Photon Antibunching

Photon antibunching is the suppression of nearby photoevents relative to events separated by a longer delay. For a stationary field, the direct temporal criterion is

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

for at least one delay $\tau$. A widely used equal-time signature is

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

Both occur for ideal resonance fluorescence from one two-level emitter, but they answer different questions. The first compares two delays and violates a stationary classical correlation bound. The second compares the equal-time factorial moment with a Poisson reference and is a sufficient optical nonclassicality witness. Temporal antibunching and sub-Poissonian counting are not generally equivalent, even for stationary light. The former compares delays; the latter depends on the occupation of a specified mode or the integral of correlations over a counting interval.

The physical mechanism is especially transparent for one emitter. Detection of a fluorescence photon leaves a two-level emitter in its ground state. It must be excited again before it can emit another photon, so the conditional emission rate starts at zero and recovers over the excitation and decay timescales.

The detected modes, timing response, and background determine how a source correlation appears in a histogram.

Required background. Correlation Functions defines normally ordered moments; HBT Interferometry explains coincidence normalization; Photon Number States separates number statistics from brightness and purity; and the Two-Level Atom supplies the source Hamiltonian.

What the Correlation Measures

Stationary normalized correlation

For one stationary detected field, write

$$G^{(2)}(\tau) ={} \left\langle \hat E^{(-)}(t) \hat E^{(-)}(t+\tau) \qquad {}\times \hat E^{(+)}(t+\tau) \hat E^{(+)}(t) \right\rangle ,$$

and

$$g^{(2)}(\tau) = \frac{ G^{(2)}(\tau) }{ \left\langle \hat E^{(-)}\hat E^{(+)} \right\rangle^2 }.$$

Normal ordering makes the numerator a joint absorption probability density in the broadband direct-detection model. Stationarity removes the arbitrary origin $t$, but it does not remove the signed delay $\tau$, detector labels, polarization, spectral filter, spatial mode, or timing response.

For a sufficiently ideal stationary counting channel,

$$g^{(2)}(\tau) = \frac{ R(\tau\mid 0) }{ R_{\mathrm{ss}} },$$

where $R(\tau\mid0)$ is the conditional event rate, allowing intervening events, and $R_{\mathrm{ss}}>0$ is the unconditional steady rate. It is not the probability density of the next photon. Thus:

• $g^{(2)}(\tau)>1$ means an event raises the conditional rate at that delay;
• $g^{(2)}(\tau)=1$ means the rate has returned to its uncorrelated baseline;
• $g^{(2)}(\tau)<1$ means an event suppresses the conditional rate.

The word “simultaneous” always refers to an idealized limit or a declared gate. A real instrument reports a time-averaged and response-convolved quantity.

Three statements that should not be collapsed

Temporal antibunching means that the correlation rises away from zero:

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

for some $\tau$ in a stationary measurement. This is a comparison between delays.

Equal-time sub-Poissonian behavior means

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

for the specified mode or sufficiently short counting gate. This is a comparison with the Poisson factorial moment.

Ideal single-photon exclusion means

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

It says that the retained field has no equal-time two-photon contribution under the model. It does not by itself state the source brightness, vacuum probability, mode purity, indistinguishability, or delivery efficiency.

In common continuous-wave fluorescence experiments all three statements may hold. Treating them as definitions of one another hides the assumptions that make them coincide.

Why Antibunching Is Nonclassical

Stationary classical delay bound

Let $I(t)\ge0$ be a stationary classical random intensity. The Cauchy–Schwarz inequality 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 second moments equal, so

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

After division by $\mathbb E[I]^2$,

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

A calibrated stationary correlation that rises away from zero violates this classical random-intensity bound. The conclusion depends on stationarity and on excluding detector recovery or other artificial short-delay suppression.

This criterion does not require $g^{(2)}(0)<1$. In principle, a curve can rise from $1.2$ at zero to $1.3$ at a later delay and still violate the stationary classical inequality.

Nonnegative Glauber–Sudarshan P bound

An optically classical single-mode state has a Glauber–Sudarshan probability measure (including delta measures), conventionally written

$$\rho = \int d^2\alpha\, P(\alpha) |\alpha\rangle\langle\alpha|, \qquad P(\alpha)\ge0.$$

For normally ordered equal-time moments, set

$$I_\alpha=|\alpha|^2.$$

Then

$$\begin{aligned} g^{(2)}(0) &= \frac{ \mathbb E_P[I_\alpha^2] }{ \mathbb E_P[I_\alpha]^2 } \\ &= 1+ \frac{ \operatorname{Var}_P(I_\alpha) }{ \mathbb E_P[I_\alpha]^2 } \ge1. \end{aligned}$$

Consequently,

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

is sufficient to rule out every nonnegative $P$ probability measure, assuming the measured normally ordered moment and calibration model are valid. It is not a necessary condition for nonclassicality: squeezed vacuum, entangled light, and many other nonclassical states can have $g^{(2)}(0)\ge1$.

Discrete clicks are not enough

Photodetectors produce discrete electronic outcomes even under weak coherent illumination. Attenuated coherent light remains Poissonian,

$$g^{(2)}(0)=1,$$

at every nonzero amplitude in the ideal model. The nonclassical evidence is the calibrated correlation inequality, not the visual fact that a detector clicks.

Relation to Photon Statistics

Factorial moments

For one mode with number operator $\hat n=\hat a^\dagger\hat a$,

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

Since

$$\langle \hat n(\hat n-1) \rangle = \operatorname{Var}(\hat n) + \langle\hat n\rangle^2 - \langle\hat n\rangle,$$

one obtains

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

The Fano factor

$$F := \frac{ \operatorname{Var}(\hat n) }{ \langle\hat n\rangle }$$

therefore satisfies

$$F = 1+ \langle\hat n\rangle \left[ g^{(2)}(0)-1 \right].$$

For this fixed mode or gate, $g^{(2)}(0)<1$ is equivalent to $F<1$. For continuous-wave light, the gate Fano factor instead integrates $g^{(2)}(\tau)-1$ with a triangular weight. A narrow dip can coexist with a broad bunching shoulder and super-Poissonian long-gate counts; see the explicit stationary example.

The Mandel parameter is

$$Q := \frac{ \operatorname{Var}(\hat n)-\langle\hat n\rangle }{ \langle\hat n\rangle } = \langle\hat n\rangle \left[ g^{(2)}(0)-1 \right].$$

Thus $Q<0$ is the same equal-time sub-Poissonian condition under these definitions.

Number states

For a number state $|n\rangle$ with $n\ge1$,

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

In particular,

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

The value for $|2\rangle$ is $1/2$, even though the state contains exactly two photons. This is one reason a threshold such as $g^{(2)}(0)<1/2$ must be interpreted within a source model rather than promoted to a universal definition of “one photon.”

Pulsed trials

For a pulsed source, let $N_k$ be the photon number in the declared output mode on trigger $k$. The per-pulse factorial moment is

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

If only zero-, one-, and two-photon sectors are appreciable,

$$\begin{aligned} \bar N &= p_1+2p_2, \\ g_{\mathrm p}^{(2)}[0] &= \frac{ 2p_2 }{ \bar N^2 }. \end{aligned}$$

A measured $g_{\mathrm p}^{(2)}[0]$ does not determine $p_2$ unless $\bar N$ or equivalent brightness information is also known. A source that emits one photon once per million triggers and vacuum otherwise has $g_{\mathrm p}^{(2)}[0]=0$, but it is not a bright on-demand source.

Independent loss

Under mode-independent Bernoulli loss with $0<\eta\le1$ and nonzero mean,

$$\begin{aligned} \langle N_{\mathrm{det}}\rangle &= \eta\langle N\rangle, \\ \langle N_{\mathrm{det}}(N_{\mathrm{det}}-1) \rangle &= \eta^2 \langle N(N-1)\rangle. \end{aligned}$$

The normalized $g^{(2)}$ is therefore unchanged by ideal independent loss. Loss lowers the event rate and enlarges statistical uncertainty. It can also lower the signal fraction relative to fixed background, which fills an observed dip.

Why One Emitter Antibunches

Nilpotent lowering operator

For a two-level emitter,

$$\hat\sigma_- = |g\rangle\langle e|, \qquad \hat\sigma_-^2=0.$$

Let the monitored radiative channel have jump operator

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

The equal-time coincidence numerator contains two successive lowering operations:

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

Whenever the steady emission rate is nonzero,

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

This operator identity is the ideal zero-delay exclusion. It is stronger and cleaner than saying that photons “repel” one another. The emitter cannot support two simultaneous excitations in the modeled transition.

State reset after a click

The normalized state immediately after a detected jump is

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

For one ideal two-level transition,

$$\rho_{\mathrm c}(0^+) = |g\rangle\langle g|.$$

The next emission requires the drive to rebuild excited-state population. This waiting process, not a force between emitted photons, creates the short-delay deficit.

Jump-superoperator expression

Define

$$\mathcal J\rho := \hat L\rho\hat L^\dagger$$

and let $\mathcal L$ be the unconditional Markovian Liouvillian. The steady count rate is

$$R_{\mathrm{ss}} = \operatorname{Tr} \left[ \mathcal J\rho_{\mathrm{ss}} \right].$$

For $\tau\ge0$, the quantum regression or conditional-jump expression is

$$g^{(2)}(\tau) = \frac{ \operatorname{Tr} \left[ \mathcal J e^{\mathcal L\tau} \left( \mathcal J\rho_{\mathrm{ss}} \right) \right] }{ R_{\mathrm{ss}}^2 }.$$

The inner jump prepares the conditional state, $e^{\mathcal L\tau}$ evolves it, and the outer jump asks for the emission rate at delay $\tau$. This formula is the natural bridge to Quantum Jump Trajectories. It assumes the same Markovian dynamics used to define $\mathcal L$; memory, frequency filtering, or delayed feedback may require an enlarged model.

A fluorescence click resets an ideal two-level emitter to $|g\rangle$, so a second click requires re-excitation. The curves use the exact models below: upward pumping $R=\Gamma$ (dashed) and resonant coherent driving $\Omega=2\Gamma$ without pure dephasing (solid). Both vanish at zero, but only the coherent case oscillates. Detector response is omitted.

Two-Level Correlation Models

Incoherent excitation

Consider upward-only incoherent pumping at rate $R>0$ and radiative decay at rate $\Gamma>0$. A separate rapidly emptied pump level can realize this effective model; a direct incoherent resonant field would also stimulate downward transitions:

$$\dot\rho = R\, \mathcal D[\hat\sigma_+]\rho + \Gamma\, \mathcal D[\hat\sigma_-]\rho,$$

where

$$\mathcal D[\hat c]\rho := \hat c\rho\hat c^\dagger - \frac12 \left\{ \hat c^\dagger\hat c,\rho \right\}.$$

Immediately after an emission, the conditional excited-state probability $p_e(\tau)$ obeys

$$\frac{d p_e}{d\tau} = R(1-p_e) - \Gamma p_e, \qquad p_e(0)=0.$$

The steady population and conditional solution are

$$\begin{aligned} p_e^{\mathrm{ss}} &= \frac{R}{R+\Gamma}, \\ p_e(\tau) &= p_e^{\mathrm{ss}} \left[ 1-e^{-(R+\Gamma)\tau} \right]. \end{aligned}$$

Since the fluorescence rate is $\Gamma p_e$,

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

This model gives a monotonic recovery. The dip narrows as the pump rate increases because re-excitation becomes faster. At very high rates, detector timing response and additional levels often become as important as this two-rate prediction.

Coherent resonant drive

For resonant coherent driving with real, nonzero Rabi frequency $\Omega$,

$$\dot\rho = -i \left[ \frac{\Omega}{2} \left( \hat\sigma_+ + \hat\sigma_- \right), \rho \right] + \Gamma \mathcal D[\hat\sigma_-]\rho.$$

With no additional pure dephasing, the steady resonance-fluorescence result can be written

$$\begin{aligned} g^{(2)}(\tau) &= 1 - e^{-3\Gamma|\tau|/4} f(\tau), \\ f(\tau) &= \cos(\nu|\tau|) + \frac{ 3\Gamma }{ 4\nu } \sin(\nu|\tau|) . \end{aligned}$$

where

$$\nu = \sqrt{ \Omega^2-\frac{\Gamma^2}{16} }.$$

If $\nu$ is imaginary, the trigonometric functions are understood by analytic continuation to hyperbolic functions. At $\nu=0$, take $f=1+3\Gamma|\tau|/4$ continuously. At strong drive the correlation shows damped Rabi oscillations: after a photon resets the emitter, the coherent drive cycles population between $|g\rangle$ and $|e\rangle$ before damping returns the system to steady state.

The exact line shape changes with detuning, pure dephasing, polarization selection, coherent laser leakage, and which fluorescence component is filtered. The reliable procedure is to propagate the post-click state with the same calibrated master equation used for the steady state. Optical Bloch Equations derives the one-time driven-state dynamics.

Shelving and blinking

A metastable dark state adds a slow conditional timescale. A common phenomenological form is

$$g^{(2)}(\tau) ={} 1 - (1+a) e^{-|\tau|/\tau_{\mathrm a}} + a e^{-|\tau|/\tau_{\mathrm b}}, \qquad \tau_{\mathrm b}\gg\tau_{\mathrm a},$$

with $a>0$. The fast term produces antibunching and the slow term produces a bunching shoulder. At zero delay the ideal expression still gives $g^{(2)}(0)=0$.

An antibunching dip and bunching at longer delay are therefore compatible. The dip diagnoses short-time excitation exclusion; the shoulder diagnoses slow switching among emissive and dark configurations. Assigning every feature to a single “lifetime” discards this information.

More than literal two-level atoms

The same conditional-exclusion logic appears in atoms, trapped ions, molecules, color centers, semiconductor quantum dots, and engineered superconducting emitters. Multilevel structure modifies the recovery. Photon blockade and other nonlinear resonators can also suppress two-photon occupation without being literal two-level material systems. The observable remains a correlation of a declared output channel.

Experimental Signatures

Two-channel measurement

The standard laboratory measurement routes one selected source field to two detectors and builds a cross-correlation of their event records. The splitter does not create the source antibunching. Under independent linear loss, it routes the source’s factorial moment into a measurable cross-coincidence.

Using two detectors is important because a single detector’s dead time creates an automatic short-delay exclusion in its autocorrelation. A cross-channel measurement avoids that particular artifact, but it does not eliminate:

• optical leakage between channels;
• electrical cross-talk or shared threshold logic;
• common timing or trigger artifacts;
• detector afterpulsing;
• source-independent correlated background;
• saturation and live-time bias.

The apparatus, lag-histogram estimator, accidental baseline, and uncertainty are developed on Hanbury Brown–Twiss Interferometry. The present question is what a normalized dip says about the source.

Continuous-wave records

For stationary continuous excitation, an ideal narrow lag bin centered at $\tau_k$ has expected coincidence count

$$\mathbb E[C_k] \simeq R_1R_2 T_{\mathrm{acq}} \Delta\tau\, g_{12}^{(2)}(\tau_k),$$

away from record-edge corrections. Here $R_j$ are live-time-corrected channel rates, $T_{\mathrm{acq}}$ is common exposure, and $\Delta\tau$ is the bin width.

The uncorrelated baseline is not necessarily the average of whichever bins look flat. It must account for:

• acquisition gaps and unequal live time;
• rate drift and intermittency;
• start–stop or all-pairs counting conventions;
• lag-dependent exposure near record edges;
• periodic triggers or modulation;
• excluded regions and fit uncertainty.

A continuous-wave antibunching fit commonly returns a correlation timescale, raw dip depth, background-corrected depth, and perhaps a slow blinking timescale. These parameters should not be called lifetimes unless the fitted physical model justifies that identification.

Pulsed records

With repetition period $T_{\mathrm r}$, a delay histogram forms peaks near

$$\tau=mT_{\mathrm r}, \qquad m\in\mathbb Z.$$

Let $A_m$ be a peak area integrated over a declared coincidence window. Under stable independent trials, equal lag exposure, and linear number-resolving counts (or controlled weak-response threshold detection),

$$\widehat g_{\mathrm p}^{(2)}[0] \simeq \frac{ A_0 }{ \langle A_m\rangle_{m\ne0} }.$$

At appreciable threshold saturation, it is a click-probability ratio rather than the source factorial moment; the exact HBT expression shows the difference. This ratio is only a starting estimator. Side peaks are not an unbiased baseline when blinking correlates neighboring pulses, the trigger train has gaps, detector dead time spans several periods, or the source drifts. One may instead fit the complete sequence of peak areas with a trial-to-trial correlation model.

The central area probes two detections associated with one trigger. It should not be replaced by the height of one histogram bin, which depends strongly on timing jitter and arbitrary bin alignment.

Re-excitation within a pulse

An emitter can radiate once, be re-excited before the pump pulse ends, and radiate again. A source may therefore have a single occupied transition and still produce a nonzero same-pulse multiphoton probability.

Suppressing re-excitation can require:

• a pump pulse short compared with the radiative cycle;
• coherent pulse-area control;
• resonant rather than above-band excitation;
• temporal filtering, with its stated efficiency cost;
• a level scheme that prevents immediate return to the pump resonance.

A small continuous-wave zero-delay dip and a small pulsed central-peak area are related diagnostics, not interchangeable numbers.

Measurement Forward Model

Independent background

Let channel $j$ have mean signal rate $S_j$, independent background counts $B_j$ added without optical interference, and signal fraction

$$\rho_j := \frac{ S_j }{ S_j+B_j }.$$

If the two backgrounds are mutually independent and independent of the signal, with constant stationary means, then

$$g_{\mathrm{raw}}^{(2)}(\tau)-1 = \rho_1\rho_2 \left[ g_{\mathrm s}^{(2)}(\tau)-1 \right].$$

For symmetric channels with $\rho_1=\rho_2=\rho$,

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

Background pulls every dip and peak toward one. For an ideal source with $g_{\mathrm s}^{(2)}(0)=0$,

$$g_{\mathrm{raw}}^{(2)}(0) = 1-\rho_1\rho_2.$$

Poisson statistics within each separate background stream are not required for this cross-channel mean; their independence is required. Coherent pump leakage in the same optical mode can interfere with fluorescence and is not an additive-count background. The correction is invalid for correlated backgrounds, source-dependent fluorescence, or a background estimate taken under a different operating condition.

Worked background example

Suppose both detector channels have signal fraction $0.90$, and the measured raw value is

$$g_{\mathrm{raw}}^{(2)}(0)=0.433.$$

Under the independent-background model,

$$\begin{aligned} g_{\mathrm s}^{(2)}(0) &= 1+ \frac{ g_{\mathrm{raw}}^{(2)}(0)-1 }{ (0.90)^2 } \\ &= 0.30. \end{aligned}$$

The inferred source value is meaningful only with uncertainties on the raw correlation and both signal fractions. It remains an inference, not another raw datum.

Timing response

Let $K(\tau)$ be the normalized relative-time response of the detector pair, including both detector jitters and any synchronization uncertainty:

$$\int d\tau\,K(\tau)=1.$$

For a stationary linear timing model,

$$g_{\mathrm{timed}}^{(2)}(\tau)-1 ={} \int d\tau'\, K(\tau-\tau') \times \left[ g_{\mathrm{raw}}^{(2)}(\tau')-1 \right].$$

Histogram binning performs another average. For a bin of width $\Delta\tau$,

$$\overline g_k^{(2)} = \frac1{\Delta\tau} \int_{\tau_k-\Delta\tau/2}^{\tau_k+\Delta\tau/2} d\tau\, g_{\mathrm{timed}}^{(2)}(\tau).$$

Finite timing resolution conserves the integrated correlation excess or deficit in this ideal convolution model but spreads it over delay. A narrow perfect source dip can therefore appear shallow or unresolved. Deconvolution without regularization and uncertainty propagation can manufacture an overconfident zero.

Multiple independent emitters

Suppose independent emitters contribute noninterfering count streams with mean rates $s_i$; for example, the detector sums orthogonal polarization or spatial channels. Both HBT branches must weight the contributions proportionally. If their individual equal-time correlations are $g_i^{(2)}(0)$, then

$$\begin{aligned} g_{\mathrm{tot}}^{(2)}(0)-1 = \frac{ \displaystyle \sum_i s_i^2 \left[ g_i^{(2)}(0)-1 \right] }{ \displaystyle \left( \sum_i s_i \right)^2 }. \end{aligned}$$

For ideal antibunched emitters, $g_i^{(2)}(0)=0$, so

$$g_{\mathrm{tot}}^{(2)}(0) = 1- \frac{ \sum_i s_i^2 }{ \left( \sum_i s_i \right)^2 }.$$

Define the effective contributor number

$$N_{\mathrm{eff}} := \frac{ \left( \sum_i s_i \right)^2 }{ \sum_i s_i^2 }.$$

Then

$$g_{\mathrm{tot}}^{(2)}(0) = 1- \frac1{N_{\mathrm{eff}}}.$$

For $N$ equal-brightness emitters, $N_{\mathrm{eff}}=N$ and

$$g_{\mathrm{tot}}^{(2)}(0) = 1-\frac1N.$$

Unequal brightness matters. Two ideal emitters with rates in the ratio $3:1$ give

$$g_{\mathrm{tot}}^{(2)}(0) = 1- \frac{3^2+1^2}{(3+1)^2} = \frac38.$$

If instead the detector coherently collects amplitudes $L=\sum_i c_i\sigma_{i-}$, indistinguishable two-emitter paths can interfere. Independence of the initial emitters alone does not justify the count-sum formula; see the solved example below and Koong et al. (2022).

The value $3/8$ is below $1/2$ even though two emitters are present. The correlation shows that one contributor dominates; by itself it does not count physical objects.

Detector artifacts

Detector behavior can create either sign of correlation:

• dead time and recovery suppress short-delay events and can imitate an antibunching dip in a same-detector record;
• afterpulsing creates a delayed excess, often on a detector-specific timescale;
• electrical or optical cross-talk creates near-zero cross-channel excess;
• saturation changes the live time and makes accidental normalization rate dependent;
• gating can impose a periodic envelope unrelated to source dynamics;
• time-tagger rollover or clock pickup can create narrow repeated structures.

Blocked-input, dark-record, split coherent-light, rate-scaling, channel-swap, and large-time-shift controls are often more informative than adding another free parameter to the source fit.

Spectral and polarization filtering

The detected field is the source field after mode selection. Filtering is not always a passive cosmetic step:

• rejecting uncorrelated background can deepen the measured dip;
• admitting several independent transitions can fill it;
• polarization selection can isolate or mix decay channels;
• a narrow spectral filter has a long impulse response and changes temporal correlations;
• coherent pump leakage can interfere with the emitted field;
• selecting one sideband of resonance fluorescence measures a different correlation from collecting the total fluorescence.

Every quoted $g^{(2)}$ therefore belongs to a stated detection channel, not to an emitter independent of its optical measurement.

What a small pair moment determines

A small normally ordered pair moment establishes nonclassical light under the calibrated detection model. It does not determine vacuum probability, mode purity, delivery efficiency, or indistinguishability. Likewise, $g^{(2)}(0)<1/2$ identifies a dominant contributor only under the independent, noninterfering ideal-emitter model above.

For pulsed emission, the exact inequality

$$P(N\ge2)\le\frac{\langle N(N-1)\rangle}{2} =\frac12g_{\rm p}^{(2)}[0]\bar N^2$$

follows because $n(n-1)\ge2$ for $n\ge2$. It becomes an equality when sectors above two are absent. Brightness $\bar N$ is essential to this bound; the normalized pair moment alone cannot reconstruct photon number.

Exercises

1. Long-gate counts from conditional recovery

For upward pumping $R$ and decay $\Gamma$, use the derived $g^{(2)}(\tau)=1-e^{-(R+\Gamma)|\tau|}$ and steady count rate $r=R\Gamma/(R+\Gamma)$ to find the long-gate Fano factor. When is its minimum reached?

Solution

For a stationary ideal counting process,

$$F_\infty=1+2r\int_0^\infty[g^{(2)}(\tau)-1]d\tau =1-\frac{2R\Gamma}{(R+\Gamma)^2}\ge\frac12.$$

Equality occurs at $R=\Gamma$. When one rate is much slower, that step dominates each emission cycle and the long-gate distribution approaches Poisson statistics, even though the exact zero-delay correlation remains zero. Independent detection with efficiency $\eta$ gives $F_{\rm det}=1-\eta+\eta F_\infty$.

2. Number states and the Fano factor

For a number state $|n\rangle$ with $n\ge1$, calculate $g^{(2)}(0)$, $F$, and $Q$. Explain why $|2\rangle$ is nonclassical even though $g^{(2)}(0)=1/2$ rather than zero.

Solution

The state has

$$\langle\hat n\rangle=n, \qquad \operatorname{Var}(\hat n)=0,$$

and

$$\langle \hat n(\hat n-1) \rangle = n(n-1).$$

Therefore

$$\begin{aligned} g^{(2)}(0) &= 1-\frac1n, \\ F&=0, \qquad Q=-1. \end{aligned}$$

For $n=2$, the normalized pair moment is $1/2$. It is still below the nonnegative-$P$ classical bound of one, and the number variance vanishes. Zero delay correlation equal to zero is special to the one-photon state, not the definition of every nonclassical number state.

3. Coherent collection is not a sum of records

Two independent two-level emitters each have a diagonal state with excited probability $p>0$. Compare the equal-time correlation when their separate count records are added with detection of the coherent amplitude $L=\sigma_{1-}+\sigma_{2-}$.

Solution

Separate records give the equal-brightness result $g^{(2)}(0)=1/2$. For coherent collection, $\langle L^\dagger L\rangle=2p$, while $L^2=2\sigma_{1-}\sigma_{2-}$, so $\langle L^{\dagger2}L^2\rangle=4p^2$ and $g^{(2)}(0)=1$. The two alternatives for assigning emitters to detected photons add as amplitudes. No initial emitter coherence is needed for this exchange term. This is an equal-time calculation for the specified product state; steady collective radiative dynamics can also change the emitter state itself.

4. Asymmetric background correction

A continuous-wave experiment has channel signal fractions

$$\rho_1=0.80, \qquad \rho_2=0.75.$$

The raw measured value is

$$g_{\mathrm{raw}}^{(2)}(0)=0.52.$$

Assuming independent Poisson background, infer the source value. What raw zero-delay value would an ideal source produce with the same background?

Solution

The signal-fraction product is

$$\rho_1\rho_2 = (0.80)(0.75) = 0.60.$$

Hence

$$\begin{aligned} g_{\mathrm s}^{(2)}(0) &= 1+ \frac{ 0.52-1 }{ 0.60 } \\ &= 0.20. \end{aligned}$$

For an ideal source,

$$g_{\mathrm{raw,ideal}}^{(2)}(0) = 1-\rho_1\rho_2 = 0.40.$$

The difference between $0.52$ and $0.40$ represents residual source coincidences under this model. Uncertainty in the two signal fractions must be propagated into the inferred value.

5. Three unequal emitters

Three independent ideal antibunched emitters contribute noninterfering count streams with detected rates in the ratio

$$s_1:s_2:s_3=6:2:1.$$

Find $g_{\mathrm{tot}}^{(2)}(0)$ and $N_{\mathrm{eff}}$. Does a value below one-half prove that only one emitter is present?

Solution

The total rate weight is $6+2+1=9$, and the squared-weight sum is

$$6^2+2^2+1^2 = 41.$$

Therefore

$$g_{\mathrm{tot}}^{(2)}(0) = 1-\frac{41}{81} = \frac{40}{81} \simeq 0.494.$$

The effective contributor number is

$$N_{\mathrm{eff}} = \frac{81}{41} \simeq 1.98.$$

All three emitters are present, but one dominates the rate. The result below one-half does not prove a literal emitter count of one; it supports a dominant-emitter interpretation under the independent ideal-emitter model.

6. Bin-width dilution

An ideal source has

$$g^{(2)}(\tau) = 1-e^{-|\tau|/\tau_{\mathrm c}}.$$

Ignore detector jitter and average the central value over a top-hat bin of width $\Delta\tau$. Derive the bin-averaged value and evaluate it for $\Delta\tau=2\tau_{\mathrm c}$.

Solution

The symmetric central-bin average is

$$\begin{aligned} \overline g_0^{(2)} &= \frac1{\Delta\tau} \int_{-\Delta\tau/2}^{\Delta\tau/2} d\tau\, \left( 1-e^{-|\tau|/\tau_{\mathrm c}} \right) \\ &= 1- \frac{ 2\tau_{\mathrm c} }{ \Delta\tau } \left[ 1-e^{-\Delta\tau/(2\tau_{\mathrm c})} \right]. \end{aligned}$$

For $\Delta\tau=2\tau_{\mathrm c}$,

$$\overline g_0^{(2)} = e^{-1} \simeq 0.368.$$

The ideal point value is zero, yet a finite bin reports a positive number. Timing jitter would dilute it further.

7. Pulsed multiphoton probability

A pulsed source is known to have negligible $p_{n\ge3}$. At a declared source plane,

$$\bar N=0.20, \qquad g_{\mathrm p}^{(2)}[0]=0.10.$$

Find $p_2$, $p_1$, and $p_0$.

Solution

Using

$$g_{\mathrm p}^{(2)}[0] = \frac{ 2p_2 }{ \bar N^2 },$$

one finds

$$p_2 = \frac12 (0.10)(0.20)^2 = 0.002.$$

Since $\bar N=p_1+2p_2$,

$$p_1 = 0.20-2(0.002) = 0.196.$$

Normalization gives

$$p_0 = 1-p_1-p_2 = 0.802.$$

The source has strong multiphoton suppression, but vacuum still dominates the trials. The correlation and mean together reveal that distinction.

References

• R. Brouri, A. Beveratos, J.-P. Poizat, and P. Grangier, “Photon Antibunching in the Fluorescence of Individual Color Centers in Diamond,” Optics Letters 25, 1294–1296 (2000), doi:10.1364/OL.25.001294.
• H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations, Springer (1999), doi:10.1007/978-3-662-03875-8.
• H. J. Carmichael and D. F. Walls, “A Quantum-Mechanical Master Equation Treatment of the Dynamical Stark Effect,” Journal of Physics B 9, 1199–1219 (1976), doi:10.1088/0022-3700/9/8/007.
• F. Diedrich and H. Walther, “Nonclassical Radiation of a Single Stored Ion,” Physical Review Letters 58, 203–206 (1987), doi:10.1103/PhysRevLett.58.203.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004), doi:10.1007/978-3-662-13573-4.
• R. J. Glauber, “The Quantum Theory of Optical Coherence,” Physical Review 130, 2529–2539 (1963), doi:10.1103/PhysRev.130.2529.
• P. Grangier, G. Roger, and A. Aspect, “Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences,” Europhysics Letters 1, 173–179 (1986), doi:10.1209/0295-5075/1/4/004.
• H. J. Kimble, M. Dagenais, and L. Mandel, “Photon Antibunching in Resonance Fluorescence,” Physical Review Letters 39, 691–695 (1977), doi:10.1103/PhysRevLett.39.691.
• H. J. Kimble and L. Mandel, “Theory of Resonance Fluorescence,” Physical Review A 13, 2123–2144 (1976), doi:10.1103/PhysRevA.13.2123.
• Z. X. Koong, M. Cygorek, E. Scerri, T. S. Santana, S.-I. Park, J. D. Song, E. M. Gauger, and B. D. Gerardot, “Coherence in Cooperative Photon Emission from Indistinguishable Quantum Emitters,” Science Advances 8, eabm8171 (2022), doi:10.1126/sciadv.abm8171.
• B. Lounis and W. E. Moerner, “Single Photons on Demand from a Single Molecule at Room Temperature,” Nature 407, 491–493 (2000), doi:10.1038/35035032.
• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995), doi:10.1017/CBO9781139644105.
• P. Michler et al., “A Quantum Dot Single-Photon Turnstile Device,” Science 290, 2282–2285 (2000), doi:10.1126/science.290.5500.2282.
• H. Paul, “Photon Antibunching,” Reviews of Modern Physics 54, 1061–1102 (1982), doi:10.1103/RevModPhys.54.1061.
• R. Short and L. Mandel, “Observation of Sub-Poissonian Photon Statistics,” Physical Review Letters 51, 384–387 (1983), doi:10.1103/PhysRevLett.51.384.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer (2008), doi:10.1007/978-3-540-28574-8.
• X. T. Zou and L. Mandel, “Photon-Antibunching and Sub-Poissonian Photon Statistics,” Physical Review A 41, 475–476 (1990), doi:10.1103/PhysRevA.41.475.