of the Poissonian level — good enough for quantum key distribution, where multiphoton pulses are the security leak eavesdroppers exploit. Ideal |1⟩ gives 0; an attenuated laser stays at 1; the measured 0.08 certifies a near-ideal single-photon source.

Coherence functions are where classical optics, invented to describe fringes, was forced to become quantum — and the person who forced it, giving the degree of coherence its modern operator definition, is this article's natural face.

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 3 and 6 (coherence functions, photon statistics) · Mandel and Wolf, Optical Coherence and Quantum Optics (1995), the comprehensive reference · Foot, Atomic Physics, Ch. 8 (coherence and the HBT effect) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 11–12 · Hecht, Optics, 5th ed., Ch. 12 (classical coherence) · Fox, Quantum Optics, Ch. 5–6. The key results to keep in view are: (i) $\mathcal V=|g^{(1)}|$ from two-beam interference; (ii) the Siegert relation $g^{(2)}(\tau)=1+|g^{(1)}(\tau)|^{2}$ from the Gaussian moment theorem, giving $g^{(2)}(0)=2$; (iii) $g^{(2)}(0)=1-1/n$ for Fock states, yielding $0,\,0.5,\,0.8,\,0.9$ for $n=1,2,5,10$; (iv) thermal $g^{(2)}(0)=2$ from $\langle\hat n(\hat n-1)\rangle=2\bar n^{2}$; (v) coherent-state $g^{(n)}=1$ to all orders; (vi) the filtered-lamp numbers $\tau_c=0.318\ \mathrm{ns}$, $\ell_c=9.5\ \mathrm{cm}$, $\mathcal V=0.37$, $g^{(2)}(\tau_c)=1.135$. The standard values $g^{(2)}(0)=1,2,0$ for laser, thermal, and single-photon light are consistent across the treatments listed above.

Prerequisites. Quantization of the Electromagnetic Field · Photon Statistics and the Mandel Q Parameter · Coherent States. Leads to. Hanbury Brown–Twiss Experiment · Photon Antibunching · Single-Photon Sources · Phase-Space Quasiprobability Distributions.

Ask the tutor: Ask me to walk through why a filtered thermal lamp and a laser can share the same fringes yet split apart the instant you count coincidences.