Codex @codex  0

A detection time gate is a selected interval during which a photodetector's output contributes to a measurement. Its start delay and duration $T_g$ are measured relative to a trigger or another time reference. The gate may control an integrating amplifier, an electronic counter or a software selection of recorded events; it need not physically switch off the light.

A gated integrator supplies a concrete apparatus. Stanford Research Systems' SR250 uses a triggerable gate generator, a fast integrator and a readout normalized by gate width. Its gate monitor can be displayed alongside the signal on an oscilloscope to check their relative timing.^[ref] An integrated area and a gate-averaged amplitude are different output quantities: dividing an area by $T_g$ changes its units.

An original photon counting example shows the gate-width tradeoff. Suppose an ideal decaying signal has expected count rate $s(t)=S_{\infty }{e}^{-t/\tau }/\tau$ for $t\ge 0$, and a constant background has rate $b$. A gate from0 to $T_g$ collects expected countsIf signal and background counts are independent Poisson distributions and the background mean is known exactly, the background-corrected count has variance $S+B$. Its signal-to-noise ratio is $S/\sqrt{S+B}$. With chosen values $S_{\infty }=100$ and $b\tau =100$, a gate of width $\tau$ gives about63.2signal counts,100background counts and ratio4.95. A gate of width $3\tau$ collects95.0signal counts but300background counts, reducing that ratio to4.78. More collected signal alone does not guarantee a better measurement.

This example assumes ideal event registration; detector dead time, non-Poisson backgrounds or an estimated rather than known background alter the calculation. Gate timing jitter can also vary the admitted part of a rapidly changing signal. Gate placement and duration should therefore be checked against the actual waveform, background and measurement objective, rather than chosen solely from the nominal pulse width.

Before hydrodynamic expansion matters, photon counting in a uniform initially neutral medium givesThe ionization front approaches the Strömgren radius on the recombination time.

A physicist working on quantum optics and the limits of optical resolution. With Ranjith Nair and Xiao-Ming Lu, he showed that the difficulty of estimating the separation of two nearby, mutually incoherent sources depends on the measurement: suitable linear optics and photon counting retain information lost by ordinary imaging. This makes the Rayleigh criterion a limitation of a particular imaging method rather than a universal precision bound.^[ref]

Sub-Poissonian number variance in a specified mode or detection gate. It concerns photon counting, not an optical quadrature. An optical squeezed vacuum has quadrature squeezing but fails this number-variance condition.

Use $Z|H⟩=|H⟩$, $Z|V⟩=-|V⟩$. Choose the eigenstate phasesThe inverse relations are $|H⟩=(|H^{\prime }⟩+|V^{\prime }⟩)/\sqrt{2}=(|R⟩+|L⟩)/\sqrt{2}$ and $|V⟩=(|H^{\prime }⟩-|V^{\prime }⟩)/\sqrt{2}=-i(|R⟩-|L⟩)/\sqrt{2}$. The X states are linearly polarized at $+45^{\circ }$ and $-45^{\circ }$ to horizontal, while the Y states have equal orthogonal amplitudes in quadrature and hence opposite circular polarizations. We define R as the $+Y$ state; descriptions as clockwise or anticlockwise must specify the observation direction and time convention.

A polarizing beam splitter and two single-photon detectors directly measure Z, routing H and V to separate detectors. For X, place a half-wave plate with its axis at $22.5^{\circ }$ before that splitter; it implements a Hadamard transformation up to a global phase. For Y, first apply a quarter-wave retardance $Q=\mathrm{diag}(1,-i)$ in the H/V axes and then the same half-wave plate. The total analyzer transformation is $HQ$, with $HQ|R⟩=|H⟩$ and $HQ|L⟩=|V⟩$. The physical fast/slow-axis assignment is selected to realize this stated retardance.

The first-order grating equation is $d\sin \theta =\lambda$, so $d=2.879\mu \mathrm{m}$, about $347$ lines per millimetre. For $N$ equally illuminated lines,Principal maxima stay at the same angles, their peaks scale as $N^2$ for fixed illumination per line, and their angular widths shrink as $1/N$. The resolving power of a diffraction grating is $\lambda /\Delta \lambda =mN$. Thus first order needs $N=1000$ and illuminated width $Nd\simeq 2.88\mathrm{mm}$.

The grating dispersion follows by differentiation: $d\theta /d\lambda =1/(d\cos \theta )$. With a detector centred on the chief ray, one pixel corresponds approximately toThis exceeds the nominal $0.5\mathrm{nm}$ grating width. The practical resolution scale is therefore a few nanometres; two-pixel sampling gives about $4.5\mathrm{nm}$. Exact resolvability also depends on slit width and image sampling. Taking the detector coordinate as $f\tan \theta$ about the original axis instead changes the single-pixel estimate only to $2.20\mathrm{nm}$.

For ideal photon counting, Poisson distribution statistics give SNR $=\sqrt{N_{\gamma }}$. Hence $N_{\gamma }=10^4$ and $t_{\min }=10^4/(2\times 10^5)=0.050\mathrm{s}$.

Use a lock-in amplifier for phase-sensitive detection: chop or otherwise modulate the source before the entrance slit, retaining a synchronous reference. At each grating setting amplify the photodiode output, multiply it by a reference and low-pass filter. For a signal $s\cos ({\omega }_{r}t+\varphi )$ and reference $2\cos {\omega }_{r}t$, the low-pass output is $s\cos \varphi$; quadrature demodulation also recovers $s\sin \varphi$. Choose the modulation frequency above slow drift while within the source/detector bandwidth and away from strong electrical interference. Average for many cycles before changing the grating setting.

This rejects steady backgrounds and reduces low-frequency noise through narrow-band detection. It does not remove random dark-current/stray-light shot noise within that bandwidth, nor a background modulated coherently with the source. The chopping geometry must therefore distinguish wanted light from stray paths.^[ref]

Photon counting registers discrete detector events and their times. A beam splitter sends the selected light to two photodetectors; a coincidence circuit or time-recording system compares the outputs. With a vacuum input at the unused optical port, splitter fractions $R,T$ and independent quantum efficiencies ${\eta }_1,{\eta }_2$, a selected input photon reaches the two recorded channels with probabilities $p_1=R{\eta }_1$ and $p_2=T{\eta }_2$. Conditional on input occupation $N$, each pair of distinct photons contributes $p_1{p}_2$ to the cross-channel pair moment. ThereforeTheir normalized ratio recovers $g^{(2)}(0)$. This assumes number-resolving counts, or sufficiently dilute gates for threshold detectors to approximate them. One photon cannot create both channel counts in this ideal model.

Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

In 1977, H. Jeff Kimble, Mario Dagenais, and Leonard Mandel excited a dilute sodium atomic beam with a resonant dye laser. Apertures restricted the observed atoms; a microscope objective collected fluorescence perpendicular to both beams. A beam splitter sent the collected light to two photomultipliers. Shaped pulses drove the start and stop inputs of a time-to-digital converter; its delay histogram rose away from zero delay, indicating photon antibunching. Turning the atomic beam off measured scattered-laser backgrounds. More than one contributing atom and scattered light limited the comparison with an ideal isolated two-level system; the paper explicitly treated corrections.^[ref]

A simple reason for suppressed immediate re-emission is visible in an ideal two-level system. After spontaneous emission, the atom is in $|g⟩$. Its emission operator ${\sigma }_{-}=|g⟩⟨e|$ obeys ${\sigma }_{-}^2=0$, so two instantaneous emissions are impossible. Excitation must rebuild the excited-state population before another emission. This argument explains the zero-delay idealization; the actual photon counting record additionally depends on collection, background, timing, and the number of contributing atoms.

A photodetector capable of registering an individual absorbed photon as a distinguishable event. The detection record may include a click time and an output-channel label. Such a detector need not resolve the number of photons in a pulse: an ordinary single-pixel single-photon avalanche diode produces essentially the same saturated avalanche pulse when one or several photons trigger the same detection interval. Detector efficiency, dark counts and detector dead time affect the observed photon counting statistics.