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An experimental physicist who coauthored a quantum nondemolition measurement of cavity photon number. Atomic probes crossed the cavity one at a time; field-induced changes in their internal evolution revealed the stored field while preserving it for further probing.^[ref]

Consider an original balanced-readout model, rather than a fit to the experiment. Let $n\in \{0,\dots ,N_{\max }\}$, with a normalized prior probability $p_0(n)\ge 0$ and ${\sum }_n{p}_0(n)=1$. A detected chemical atom has outcome $s\in \{e,g\}$. Define the conditional probabilitiesHere $\chi$ is the accumulated quantum phase per photon, $\varphi$ the controlled analysis quantum phase, and $C$ the fringe contrast. The two probabilities sum to one for every $n$. Actual calibration may also require an offset and asymmetric detection errors; $C$ alone does not specify a quantum instrument.

Assume no photon number change during the record and fresh independently prepared atomic meters. For observed outcomes $s_1,\dots ,s_k$ at recorded settings ${\varphi }_1,\dots ,{\varphi }_k$, the Born rule and Bayes' theorem giveThe posterior probability is defined only for $Z_k>0$. Summing its numerator over $n$ proves normalization. This is also the update of diagonal elements of a field density operator under a measurement in quantum mechanics; it does not assume that an initial quantum superposition was an unknown pre-existing integer. The observed settings are part of the record, especially when selected adaptively.

For an exact finite example take $p_0=(1,1,1,1)/4$, $C=4/5$, and $\chi =\pi /2$. An $e$ outcome at $\varphi =0$ has likelihood $(9,5,1,5)/10$, giving $p_1=(9,5,1,5)/20$. Another $e$ at $\varphi =-\pi /2$ gives $p_2=(9,9,1,1)/20$. A subsequent $g$ at $\varphi =0$ givesThus three atomic records raise the posterior probability of $n=1$ from $1/4$ to $45/68$, while leaving finite uncertainty. No one of these outcomes identifies $n$ with certainty.

The periodic likelihood function imposes an identifiability limit. If $\chi =2\pi /q$ for an integer $q>0$, then $L_s(n+q;\varphi )=L_s(n;\varphi )$ for every setting: the posterior odds between $n$ and $n+q$ never change. Varying $\varphi$ can resolve a cosine reflection ambiguity within a period, but cannot remove this exact period. Changing $\chi$ can supply additional information. A finite cutoff is therefore a physical assumption to justify, not evidence that excluded high photon numbers were absent. At $C=0$ no outcome changes the prior probability; at $0<C<1$ neither binary outcome has an exactly zero likelihood.

The no-loss assumption also has a time limit. In an original zero-temperature loss model, let $\kappa =1/T_c$. A Fock state $|n⟩$ loses its first photon at rate $n\kappa$, so its survival probability is $\exp (-n\kappa t)$ and its mean lifetime is $T_c/n$ for $n>0$. Several records must be acquired within that lifetime to identify a plateau. When losses matter, first predict $p_k^{-}(n)={\sum }_m{T}_{nm}(\Delta t)p_{k-1}(m)$ using a normalized transition matrix, then apply the atomic Bayesian inference update to $p_k^{-}$. Thermal excitation requires upward transitions too. A delayed or broadened inference after a jump need not indicate that an atom caused the jump.

Serge Haroche developed experiments that probe photons stored in a cavity without immediately absorbing them. Passing chemical atoms interact with the cavity field and carry information about its photon number to an external detector.

Repeated nondestructive probes can progressively narrow the possible photon number, allowing the evolution of a measurement result to be followed. The approach separates the system carrying the stored energy from the successive particles carrying information about it, enabling experimental studies of quantum measurement and the quantum-to-classical transition.^[ref]

A single-photon detector based on a reverse-biased photodiode operated above its breakdown voltage. A carrier generated by photon absorption can trigger a self-sustaining impact ionization avalanche. Quenching lowers the bias to stop that avalanche; recovery creates detector dead time. Random internally generated carriers can also trigger indistinguishable pulses, producing dark counts. Single-photon avalanche diodes ordinarily detect an event rather than resolve the incident photon number. ^[ref]

A single-photon state lies in the one-excitation sector of the quantized electromagnetic field. It can be a quantum superposition over spatial ports or temporal modes while its total photon number remains one. Squared coefficients give exclusive event probabilities only when the corresponding wavepacket modes are orthonormal; overlapping temporal histories must be added coherently.

For two equally coupled atoms initially in $|ee;0⟩$, excitation conservation confines the interaction picture to $|ee;0⟩$, $(|eg⟩+|ge⟩)|1⟩/\sqrt{2}$ and $|gg;2⟩$. The Hamiltonian operator divided by $\hslash$ has off-diagonal entries $\sqrt{2}g$ and $2g$. Its eigenvalues are $0,\pm \sqrt{6}g$, so the amplitudes are $(2+\cos \sqrt{6}gt)/3$, $-i\sin (\sqrt{6}gt)/\sqrt{3}$ and $\sqrt{2}(\cos \sqrt{6}gt-1)/3$. An ideal one-photon number projection therefore heralds a symmetric atomic Bell state with probability ${\sin }^2(\sqrt{6}gt)/3$, at most $1/3$.

A vacuum fluctuation is a nonzero quantum variance in a vacuum state. For a single electromagnetic normal mode, dimensionless optical quadratures $a+a^{†}$ and $-i(a-a^{†})$ have zero means and unit variances in the vacuum state, despite zero photon number. The mode's zero-point energy is $\hslash \omega /2$.