Optical Correlation Functions

Optical coherence is not one number and not a synonym for narrow linewidth. It is a hierarchy of correlation functions that predicts how field samples at different positions, times, polarizations, and detector channels interfere or produce joint photoevents. First-order coherence controls amplitude interference. Second-order coherence controls ideal coincidence statistics. Higher orders control increasingly detailed multiphoton records.

The hierarchy is operational. In Glauber photodetection theory, an absorbing detector selects a positive-frequency field operator at each event, and an $n$-fold coincidence samples a normally ordered $2n$-operator expectation value. The ordering is part of the physics, not decorative notation. Symmetrized noise, time-ordered Green functions, and retarded response functions answer different questions.

The selected modes, ordering, time arguments, and detector gates must be specified together. One value such as $g^{(2)}(0)=1$ does not identify a state.

Required background. Quantized Electromagnetic Modes defines positive-frequency fields, Photon Counting connects them to detection probabilities, and Correlations and Covariance introduces statistical moments. Hanbury Brown–Twiss Interferometry develops coincidence measurements; Bath Correlations and Many-Body Correlations explain other operator orderings and response functions.

Field and Detector Notation

Detection coordinate

Write

$$x=(\mathbf r,t,\lambda),$$

where $\lambda$ abbreviates polarization, detector channel, spectral filter, or other resolved labels. Let

$$\hat E^{(+)}(x)$$

be the scalar positive-frequency field after projection onto the detector response. Its adjoint is

$$\hat E^{(-)}(x) = \left[ \hat E^{(+)}(x) \right]^\dagger.$$

Vector fields require polarization contractions or a coherence matrix. The scalar notation below assumes that those choices have already been made.

Constants containing detector responsivity can be restored when absolute rates are needed. Correlation ratios often cancel a common constant, but they do not cancel mode mismatch or channel-dependent response automatically.

Intensity notation

The normally ordered optical intensity operator at $x$ is proportional to

$$\hat I(x) = \hat E^{(-)}(x)\hat E^{(+)}(x).$$

Its expectation,

$$I(x) = \left\langle\hat I(x)\right\rangle,$$

sets the ideal first-order absorption rate. The symbol $I$ below denotes this normally ordered mean unless stated otherwise.

Vacuum has a nonzero symmetrized field variance but

$$\langle0| \hat E^{(-)}(x)\hat E^{(+)}(x) |0\rangle = 0$$

in an ideal empty mode. This is one reason ordering conventions cannot be interchanged casually.

First-Order Coherence

Mutual coherence function

The first-order optical correlation is

$$G^{(1)}(x_1,x_2) = \left\langle \hat E^{(-)}(x_1) \hat E^{(+)}(x_2) \right\rangle.$$

Its diagonal is the mean intensity:

$$G^{(1)}(x,x)=I(x).$$

The kernel is Hermitian,

$$G^{(1)}(x_2,x_1) = \left[ G^{(1)}(x_1,x_2) \right]^*,$$

and positive semidefinite. For arbitrary complex coefficients $c_j$,

$$\sum_{j,k} c_j^*c_k G^{(1)}(x_j,x_k) \ge0.$$

This follows because the left side is

$$\left\langle \hat A^\dagger\hat A \right\rangle, \qquad \hat A = \sum_k c_k\hat E^{(+)}(x_k).$$

Normalized degree of coherence

When both local intensities are nonzero, define

$$g^{(1)}(x_1,x_2) = \frac{ G^{(1)}(x_1,x_2) }{ \sqrt{I(x_1)I(x_2)} }.$$

The Cauchy–Schwarz inequality gives

$$\left| g^{(1)}(x_1,x_2) \right| \le1.$$

The magnitude measures normalized amplitude coherence between the two samples; the argument gives their correlation phase. If either intensity vanishes, the normalized ratio is undefined even though the unnormalized kernel is well defined.

Interference visibility

Suppose two sampled fields are recombined with a controllable phase $\phi$. After absorbing fixed transmission factors into $I_1$ and $I_2$, the mean output intensity is

$$I(\phi) ={} I_1+I_2 + 2\sqrt{I_1I_2}\, \operatorname{Re} \left[ e^{i\phi}g_{12}^{(1)} \right].$$

The fringe visibility

$$\mathcal V \equiv \frac{I_{\max}-I_{\min}} {I_{\max}+I_{\min}}$$

is therefore

$$\mathcal V = \frac{ 2\sqrt{I_1I_2} }{ I_1+I_2 } \left|g_{12}^{(1)}\right|.$$

For balanced intensities,

$$\mathcal V = \left|g_{12}^{(1)}\right|.$$

Unequal intensity lowers visibility even for perfect mutual coherence. Conversely, high visibility establishes first-order coherence for the selected samples but does not establish Poisson statistics or a coherent quantum state.

Coherence matrix

For several polarization or spatial components, define

$$\Gamma_{\mu\nu}^{(1)}(x_1,x_2) = \left\langle \hat E_\mu^{(-)}(x_1) \hat E_\nu^{(+)}(x_2) \right\rangle.$$

This matrix carries intensity, polarization, and mutual coherence together. Tracing over unresolved components can reduce measured visibility. A scalar $g^{(1)}$ is meaningful only after specifying the projections or contractions used by the optical system and detector.

Stationarity and Spectrum

Time-translation invariance

For a stationary field,

$$G^{(1)}(t_1,t_2) = G^{(1)}(t_2-t_1).$$

With the convention

$$G^{(1)}(\tau) = \left\langle \hat E^{(-)}(t) \hat E^{(+)}(t+\tau) \right\rangle,$$

stationarity implies

$$G^{(1)}(-\tau) = \left[ G^{(1)}(\tau) \right]^*.$$

Stationarity is an ensemble property. A finite record can drift, blink, age, or mix several operating points even when a time-averaged plot appears smooth.

Wiener–Khinchin relation

Choose the spectral convention

$$S(\omega) = \frac1{2\pi} \int_{-\infty}^{\infty} d\tau\, e^{i\omega\tau} G^{(1)}(\tau).$$

Then

$$G^{(1)}(\tau) = \int_{-\infty}^{\infty} d\omega\, e^{-i\omega\tau}S(\omega).$$

For a stationary positive-frequency field, the spectral measure is nonnegative and supported at $\omega\ge0$. It may include delta peaks; an integrable density and a finite decay time are additional assumptions. Carrier-envelope approximations extend detunings to the whole real line. The spectral width and temporal decay of $g^{(1)}$ are Fourier related, but “coherence time” has several definitions.

A common integral convention is

$$\tau_c = \int_{-\infty}^{\infty} d\tau\, \left|g^{(1)}(\tau)\right|^2.$$

Other conventions use a $1/e$ width, half width at half maximum, or integral of $|g^{(1)}|$. Numerical coherence times should therefore be accompanied by their definition. Linewidth and Coherence applies these alternatives to laser phase diffusion, frequency-noise PSDs, and measured cw line shapes.

Two line-shape examples

For a Lorentzian spectrum centered at $\omega_0$ with field-correlation decay rate $\gamma$,

$$g^{(1)}(\tau) = e^{-i\omega_0\tau} e^{-\gamma|\tau|}.$$

For a Gaussian spectrum with rms width $\sigma_\omega$,

$$g^{(1)}(\tau) = e^{-i\omega_0\tau} e^{-\sigma_\omega^2\tau^2/2}.$$

These line-shape formulas extend detunings to the real line and approximate an optical positive-frequency spectrum when $\gamma,\sigma_\omega\ll\omega_0$. The Lorentzian angular-frequency HWHM is $\gamma$. The oscillatory carrier and decaying envelope contain different information. Removing a carrier in a rotating frame does not lengthen the physical coherence envelope.

Second-Order Coherence

Joint photodetection function

The normally ordered second-order function is

$$G^{(2)}(x_1,x_2) ={} \bigl\langle \hat E^{(-)}(x_1) \hat E^{(-)}(x_2) \times \hat E^{(+)}(x_2) \hat E^{(+)}(x_1) \bigr\rangle.$$

It is nonnegative because it has the form

$$G^{(2)}(x_1,x_2) = \left\langle \hat A^\dagger\hat A \right\rangle,$$

with

$$\hat A = \hat E^{(+)}(x_2) \hat E^{(+)}(x_1).$$

For ideal weak absorbers, $G^{(2)}$ is proportional to the joint density for two photoevents at the declared coordinates. Detector response and finite gates turn this density into integrated probabilities.

Normalized second-order coherence

When $I(x_1)I(x_2)\ne0$, define

$$g^{(2)}(x_1,x_2) = \frac{ G^{(2)}(x_1,x_2) }{ I(x_1)I(x_2) }.$$

For one stationary channel,

$$g^{(2)}(\tau) = \frac{G^{(2)}(\tau)}{I^2},$$

where

$$G^{(2)}(\tau) ={} \bigl\langle \hat E^{(-)}(t) \hat E^{(-)}(t+\tau) \times \hat E^{(+)}(t+\tau) \hat E^{(+)}(t) \bigr\rangle.$$

At large delays, independent stationary samples often give

$$g^{(2)}(\tau)\longrightarrow1,$$

but this limit can fail for nonergodic mixtures, blinking, periodic driving, long memory, or conditioning.

Conditional-rate interpretation

For a stationary ideal point process with mean detection rate $\bar R$,

$$\bar R\,g^{(2)}(\tau)$$

is the conditional event rate at delay $\tau$ given an event at the origin, for distinct events under ideal detection assumptions. This conditional rate includes histories with intervening events; it is not the waiting-time density for the next event and need not integrate to one. Thus:

• $g^{(2)}(\tau)>1$ means enhanced conditional detection;
• $g^{(2)}(\tau)<1$ means suppressed conditional detection;
• $g^{(2)}(\tau)=1$ means no second-order excess at that delay.

This interpretation concerns events in the selected channel. It does not imply a force between photons, a literal packet size, or a complete description of the field state.

Normal ordering and shot noise

At one point,

$$G^{(2)}(x,x) = \left\langle : \hat I(x)^2 : \right\rangle,$$

not the unordered $\langle\hat I^2\rangle$. For one mode,

$$\hat a^\dagger\hat a^\dagger\hat a\hat a = \hat N(\hat N-1).$$

The falling factorial excludes pairing one registered event with itself. When ordinary count variance is reconstructed, the missing diagonal returns as the Poisson shot-noise term.

Pulsed and Gated Correlations

One selected pulse mode

For a pulse described by one selected mode with number operator $\hat N$, the zero-delay pulsed quantity is

$$g_{\mathrm p}^{(2)}[0] = \frac{ \langle\hat N(\hat N-1)\rangle }{ \langle\hat N\rangle^2 }.$$

Square brackets emphasize a discrete pulse lag. This quantity is not automatically the same as an infinitesimal continuous-time $g^{(2)}(0)$. It integrates all temporal structure accepted within the pulse and detection gate.

Factorial moments in a gate

Let $N_G$ be the ideal count in a gate $G$. Under linear photodetection,

$$\left\langle (N_G)_m \right\rangle ={} \int_G dx_1\cdots \int_G dx_m \times G^{(m)}(x_1,\ldots,x_m),$$

up to the declared responsivity factors. Here

$$(N_G)_m = N_G(N_G-1)\cdots(N_G-m+1).$$

Changing the gate changes the measured mode mixture and therefore can change the normalized correlation even when the source is unchanged.

Count variance from temporal coherence

For one stationary stream with mean ideal rate $\bar R$ and a gate of duration $T$,

$$\langle N_T\rangle = \bar R T.$$

The general factorial second moment is

$$\langle N_T(N_T-1)\rangle ={} \bar R^2 \int_0^Tdt \int_0^Tdt' \times g^{(2)}(t'-t).$$

When the same-channel stationary correlation is even in delay,

$$\mathcal C_T := \int_0^T d\tau\, (T-\tau) \left[ g^{(2)}(\tau)-1 \right].$$

The count variance is then

$$\operatorname{Var}(N_T) = \bar R T + 2\bar R^2\mathcal C_T.$$

The first term is shot noise. Positive integrated excess produces super-Poissonian count variance; negative integrated excess can produce sub-Poissonian variance. A local dip at zero delay does not by itself fix the long-gate Fano factor because the complete correlation area matters.

Higher-Order Glauber Correlations

Detection-diagonal hierarchy

For $m$ detection coordinates, define the ordered absorption product

$$\hat A_m(\mathbf x) := \hat E^{(+)}(x_m)\cdots \hat E^{(+)}(x_1),$$

where $\mathbf x=(x_1,\ldots,x_m)$. The detection-diagonal correlation is

$$G^{(m)}(\mathbf x) := \left\langle \hat A_m^\dagger(\mathbf x) \hat A_m(\mathbf x) \right\rangle \geq 0.$$

Expanding $\hat A_m^\dagger\hat A_m$ reverses the positive-frequency factors relative to the negative-frequency factors. The positive-operator form also makes the nonnegativity of an ideal joint photodetection rate explicit.

The normalized detection-diagonal function is

$$g^{(m)}(x_1,\ldots,x_m) = \frac{ G^{(m)}(x_1,\ldots,x_m) }{ \prod_{j=1}^{m}I(x_j) },$$

when every denominator is nonzero.

Full two-set kernel

The more general coherence kernel keeps separate negative- and positive-frequency coordinates:

$$G^{(m)}(\mathbf x;\mathbf y) := \left\langle \hat A_m^\dagger(\mathbf x) \hat A_m(\mathbf y) \right\rangle.$$

Here $\mathbf y=(y_1,\ldots,y_m)$. Expanding the compact notation gives negative-frequency factors at $\mathbf x$ followed by the reversed positive-frequency factors at $\mathbf y$. The detection-diagonal function sets $y_j=x_j$. Keeping the full kernel is important in coherence theory, propagation, and interferometric transformations.

Coherence order

A field is first-order coherent over a domain when its first-order kernel factorizes there:

$$G^{(1)}(x,y) = \mathcal E^*(x)\mathcal E(y).$$

Full Glauber coherence requires factorization of the entire normally ordered hierarchy:

$$G^{(m)}(\mathbf x;\mathbf y) ={} \prod_{j=1}^{m} \mathcal E^*(x_j) \times \prod_{j=1}^{m} \mathcal E(y_j)$$

for all relevant orders and coordinates. An ideal multimode coherent state satisfies this condition. Demonstrating one value such as $g^{(2)}(0)=1$ does not demonstrate full coherence.

Correlation hierarchy is not always state tomography

Photodetection correlations are invariant under a global optical phase. A coherent state with unknown uniformly distributed global phase can reproduce the same phase-insensitive normally ordered number correlations as a fixed coherent amplitude. Low-order correlation data also leave many distinct states compatible with the observations.

The complete state requires an informationally complete measurement, often using a phase reference. Phase-Space Distributions and homodyne tomography address that larger reconstruction problem.

Standard State Examples

Coherent light

For a coherent state, each positive-frequency field annihilation operator acts by its classical complex amplitude. Therefore all normally ordered correlations factorize. Wherever the intensity is nonzero,

$$g_{\mathrm{coh}}^{(m)} = 1.$$

The result holds across coordinates for an ideal coherent field after deterministic linear propagation. Technical amplitude or phase fluctuations can produce a mixture whose measured coherence is lower or whose intensity correlations differ from one.

Thermal light in a selected channel

For one scalar detected field with zero-mean circular Gaussian statistics (no anomalous pair correlator), Gaussian moment factorization gives the Siegert relation

$$g_{\mathrm{th}}^{(2)}(x_1,x_2) = 1+ \left| g_{\mathrm{th}}^{(1)}(x_1,x_2) \right|^2.$$

At one coordinate,

$$g_{\mathrm{th}}^{(2)}(0)=2.$$

A freely evolving single-frequency thermal oscillator has $|g^{(1)}(\tau)|=1$ and hence $g^{(2)}(\tau)=2$ at every delay. A finite-bandwidth traveling field can instead have a decaying correlation. The scalar detection channel and the single-frequency oscillator are different uses of the word mode. More generally, for one thermal mode with nonzero occupation,

$$g_{\mathrm{th}}^{(m)}(0)=m!.$$

The Siegert relation requires Gaussian chaotic statistics. It is not a definition of thermal light and need not hold for arbitrary bunched, non-Gaussian, displaced, or technically noisy fields.

One-mode number state

For $|n\rangle$ with $n>0$ and $m\le n$,

$$\left\langle (\hat a^\dagger)^m\hat a^m \right\rangle = \frac{n!}{(n-m)!}.$$

Thus

$$g_{|n\rangle}^{(m)}(0) = \frac{n!} {n^m(n-m)!},$$

and the correlation vanishes for $m>n$. In particular,

$$g_{|n\rangle}^{(2)}(0) = 1-\frac1n.$$

A freely evolving one-mode number state has $|g^{(1)}(\tau)|=1$ and constant $g^{(2)}(\tau)=1-1/n$. It has sub-Poissonian occupation without a temporal rise away from zero delay. Perfect first-order coherence therefore does not imply coherent-state number statistics.

Equal multimode thermal mixture

For $M$ independent thermal modes with equal mean occupation and a detector that sums their intensities,

$$g^{(2)}(0) = 1+\frac1M.$$

As more unresolved modes are collected, the bunching excess decreases. This approach toward one is mode averaging, not proof that the light became a coherent state.

For unequal modal means $\mu_j$, define

$$M_{\mathrm{eff}} = \frac{ \left(\sum_j\mu_j\right)^2 }{ \sum_j\mu_j^2 }.$$

Then

$$g^{(2)}(0) = 1+\frac1{M_{\mathrm{eff}}}$$

for independent thermal modes whose occupations are summed. A coherent projection onto one superposed Gaussian mode is a different measurement and still gives $g^{(2)}=2$; see Thermal Light.

Number Statistics and the Mandel Parameter

For one selected mode with finite nonzero mean,

$$g^{(2)}(0) = \frac{ \langle N(N-1)\rangle }{ \langle N\rangle^2 }.$$

Using

$$N^2=N(N-1)+N,$$

one obtains

$$g^{(2)}(0) = 1+ \frac{ \operatorname{Var}(N)-\langle N\rangle }{ \langle N\rangle^2 }.$$

The Mandel parameter

$$Q = \frac{ \operatorname{Var}(N)-\langle N\rangle }{ \langle N\rangle }$$

therefore satisfies

$$Q = \langle N\rangle \left[ g^{(2)}(0)-1 \right].$$

For this one-mode, one-gate setting:

• $g^{(2)}(0)<1$ is equivalent to sub-Poissonian number variance;
• $g^{(2)}(0)=1$ gives Poisson variance, but not necessarily a Poisson distribution;
• $g^{(2)}(0)>1$ gives super-Poissonian variance.

For continuous-time light, a point value $g^{(2)}(0)$ and a finite-window Fano factor are not equivalent unless the gate and temporal correlation structure are included.

Classical and Quantum Bounds

Positive Glauber–Sudarshan representation

A field is classical in the standard optical coherence sense if its density operator is a mixture of coherent states with a nonnegative Glauber–Sudarshan probability measure:

$$\rho = \int d^{2M}\boldsymbol{\alpha}\, P(\boldsymbol{\alpha}) |\boldsymbol{\alpha}\rangle \langle\boldsymbol{\alpha}|,$$

with

$$P(\boldsymbol{\alpha})\ge0.$$

Delta measures are allowed; a coherent state itself has one. Singularity alone is not a nonclassicality test. With finite moments, normally ordered expectations become ordinary classical averages over complex amplitudes. This gives useful inequalities. Violation proves that no such positive classical mixture explains the selected-mode correlations.

Equal-time lower bound

For classical random intensity $I\ge0$,

$$g_{\mathrm{cl}}^{(2)}(0) = \frac{\mathbb E[I^2]}{\mathbb E[I]^2} = 1+ \frac{\operatorname{Var}(I)} {\mathbb E[I]^2} \ge1.$$

Therefore

$$g^{(2)}(0)<1$$

is a sufficient nonclassicality witness under the standard calibrated photodetection assumptions.

The converse is false. Squeezed vacuum, entangled light, and other nonclassical states can have $g^{(2)}(0)>1$.

Stationary classical delay bound

For a stationary classical intensity process, Cauchy–Schwarz gives

$$\left| \mathbb E[I(t)I(t+\tau)] \right| \le \mathbb E[I(t)^2].$$

After normalization,

$$g_{\mathrm{cl}}^{(2)}(\tau) \le g_{\mathrm{cl}}^{(2)}(0).$$

An observed rise away from zero delay,

$$g^{(2)}(\tau)>g^{(2)}(0)$$

for some $\tau$, is the operational antibunching pattern when stationarity and detector artifacts have been controlled. It is a distinct test from $g^{(2)}(0)<1$; neither universally implies the other. If the additional condition $g^{(2)}(\tau)\to1$ holds at long delay, sub-unity zero-delay correlation does imply a rise somewhere. It still does not determine the finite-gate Fano factor, which depends on the whole correlation area. Zou and Mandel discuss the distinction between antibunching and sub-Poissonian counting statistics.

Two different nonclassicality tests. The free $|2\rangle$ mode has sub-Poissonian occupation but no temporal rise. The solid curve is the independent sum of an antibunched emitter stream and a slightly brighter chaotic stream, calculated in Exercise 9. It rises away from zero despite $g^{(2)}(0)>1$, and its long-gate counts are super-Poissonian. The dotted line marks unity, not a universal long-delay limit for every field.

Two-channel Cauchy–Schwarz bound

For classical intensities $I_a$ and $I_b$ at equal time,

$$\left[ g_{ab}^{(2)} \right]^2 \le g_{aa}^{(2)} g_{bb}^{(2)}.$$

Violation is a nonclassical cross-correlation witness. The auto- and cross-correlations must use compatible gates, modes, backgrounds, and normalizations. The channels may have different filters, but each autocorrelation must use the same filtered field that appears in the cross-correlation. Substituting an autocorrelation measured with a different filter compares different random variables and invalidates the bound.

Bounds that are not nonclassicality tests

The universal first-order inequality

$$|g^{(1)}|\le1$$

follows from quantum positivity as well as classical Cauchy–Schwarz. It is not a classicality criterion.

For any state with nonzero intensity,

$$g^{(2)}(0)\ge0,$$

because the numerator is an expectation of $\hat A^\dagger\hat A$. There is no state-independent finite upper bound: rare bright events can make the normalized correlation arbitrarily large.

Ordering Is Part of the Observable

Normal ordering

Absorptive direct photodetection produces products with all negative-frequency operators to the left of all positive-frequency operators. For one mode,

$$\left\langle (\hat a^\dagger)^m\hat a^m \right\rangle = \left\langle : \hat N^m : \right\rangle.$$

This is a factorial moment. It differs from the ordinary power $\langle\hat N^m\rangle$ by commutator terms.

Symmetric ordering

Quadrature detectors and phase-space representations often involve symmetrically ordered moments. For example,

$$\frac12 \left\langle \hat a\hat a^\dagger + \hat a^\dagger\hat a \right\rangle = \langle\hat N\rangle+\frac12.$$

The extra $1/2$ is the vacuum contribution in this quadrature convention. It does not imply that an ideal absorption counter clicks in vacuum.

Time ordering and retarded response

A time-ordered propagator has the structure

$$\left\langle \mathcal T \hat E(x_1)\hat E(x_2) \right\rangle,$$

while a retarded response involves a causal commutator. Neither is generally equal to $G^{(1)}$ or $G^{(2)}$. Green Functions in Many-Body QM and Retarded and Advanced Response develop those orderings.

An experiment fixes an ordering through its coupling, detection, and signal processing. Reordering operators after the fact changes the predicted observable.

Propagation and Mode Selection

Linear optical propagation

A deterministic passive linear optical system, with all additional input ports in independent vacuum, maps the selected field to

$$\hat E_{\mathrm{out}}^{(+)}(x) = \int dy\, h(x,y)\hat E_{\mathrm{in}}^{(+)}(y)+\hat E_{\mathrm{vac}}^{(+)}(x).$$

The auxiliary vacuum operators preserve commutators when the selected transfer kernel contains loss. Their normally ordered contributions vanish, so the first-order kernel propagates bilinearly:

$$G_{\mathrm{out}}^{(1)}(x,x') ={} \int dy \int dy'\, h^*(x,y)h(x',y') \times G_{\mathrm{in}}^{(1)}(y,y').$$

Thermal, squeezed, or coherently driven auxiliary ports add their own correlations; an active amplifier also mixes creation operators. Under the stated passive vacuum-input assumptions, the second-order function carries four copies of the transfer kernel, and the $m$th-order function carries $2m$. Apertures, dispersion, polarization elements, spectral filters, fibers, and interferometers therefore reshape the arguments and mode weights of measured correlations.

Beam Splitters and Interferometers derive the corresponding two-mode transformations and phase conventions.

Uniform independent loss

For one selected channel with uniform efficiency $0<\eta\le1$ and a vacuum loss environment,

$$G_{\mathrm{det}}^{(m)} = \eta^mG_{\mathrm{in}}^{(m)},$$

while each intensity in the denominator contributes one factor of $\eta$. Therefore

$$g_{\mathrm{det}}^{(m)} = g_{\mathrm{in}}^{(m)}.$$

This useful invariance assumes linear independent loss, no additive background, no saturation, and the same efficiency weighting in numerator and denominator. Loss still reduces sample size and increases uncertainty.

Mode-dependent loss can change a correlation by changing the mixture of accepted modes. A spectral filter that selects one thermal mode from many can increase $g^{(2)}(0)$ toward $2$ even while reducing the total count rate.

Independent background

Let a signal of mean rate $S$ have second-order function $g_{\mathrm s}^{(2)}(\tau)$, and let independent Poisson background of rate $B$ have $g_{\mathrm b}^{(2)}=1$. Assume the detector sums independent counts without interference between signal and background fields. The observed correlation is

$$\begin{aligned} g_{\mathrm{obs}}^{(2)}(\tau)-1 ={}& \rho^2 \left[ g_{\mathrm s}^{(2)}(\tau)-1 \right], \\ \rho ={}& \frac{S}{S+B}. \end{aligned}$$

Background pulls both bunching peaks and antibunching dips toward one. A background-corrected value inherits uncertainty from $S$, $B$, and any assumption that the background is Poisson and independent. A coherent displacement in the same optical mode instead produces field interference and generally follows a different formula.

Timing response

Let $h_{\mathrm{rel}}(\tau)$ be the normalized relative-time response of two detector channels. In a stationary approximation,

$$g_{\mathrm{obs}}^{(2)}(\tau)-1 ={} \int d\tau'\, h_{\mathrm{rel}}(\tau-\tau') \times \left[ g_{\mathrm{true}}^{(2)}(\tau')-1 \right].$$

A nonnegative normalized jitter kernel averages neighboring delays: it cannot exceed the original maximum or fall below its minimum. It broadens an isolated narrow peak or dip while preserving the signed excess area over the full delay line. It need not simply lower every extremum of an oscillatory or overlapping-feature curve. Finite histogram bins apply an additional average. A displayed bin width is not the detector timing resolution.

Detector memory

Dead time suppresses nearby same-channel events and can create a false dip. Afterpulsing creates excess events following a click. Cross-talk creates short-delay or neighboring-channel excess. These are correlations of the detector record, not necessarily of the incident field. The response models are developed in Photon Counting.

Estimating Correlations from Data

Preserve the raw record

Time tags, pulse indices, detector labels, and live-time intervals should be retained whenever possible. A histogram is a derived statistic. Once events are irreversibly binned or detector labels are discarded, later analyses cannot recover the lost timing or channel information.

For a stationary two-detector record, a coincidence histogram estimates the pair density as a function of delay. Normalization divides by the pair exposure expected from independent streams, including acquisition duration, bin width, and detector live time. The exact estimator depends on whether the source is continuous, pulsed, triggered, periodic, or nonstationary.

Continuous and pulsed baselines differ

In a continuous stationary measurement, large delays can provide an independent-event baseline if the record is long compared with every source correlation time. In a pulsed experiment, side peaks correspond to different pulse pairs and the zero-pulse peak to the same pulse. Pulse-to-pulse drift, missing triggers, and unequal gate acceptance can bias a simple peak-area ratio.

Uncertainty is not independent-bin Poisson by default

One event contributes to several delay pairs, normalization is estimated from the same record, and blinking creates long-range dependence. Histogram bins can therefore be correlated. Suitable uncertainty methods include:

• likelihoods for the underlying time-tag or count process;
• block bootstrap with blocks longer than relevant memory;
• repeated independent acquisitions;
• propagation of background and timing calibrations;
• model comparison using held-out time intervals.

The measurement should be repeated at several bin widths and normalization windows. A physical feature should transform according to the convolved model, not disappear unpredictably under modest analysis choices.

Computing Output Correlations

Input–output field

For a Markovian system coupled to one traveling output channel,

$$\hat b_{\mathrm{out}}(t) = \hat b_{\mathrm{in}}(t)+\hat L(t).$$

Input–Output Theory derives the boundary relation, normalization, coherent drives, and multiport extensions. With vacuum input and normally ordered output detection, the source contribution often reduces to correlations of the system coupling operator $\hat L$.

Quantum regression bridge

Let the system obey a time-independent Markov master equation

$$\dot\rho = \mathcal L\rho$$

with steady state $\rho_{\mathrm{ss}}$. For $\tau\ge0$, the quantum regression construction gives

$$\begin{aligned} G^{(1)}(\tau) ={}& \operatorname{Tr} \left[ \hat L e^{\mathcal L\tau} \left( \rho_{\mathrm{ss}}\hat L^\dagger \right) \right], \end{aligned}$$

and

$$\begin{aligned} G^{(2)}(\tau) ={}& \operatorname{Tr} \left[ \hat L^\dagger\hat L\, e^{\mathcal L\tau} \left( \hat L\rho_{\mathrm{ss}}\hat L^\dagger \right) \right]. \end{aligned}$$

The second formula has a transparent conditional structure:

1. a detection applies $\hat L$ at the origin;
2. the unnormalized conditional state evolves for time $\tau$;
3. $\hat L^\dagger\hat L$ evaluates the later emission rate.

Normalize by

$$\bar R = \operatorname{Tr} \left( \hat L^\dagger\hat L\rho_{\mathrm{ss}} \right)$$

to obtain

$$g^{(2)}(\tau) = \frac{G^{(2)}(\tau)}{\bar R^2}.$$

These formulas assume Markovian reduced dynamics, the relevant regression conditions, a stationary state, vacuum input after coherent amplitudes are handled, and ideal output detection. Structured reservoirs, feedback, propagation delay, or coherent input interference require the appropriate extra degrees of freedom or output-field terms. For a time-dependent Markov generator satisfying regression assumptions, replace the stationary state by $\rho(t)$ and $e^{\mathcal L\tau}$ by its two-time propagator; nonstationary driving alone does not invalidate that construction.

Single-emitter preview

For a two-level emitter with

$$\hat L=\sqrt{\Gamma}\,\hat\sigma_-,$$

a detected emission prepares the emitter in its ground state in the ideal model. Because

$$\hat\sigma_-^2=0,$$

the immediate second emission is forbidden and the ideal source has

$$g^{(2)}(0)=0.$$

Driving must repopulate the excited state before another photon can be emitted. The complete resonance-fluorescence shape and experimental qualification are developed in Photon Antibunching.

Exercises

1. First-order Cauchy–Schwarz bound

Prove

$$\left| G^{(1)}(x_1,x_2) \right|^2 \le I(x_1)I(x_2)$$

and hence $|g^{(1)}|\le1$.

Solution

Define vectors in the Hilbert–Schmidt inner-product space by

$$|v_j) = \hat E^{(+)}(x_j)\sqrt{\rho}.$$

Their inner product is

$$\begin{aligned} (v_1|v_2) &= \operatorname{Tr} \left[ \sqrt{\rho}\, \hat E^{(-)}(x_1) \hat E^{(+)}(x_2) \sqrt{\rho} \right] \\ &= G^{(1)}(x_1,x_2). \end{aligned}$$

Their squared norms are

$$(v_j|v_j) = I(x_j).$$

Hilbert-space Cauchy–Schwarz therefore gives

$$|G^{(1)}(x_1,x_2)|^2 \le I(x_1)I(x_2).$$

Dividing by the positive denominator proves $|g^{(1)}|\le1$. Equality means the two detector-projected field vectors are linearly dependent in the state, which is the rank-one condition behind perfect normalized fringe contrast.

2. Visibility with unequal intensities

Two interferometer arms have

$$I_2=4I_1$$

and

$$|g_{12}^{(1)}|=0.90.$$

Find the fringe visibility. Would balancing the intensities change the intrinsic $|g^{(1)}|$?

Solution

Use

$$\mathcal V = \frac{2\sqrt{I_1I_2}}{I_1+I_2} |g_{12}^{(1)}|.$$

Since $\sqrt{I_1I_2}=2I_1$,

$$\begin{aligned} \mathcal V &= \frac{4I_1}{5I_1}(0.90) \\ &= 0.72. \end{aligned}$$

Balancing the detected intensities would raise the measured visibility to $0.90$, but it would not change the intrinsic normalized mutual coherence if the balancing attenuation were uniform over the accepted modes. The original visibility was limited by both imbalance and imperfect coherence.

3. Higher-order correlations of a three-photon state

For the one-mode state $|3\rangle$, calculate

$$g^{(2)}(0), \qquad g^{(3)}(0), \qquad g^{(4)}(0).$$

Solution

For $m\le n$,

$$g_{|n\rangle}^{(m)}(0) = \frac{n!}{n^m(n-m)!}.$$

Thus

$$g^{(2)}(0) = \frac{3\cdot2}{3^2} = \frac23,$$

and

$$g^{(3)}(0) = \frac{3!}{3^3} = \frac29.$$

Four annihilations acting on $|3\rangle$ give zero, so

$$g^{(4)}(0)=0.$$

The hierarchy records the finite occupation cutoff in a way that $g^{(2)}$ alone cannot.

4. Thermal correlations and long-gate count noise

A stationary thermal field has

$$g^{(2)}(\tau) = 1+e^{-2\gamma|\tau|}$$

and mean ideal rate $\bar R$. Derive the count Fano factor for a gate $T$ and its limit for $\gamma T\gg1$.

Solution

Insert the excess correlation into

$$\mathcal C_T := \int_0^T d\tau\, (T-\tau)e^{-2\gamma\tau}.$$

Then

$$\operatorname{Var}(N_T) = \bar RT + 2\bar R^2\mathcal C_T.$$

For $a=2\gamma$,

$$\int_0^T d\tau\, (T-\tau)e^{-a\tau} = \frac{T}{a} - \frac{1-e^{-aT}}{a^2}.$$

Because $\langle N_T\rangle=\bar RT$, the Fano factor is

$$F_T ={} 1+ \frac{\bar R}{\gamma} - \frac{\bar R} {2\gamma^2T} \left( 1-e^{-2\gamma T} \right).$$

For $\gamma T\gg1$,

$$F_T \longrightarrow 1+\frac{\bar R}{\gamma}.$$

The long gate contains many coherence intervals, but each interval contributes thermal excess noise. The normalized variance depends on the count rate per coherence time.

5. Effective number of thermal modes

Three independent thermal modes have mean occupations

$$\mu_1=2, \qquad \mu_2=1, \qquad \mu_3=1.$$

Find $M_{\mathrm{eff}}$ and the summed zero-delay $g^{(2)}(0)$.

Solution

The total mean is $4$, and the sum of squared modal means is

$$\sum_j\mu_j^2 = 4+1+1 = 6.$$

Therefore

$$M_{\mathrm{eff}} = \frac{4^2}{6} = \frac83.$$

The correlation is

$$\begin{aligned} g^{(2)}(0) &= 1+\frac1{M_{\mathrm{eff}}} \\ &= 1+\frac38 = \frac{11}{8}. \end{aligned}$$

The effective mode number need not be an integer because unequal modal weights interpolate continuously between equal-mode cases.

6. Classical bounds from intensity fluctuations

Let $I(t)\ge0$ be a stationary classical random intensity. Prove both

$$g^{(2)}(0)\ge1$$

and

$$g^{(2)}(\tau)\le g^{(2)}(0).$$

Solution

At equal time,

$$\begin{aligned} g^{(2)}(0)-1 &= \frac{ \mathbb E[I^2]-\mathbb E[I]^2 }{ \mathbb E[I]^2 } \\ &= \frac{\operatorname{Var}(I)} {\mathbb E[I]^2} \ge0. \end{aligned}$$

For separated times, Cauchy–Schwarz gives

$$\left| \mathbb E[I(t)I(t+\tau)] \right|^2 \le{} \mathbb E[I(t)^2] \times \mathbb E[I(t+\tau)^2].$$

Stationarity makes the two factors on the right equal. Since the intensities are nonnegative,

$$\mathbb E[I(t)I(t+\tau)] \le \mathbb E[I^2].$$

Dividing by $\mathbb E[I]^2$ proves the delay bound. A calibrated stationary record whose correlation rises away from zero cannot be represented by this classical random-intensity model.

7. Background dilution of antibunching

A signal has

$$g_{\mathrm s}^{(2)}(0)=0.20.$$

Independent Poisson background is present, and the signal fraction of the total mean rate is $\rho=0.80$. Find the raw observed $g_{\mathrm{obs}}^{(2)}(0)$. Does uniform signal loss alone change the ideal signal correlation?

Solution

Use

$$g_{\mathrm{obs}}^{(2)}(0)-1 = \rho^2 \left[ g_{\mathrm s}^{(2)}(0)-1 \right].$$

Therefore

$$\begin{aligned} g_{\mathrm{obs}}^{(2)}(0) &= 1+(0.80)^2(0.20-1) \\ &= 1-0.512 \\ &= 0.488. \end{aligned}$$

The background substantially fills the dip. Uniform independent signal loss by itself multiplies the signal numerator and denominator by the same $\eta^2$ and leaves its ideal normalized $g^{(2)}$ unchanged. In practice, loss lowers the signal fraction relative to fixed background and can indirectly worsen the raw value.

8. Immediate correlation after a two-level emission

A Markovian two-level emitter has output coupling

$$\hat L=\sqrt{\Gamma}\,\hat\sigma_-.$$

Use the conditional form of $G^{(2)}(0)$ to show that the ideal zero-delay correlation vanishes whenever the steady emission rate is nonzero.

Solution

At zero delay,

$$G^{(2)}(0) = \operatorname{Tr} \left[ \hat L^\dagger\hat L \hat L\rho_{\mathrm{ss}}\hat L^\dagger \right].$$

By cyclicity of the trace,

$$\begin{aligned} G^{(2)}(0) &= \operatorname{Tr} \left[ \hat L^\dagger\hat L^\dagger \hat L\hat L \rho_{\mathrm{ss}} \right]. \end{aligned}$$

For a two-level lowering operator,

$$\hat L^2 = \Gamma\hat\sigma_-^2 = 0.$$

Hence

$$G^{(2)}(0)=0.$$

If

$$\bar R = \operatorname{Tr} \left( \hat L^\dagger\hat L\rho_{\mathrm{ss}} \right) >0,$$

then normalization gives

$$g^{(2)}(0)=0.$$

The result expresses the finite excitation capacity of one ideal two-level emitter. Background, timing averaging, multiple emitters, and detector artifacts can change the measured minimum; detector exclusion can also create or deepen a false dip.

9. Antibunching above unity

Sum the counts from two independent, noninterfering stationary channels. Channel $A$ has rate $R_A=\gamma$ and $g_A^{(2)}(\tau)=1-e^{-10\gamma|\tau|}$; channel $B$ is chaotic light with rate $R_B=1.1\gamma$ and $g_B^{(2)}(\tau)=1+e^{-\gamma|\tau|}$. Find the summed correlation at zero delay and its initial slope. Does its long-gate count variance have to be sub-Poissonian?

Solution

Independence makes the cross-channel excess vanish. With $x=\gamma|\tau|$,

$$g^{(2)}(\tau)-1 =\frac{R_A^2(g_A^{(2)}-1)+R_B^2(g_B^{(2)}-1)}{(R_A+R_B)^2} =\frac{1.21e^{-x}-e^{-10x}}{4.41}.$$

Thus $g^{(2)}(0)=1+0.21/4.41\simeq1.0476$, but the initial slope with respect to $x$ is $(10-1.21)/4.41>0$. It satisfies the temporal antibunching test while remaining above unity. This model is physically realizable: incoherent pumping and decay of a two-level emitter can give the first correlation, and a Lorentzian chaotic field gives the second.

The integrated excess is $(1.21-0.1)/(4.41\gamma)>0$. Consequently

$$F_{T\to\infty}=1+2(R_A+R_B)\int_0^\infty d\tau\,[g^{(2)}(\tau)-1] =1+\frac{4.2(1.11)}{4.41}\simeq2.0571.$$

Temporal antibunching therefore coexists with super-Poissonian long-gate counts. Conversely, a free single-frequency $|2\rangle$ mode has $g^{(2)}(\tau)=1/2$ at every delay and no temporal antibunching rise.

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