Photon Counting

Photon counting is direct detection of quanta in an optical, microwave, or other bosonic output channel. Instead of a continuous quadrature current, the record is a sequence of events: detector clicks, time tags, or counts in finite bins. Calibrated optical number-resolving and threshold POVMs, efficiency, dark counts, recovery, and detector tomography are developed on the quantum-optics Photon Counting page. This page takes the detector model as given and concentrates on the conditional record produced by monitoring an open-system output.

In the ideal Markovian limit, photon counting is modeled by a counting process $N_t$ with increment

$$dN_t\in\{0,1\}, \qquad dN_t^2=dN_t.$$

The conditional mean of the increment is set by the instantaneous detected flux. For an output channel with coupling operator $L$,

$$\mathbb E[dN_t|\rho_c(t)] = \eta\, \langle L^\dagger L\rangle_c\,dt$$

in a common ideal convention, where $\eta$ is the total detection efficiency. This page explains the record and detector model. The conditioned state dynamics associated with clicks and no-click intervals is the canonical subject of Quantum Jump Trajectories.

Direct Detection

Photon counting is sensitive to intensity, not field quadrature. In input–output language, a monitored port satisfies

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

For vacuum input with known coherent amplitudes subtracted or accounted for, the emitted photon flux associated with the system channel is

$$\Phi(t) = \langle L^\dagger L\rangle.$$

The operator $L$ is often a cavity-loss operator such as $\sqrt{\kappa}a$, or a spontaneous-emission operator such as $\sqrt{\Gamma}\,\sigma_-$. The same $L$ contributes the unconditional dissipator $\mathcal D[L]\rho$ when the output is not retained.

Photon counting asks a different question from homodyne detection. It samples emitted quanta. Homodyne detection interferes the output with a strong local oscillator and samples a quadrature. Both may monitor the same physical port, but they produce different records and different conditioned trajectories.

Counting Records

The cumulative count

$$N_t$$

is a nondecreasing integer-valued process. In a small interval $[t,t+dt]$, $dN_t=1$ means one detected event and $dN_t=0$ means no detected event. For a time bin of finite width $\Delta t$,

$$\Delta N_k = N_{t_{k+1}}-N_{t_k}$$

may be $0$, $1$, or larger if the bin is wide enough or the flux is high enough.

Equivalently, an ideal time-tagging detector reports event times

$$0\lt t_1\lt t_2\lt\cdots\lt t_m\le T.$$

The two descriptions are related by binning. Fine time tags preserve more information than coarse bins, but real detector timing jitter and dead time set a practical limit on time resolution.

Conditional Intensity

The conditional intensity or hazard rate $\lambda_t$ is the rate predicted from the current conditional state:

$$\mathbb E[dN_t|\rho_c(t)] = \lambda_t\,dt.$$

For one ideal detected channel,

$$\lambda_t = \eta\,\operatorname{Tr}(L^\dagger L\rho_c(t)).$$

With dark counts, a simple model is

$$\lambda_t = \eta\,\operatorname{Tr}(L^\dagger L\rho_c(t)) + \lambda_{\mathrm d},$$

where $\lambda_{\mathrm d}$ is a background count rate. The dark-count branch carries little or no information about the system, so a click should not always be interpreted as a system quantum being emitted.

If several detectors monitor distinct channels $L_j$, the record has several counting processes $N_{j,t}$ with intensities

$$\lambda_{j,t} = \eta_j\, \operatorname{Tr}(L_j^\dagger L_j\rho_c(t)) + \lambda_{j,\mathrm d}.$$

The detector label is part of the record. Losing it is a coarse graining that changes the conditioned state update.

Click and No-Click Information

A click is an outcome, but so is no click. In an interval $dt$, the two ideal alternatives are:

click:     dN_t = 1
no click:  dN_t = 0

If a click occurs in channel $L$, the normalized state update has the jump form

$$\rho_c \longmapsto \frac{L\rho_cL^\dagger} {\operatorname{Tr}(L^\dagger L\rho_c)}.$$

If no click occurs, the state is also updated because the absence of an expected event is information. The no-click update involves the effective non-Hermitian Hamiltonian and normalization by the probability of no event. The detailed formulas, waiting-time distributions, and ensemble recovery are treated in Quantum Jump Trajectories.

The important record-level lesson is that a data file containing no clicks is not the same as no measurement. It may be strong evidence that the emitting system was dark, in the ground state, outside the collection mode, or hidden by detector inefficiency.

Detector Imperfections

Real photon counters require more than the ideal point-process model. Common effects include:

• finite efficiency $\eta$;
• dark counts or background light;
• dead time after a detection event;
• timing jitter;
• finite time-bin width;
• afterpulsing or detector recovery artifacts;
• number resolution or lack of number resolution;
• saturation at high flux;
• loss before the detector;
• mode mismatch and imperfect filtering.

Some imperfections can be represented by adding unmonitored loss channels or background counting branches. Others, such as dead time and afterpulsing, create detector memory and can invalidate a simple Markovian counting-process model unless the detector state is included explicitly.

Normal Ordering Preview

Ideal photodetection is naturally connected with normally ordered field correlations. For a positive-frequency field operator $E^{(+)}(t)$, the single-time counting rate is proportional to

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

Two-count correlations involve

$$G^{(2)}(t_1,t_2) = \left\langle E^{(-)}(t_1)E^{(-)}(t_2) E^{(+)}(t_2)E^{(+)}(t_1) \right\rangle,$$

up to detector response and normalization conventions. This is the origin of photon-correlation measurements such as photon antibunching and Hanbury Brown–Twiss interferometry.

In Markovian input–output notation, $E^{(+)}$ is represented by the relevant output field. For a vacuum-input system channel, the normally ordered flux reduces to the system expectation $\langle L^\dagger L\rangle$ after the input-output relation is applied.

This page only gives the preview. A quantum-optics treatment owns the full theory of optical coherence functions.

Prepared optical Fock-state distributions, loss-thinned number statistics, phase properties, and source-quality diagnostics are developed on Photon Number States.

Examples

For a two-level atom with spontaneous emission operator

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

the ideal click rate is

$$\lambda_t = \Gamma \langle\sigma_+\sigma_-\rangle_c = \Gamma p_e(t).$$

A click indicates an emitted photon and updates the atom toward the ground state. A long no-click interval makes excitation less likely, provided the detector efficiency is high and background counts are low.

For a leaking cavity with

$$L=\sqrt{\kappa}a,$$

the ideal count rate is

$$\lambda_t = \kappa\langle a^\dagger a\rangle_c.$$

The count record probes the output photon flux, not a nondestructive record of the intracavity photon number at every instant. Cavity loss removes photons from the mode.

Common Mistakes

• Treating no-click intervals as absence of measurement rather than no-click outcomes.
• Interpreting every click as a system photon when dark counts or background light are present.
• Forgetting that inefficiency leaves unobserved emissions that still decohere the system.
• Using a jump trajectory when the experiment actually measures a homodyne or heterodyne current.
• Confusing detected output flux with intracavity photon number.
• Ignoring detector dead time at high count rates.
• Applying infinitesimal $dN_t\in\{0,1\}$ rules to large time bins without allowing multiple counts.
• Reading physical meaning into a collapse operator without specifying the monitored port and detector.

Cross-Links

• Measurement Records for time tags, bins, intensities, and detector imperfections.
• Optical Correlation Functions for normally ordered output-field moments and normalized $g^{(m)}$ observables estimated from those records.
• Hanbury Brown–Twiss Interferometry for two-channel coincidence histograms, accidental baselines, thermal bunching, and spatial intensity interferometry.
• Quantum Jump Trajectories for click/no-click state updates and waiting times.
• Unravelings for comparison with homodyne and heterodyne monitoring of the same output channel.
• Input–Output Theory for output fields and fluxes.
• Quantum Optical Master Equation for the unconditional dynamics behind many photon-counting models.
• Homodyne Detection and Heterodyne Detection for quadrature-record alternatives.
• Quantum Optics for platform-level context.

References

• R. J. Glauber, “The quantum theory of optical coherence,” Physical Review 130, 2529–2539, 1963.
• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Reviews of Modern Physics 70, 101–144, 1998.

Exercises

1. For a constant counting intensity $\lambda$, compute the probability of no clicks in time $T$.

Solution

For a Poisson process with constant rate $\lambda$, the no-count probability is

$$P_0(T)=e^{-\lambda T}.$$

Equivalently, multiplying no-click probabilities over short bins gives

$$\lim_{N\to\infty} \left( 1-\lambda\frac{T}{N} \right)^N = e^{-\lambda T}.$$

2. A two-level atom has $L=\sqrt{\Gamma}\sigma_-$ and conditional excited-state population $p_e$. With detection efficiency $\eta$ and dark-count rate $\lambda_{\mathrm d}$, write the observed click intensity.

Solution

Since

$$L^\dagger L = \Gamma\sigma_+\sigma_-,$$

one has

$$\operatorname{Tr}(L^\dagger L\rho_c) = \Gamma p_e.$$

The observed intensity in the simple dark-count model is

$$\lambda_{\mathrm{obs}} = \eta\Gamma p_e + \lambda_{\mathrm d}.$$

3. Suppose a detector reports only the total count from two unresolved channels $L_1$ and $L_2$. What information is lost compared with a detector that labels which channel clicked?

Solution

The labeled record distinguishes the alternatives $L_1\rho L_1^\dagger$ and $L_2\rho L_2^\dagger$. If the detector reports only the total count, the click operation is the coarse-grained sum

$$\mathcal I_{\mathrm{click}}(\rho) = L_1\rho L_1^\dagger + L_2\rho L_2^\dagger,$$

with the appropriate rates absorbed into $L_1$ and $L_2$. Coarse graining loses information about which channel carried the emission and can leave the conditional state more mixed.

4. Why is a large-bin count $\Delta N_k$ not always well represented by an infinitesimal increment $dN_t$?

Solution

The infinitesimal rule keeps only terms through first order in $dt$, so the probability of two or more counts is $O(dt^2)$ and is neglected. In a finite bin $\Delta t$, especially at high flux, multiple events can have appreciable probability. Then $\Delta N_k$ should be treated as a finite-bin count with Poisson or detector-specific statistics, not as a binary increment.