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A registered detector event occurring without an incident signal photon. For a single-photon avalanche diode, internally generated carriers can trigger the same avalanche as an absorbed photon. The dark-count rate $r_{\mathrm{dark}}$ is the mean number of these events per unit time under specified operating conditions. Such events can contaminate coincidence counting and increase the quantum bit error rate in quantum key distribution.

Eduardo García-Toraño is a physicist whose work includes precision measurements of radioactive decay. His 2010 coauthored study of fluorine-18 lists an affiliation with an ionizing radiation metrology laboratory in Madrid. The experiment followed radioactive decay using three systems: ionization chambers, germanium detectors, and coincidence counting with fast scintillators. Measurements extended over as many as nine half-lives, providing several instrumental routes to the same physical parameter.^[ref]

The combined result was a half-life of $1.82871(18)\mathrm{h}$, where the parenthetical notation represents $0.00018\mathrm{h}$ in the last quoted digits. This is the result reported by that experiment; the distinction matters when comparing a historical measurement with a later evaluated half-life.^[ref]

A simple model illustrates why the long observation interval is demanding. Suppose a detector's observed event rate iswhere $\lambda$ is the radioactive decay constant and $B$ is a constant instrumental background. This illustrative model assumes stable detection efficiency and background. After nine half-lives, the decaying contribution is $R_0/512$. An initial contribution of $512$ events per second would therefore fall to just $1$ event per second: even a small background can then noticeably distort an exponential decay fit if it is omitted. Independent detector systems make a valuable comparison because their instrumental backgrounds and responses need not behave identically.

Fluorine-18 is a radioactive isotope of fluorine with 9 protons and 9 neutrons. It decays to oxygen-18, mainly by emitting a Positron and also by electron capture. Its approximately 110-minute half-life makes its radioactivity change appreciably over an ordinary laboratory session. It is used in positron emission tomography.^[ref]

The half-life can be measured by following the falling radioactivity, rather than by watching one nucleus wait to decay. A 2010 experiment used ionization chambers, germanium detectors and coincidence counting with fast scintillators, following decay for up to nine half-lives. The combined result was $1.82871(18)\mathrm{h}$. Comparing different detector systems tests whether the inferred decay rate depends on one measurement technique.^[ref]

For a closed sample governed only by radioactive decay, the surviving parent fraction is $N(t)/N(0)=2^{-t/T_{1/2}}$. Taking 110 minutes as an illustrative rounded half-life, 220 minutes leaves one quarter of the original fluorine-18 nuclei. Four half-lives leave $1/16$, not zero. This calculation concerns nuclear survival; movement of a labelled molecule into or out of an observed region is an additional process.

Coauthored the electron antineutrino experiment using coincidence counting in layered scintillators, with prompt positron annihilation followed by delayed neutron capture in cadmium.^[ref]

Use the two-pair component of polarization-entangled down-conversion to account for identical bosons in the same collected modes. Write $a_H^{†}$ for a creation operator in mode $a$ with horizontal polarization, and similarly for the other modes. For the plus Bell state, the pair-creation operator is $K^{†}=a_H^{†}{b}_V^{†}+a_V^{†}{b}_H^{†}$. The normalized four-photon state isThese are normalized Fock states. In particular, $(a_H^{†}{)}^2|0⟩=\sqrt{2}|2H_a⟩$; omitting this factor would misnormalize the four-photon input.

The first polarizing beam splitter sends $a_H^{†}$ to $T_H^{†}$ and $a_V^{†}$ to $c_V^{†}$. Only the middle term can give exactly one trigger photon and leave another photon in $c$. In the first term both $a$ photons reach $T$ and none reaches $c$; in the last there is no trigger. Thus the relevant, still unnormalized component is

For the selected paths of the half-wave plate and beam splitter, takeHere $e$ labels the output towards $D_3$. An immaterial common reflection phase has been absorbed in the mode definitions. Keeping one photon at $D_3$ leavesAt the final polarizing beam splitter, the $H_{c^{\prime }}{H}_d$ alternative yields one $H$ at each of $D_1,D_2$, while $V_{c^{\prime }}{V}_d$ yields one $V$ at each. The two opposite-polarization alternatives send both photons to the same detector and fail fourfold coincidence counting. The accepted component is thereforeAfter conditioning and normalization, the three detected output modes haveThe optical mode matching must erase any timing, frequency or other record that distinguishes the two alternatives. Otherwise the same polarization populations could describe an incoherent mixed quantum state, rather than this quantum superposition. The Bouwmeester three-photon GHZ experiment uses this principle. The photonic GHZ preparation by fourfold coincidence is a measurement in the coincidence basis: detecting all four photons projects and consumes them. A click at $T$ by itself does not herald a guaranteed freely available three-photon state.

Walther Bothe developed the coincidence counting for connecting signals from separate particle detectors. Requiring pulses within a short time window helps distinguish a common physical event from unrelated counts. For independent rates $r_1$ and $r_2$, accidental coincidences scale with $r_1{r}_2$ and the window duration; narrowing that duration therefore reduces a calculable background. The method turns timing information into evidence about particle paths and correlated processes, rather than relying on the count in one detector alone.^[ref]

A physicist.^[ref]