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Thermal photon number density is proportional to $T^3$. A sudden photon heating factor $(1+r{)}^{1/4}$ therefore raises photon number by $(1+r{)}^{3/4}$ and dilutes the baryon-to-photon ratio by its inverse when baryon number is conserved. Comparing Big Bang nucleosynthesis and cosmic microwave background baryon-density determinations can test entropy injection between the two epochs, subject to changes in the expansion rate and nuclear abundances.

An experimental physicist who coauthored a 2007 demonstration of progressive quantum measurement of cavity photon number. Successive chemical atoms acquired information from the stored field without absorbing its photons, allowing repeated measurements to narrow the possible photon number.^[ref]

An ideal two-mode intensity-intensity coupling makes the phase of each mode depend on the other's photon number. With this sign convention, evolution for time $t$ sends a mode-1 coherent amplitude to $\alpha e^{-i\chi t{n}_2}$ when mode 2 has definite occupation. This single-mode unitary model does not by itself establish a strong, noiseless interaction between real travelling single photons.

The normalized sum $N_{+}(|\alpha ⟩+|-\alpha ⟩)$ contains only even photon numbers, since the two coherent-state expansions cancel every odd term. It tends to vacuum at small amplitude. With real $\alpha$, its second quadrature can be squeezed below the vacuum variance.

In the 2007 cavity quantum electrodynamics experiment of Christine Guerlin and collaborators, including Sébastien Gleyzes, Michel Brune, Jean-Michel Raimond and Serge Haroche, an injected coherent state occupied a 51.1 GHz microwave cavity with niobium-coated mirrors exhibiting superconductivity. Velocity-selected rubidium chemical atoms were prepared as circular Rydberg atoms in levels with principal quantum numbers 50 and 51. Repeated atomic readouts progressively narrowed the inferred photon number to a Fock state; later records revealed jumps. The cavity damping time was $T_c=0.130\mathrm{s}$. Empty-cavity Ramsey interferometry scans fixed the analysis-phase origin; weak injected fields refined the response calibration. Finite fringe contrast required calibrated likelihoods; photon losses could interrupt convergence.^[ref]

The functional sequence is preparation, first Ramsey interferometry zone $R_1$, dispersive cavity $C$, second zone $R_2$, and state-selective detection. The first $\pi /2$ microwave pulse prepares an atomic quantum superposition; dispersive atom-cavity coupling adds a photon number-dependent relative quantum phase while suppressing energy exchange. The second pulse converts that quantum phase into atomic populations; state-selective field ionization reads $e$ or $g$. Successive chemical atoms carry the meter records away while the same field remains available.^[ref]

This arrangement makes quantum nondemolition measurement useful for following a field repeatedly and preparing a Fock state by conditioning. The photon number is inferred from a sequence of physical records; it is not a directly visible label attached to a photon. Each atomic probe makes a weak, partial measurement in quantum mechanics of photon number, and the accumulated records can approach a resolved count. A single binary meter need not identify one among many possible Fock states, as perfect discrimination of pure states requires orthogonality.

An initially excited atom interacting resonantly with a coherent state has inversion ${\sum }_n{P}_n\cos (2g\sqrt{n+1}t)$, where $P_n$ is its Poisson distribution of photon number. Different Rabi frequencies dephase on a time scale $1/g$ and approximately rephase near $2\pi \sqrt{\bar{n}}/g$. The revival estimate follows from the adjacent-component frequency difference $g/\sqrt{\bar{n}}$. Large mean photon number makes the early oscillations nearly classical without removing the discrete spectrum responsible for revivals.

An experimental physicist whose collaborative cavity experiments explored quantum measurement. In a 2007 study, repeated nondestructive probes progressively determined an initially uncertain photon number, making the gradual emergence of a definite measurement result experimentally accessible.^[ref]

A laser rate equation approximates coupled evolution of population inversion and intracavity photon number, including pumping, spontaneous emission, stimulated emission and optical cavity losses. A simple model is $\dot{S}=[g(N)-1/{\tau }_c]S$, $\dot{N}=R_p-N/{\tau }_u-g(N)S$, where compatible normalizations absorb mode-volume factors. Large excess optical gain drives rapid field growth until inversion depletion reduces it below threshold.

Small perturbations of a class-B laser exchange excitation between the upper-state population and cavity photon number. Linearization about a four-level lasing state gives $\ddot{w}+(r/{\tau }_2)\dot{w}+(r-1)w/({\tau }_c{\tau }_2)=0$. Its damped oscillation frequency is $(2\pi {)}^{-1}\sqrt{(r-1)/({\tau }_c{\tau }_2)-r^2/(4{\tau }_2^2)}$. Long population lifetime compared with photon lifetime permits underdamped relaxation.

An experimental physicist who coauthored measurements of a field stored in a cavity using successive atomic probes. Correlations between repeated readings demonstrated reduced uncertainty in photon number, testing the repeatability aspect of quantum measurement without destroying the field at each observation.^[ref]

The difference $N_{-}(|\alpha ⟩-|-\alpha ⟩)$ contains only odd photon numbers and tends to a one-photon state as $\alpha \to 0$. Exactly at zero amplitude the unnormalized difference vanishes; the normalized limiting state is obtained only by taking a limit.

A single-mode coherent state is an annihilation operator eigenstate:Its photon statistics are Poisson distribution: $⟨n⟩=\mathrm{Var}n=|\alpha {|}^2$. For optical quadratures $X=(a+a^{†})/2$, $P=(a-a^{†})/(2i)$, one has $[X,P]=i/2$ and $\mathrm{Var}X=\mathrm{Var}P=1/4$. Its field mean oscillates classically while retaining vacuum fluctuations.^[ref][ref]

A classical mixture of coherent states has a nonnegative Glauber-Sudarshan representation. A nonclassical state of light cannot be represented that way. Examples are a squeezed vacuum state, with quadrature variances $e^{-2r}/4$ and $e^{2r}/4$, and a nonvacuum number state, with zero photon number variance. Noise below the vacuum level in one quadrature, sub-Poisson number noise, or photon antibunching can witness nonclassicality. For $|n⟩$, $g^{(2)}(0)=1-1/n<1$; coherent light has $g^{(2)}(0)=1$. Thermal light has excess noise but is a classical mixture: noncoherent does not by itself mean nonclassical.

Choose the baseline vector from the first antenna to the second. Apart from a sign convention for receiver delay, the geometric delay is ${\tau }_g=\mathbf{b}\cdot \mathbf{s}/c$, and the phase difference is $\varphi =2\pi \nu {\tau }_g$. For equal received source powers $P_a$, adding the voltages and averaging over many wave periods givesUncorrelated receiver noise adds a mean level but not the source's coherent cross term. For an east-west baseline and one choice of hour-angle sign, ${\tau }_g=(b/c)\cos \delta \sin H(t)$. Earth rotation therefore sweeps the phase; locally the fringe frequency is $(b/\lambda )\cos \delta \cos H\dot{H}$. Tracking the antenna pointing keeps the source in the primary beam. Electronic fringe stopping is a separate operation: it can remove this phase evolution, making a source at the phase centre constant in the corrected complex visibility. The oscillating power above is the geometrical fringe before that correction.

For a small angular displacement $d\theta$ along the projected baseline, $d\varphi =2\pi b_{p}d\theta /\lambda$. One full fringe therefore spans $\Delta \theta \sim \lambda /b_p$. Here $b_p=|\mathbf{b}-(\mathbf{b}\cdot \mathbf{s})\mathbf{s}|$. This is the angular period of the measured Fourier component and the characteristic resolution scale when such long baselines are synthesized; precise beam width and image ambiguity also depend on baseline coverage and weighting. A single pair by itself does not give an unambiguous complete image.

Writing $x=h\nu /(k_{B}T)$, the expansion $e^x-1=x+O(x^2)$ immediately gives the Rayleigh-Jeans law,The factor two counts the two independent modes of polarization of waves. A small brightness-temperature perturbation thus has $\Delta I_{\nu }=2k_B{\nu }^2\Delta T_{\mathrm{RJ}}/c^2$ in this limit.

Hot intracluster electrons inverse-Compton scatter the microwave-background photons. The scattering optical depth is $d\tau =n_e{\sigma }_{T}dl$ and the characteristic thermal fractional energy transfer is of order $k_B{T}_e/(m_e{c}^2)$. The integrated distortion is therefore controlled byScattering conserves photon number while shifting photons toward higher frequencies, giving a low-frequency decrement and a high-frequency increment. Thus the printed estimate correctly gives a negative change of order $y{T}_{\mathrm{CMB}}$. Its numerical coefficient, however, is not the standard Rayleigh-Jeans coefficient. For nonrelativistic electrons and small $y$, the Kompaneets equation gives $\Delta n=y{x}^{-2}{\partial }_x(x^4{\partial }_x{n}_0)$. At small $x$, $n_0\simeq 1/x$, so $\Delta n\simeq -2y/x$ andEquivalently the thermodynamic-temperature spectral factor is $x\coth (x/2)-4\to -2$. The original PDF really prints the estimate with $-y$, not $-2y$; both numerical conventions are retained below. Relativistic corrections at the supplied $T_e$ are small compared with this factor-of-two difference.^[ref]

The uniform cube has a constant line-of-sight depth $L=6\times 10^{22}\mathrm{m}$. Using the supplied cross-section rather than replacing it with a more precise value givesAdopt $T_{\mathrm{CMB}}=2.72548\mathrm{K}$, explicitly an external measured constant, not a temperature supplied in the question. The values $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and $m_e\simeq 9.10938\times 10^{-31}\mathrm{kg}$ set the physical units. Hence the printed relation gives $\Delta T\simeq -22\mu \mathrm{K}$; the standard Rayleigh-Jeans result gives $\Delta T\simeq -44\mu \mathrm{K}$. Keeping the answer as $-8.14\times 10^{-6}{T}_{\mathrm{CMB}}$, or twice that, avoids needing a numerical background temperature.^[ref]

The face is a square, so its solid angle is $\Omega =(6\pi /10800{)}^2=3.046\times 10^{-6}\mathrm{sr}$, not the area of a circular six-arcminute aperture. At $15\mathrm{GHz}$,Thus $\Delta S_{\nu }\simeq -4.67\times 10^{-4}\mathrm{Jy}=-0.47\mathrm{mJy}$ using the printed estimate, or $-0.93\mathrm{mJy}$ using the standard factor two. The sign denotes missing brightness relative to the surrounding microwave background. At this frequency $h\nu /(k_B{T}_{\mathrm{CMB}})\simeq 0.264$; the exact nonrelativistic spectral intensity differs from the strict Rayleigh-Jeans approximation by about one percent. This precision does not affect the estimates.

For the array noise, the system temperature sets the receiver noise scale; let $A_e={\eta }_A\pi d^2/4$ be the effective area of each dish, and let ${\eta }_c$ describe correlation losses. No efficiencies or polarization of waves count are supplied, so first take ${\eta }_A={\eta }_c=1$ and a single independent polarization of waves. An unpolarized point source produces antenna temperature $T_a=S_{\nu }{A}_e/(2k_B)$, and the system-equivalent flux density is $\mathrm{SEFD}=2k_B{T}_{\mathrm{sys}}/A_e$. The interferometric radiometer equation gives one baseline an rms flux error $\mathrm{SEFD}/\sqrt{2\Delta \nu t}$. There are $N(N-1)/2=45$ cross-correlation baselines; averaging their independent thermal noise for a point source givesThe ideal single-dish effective area is $4\pi {\mathrm{m}}^2$, giving $\mathrm{SEFD}=6.59\times 10^3\mathrm{Jy}$ andCombining two independent parallel-hand polarization of waves measurements gives $6.3\mu \mathrm{Jy}$. Nonunit efficiencies increase these estimates by $1/({\eta }_A{\eta }_c)$. Here signal equal to noise means a weak-source signal-to-noise ratio of one after integration; it does not mean equating the source to the entire unaveraged receiver-noise power.^[ref]

At the band centre, $\lambda =0.01999\mathrm{m}$. Taking a representative maximum projected baseline of $20\mathrm{m}$ gives ${\theta }_{\mathrm{res}}\sim 3.4$ arcminutes. A corner-to-corner baseline of $20\sqrt{2}\mathrm{m}$ would give about $2.4$ arcminutes before projection and weighting. The exact antenna locations and source declination are not supplied, so neither a unique synthesized beam shape nor an exact FWHM is determined. The individual-dish primary-beam scale, roughly $\lambda /4\mathrm{m}\sim 17$ arcminutes, is the field of view, not the synthesized resolution.

To extract the cluster signal, retain frequency-resolved complex visibilities and fit an extended SZ template together with contaminating compact sources. For the ideal uniform square centred at the phase centre and with a nearly constant primary beam across it, the Fourier transform isThis is why the integrated decrement must not be treated as a point-source flux on every baseline. The first axial visibility null occurs at $b=\lambda /\Theta \simeq 11.5\mathrm{m}$. Short baselines carry most of the large-scale decrement; the longest baselines are useful for separating unresolved positive radio sources. natural visibility weighting or a suitable taper gives a clearer extended-source image than emphasizing the longest baselines, while a visibility-domain matched filter uses all samples with weights proportional to template amplitude divided by noise variance. Its amplitude estimate is $\hat{a}=\mathrm{Re}(T^{†}{C}^{-1}d)/(T^{†}{C}^{-1}T)$ for a real template amplitude. A joint fit propagates radio-source uncertainty instead of subtracting exact, noise-free source models.

Earth-rotation and multifrequency synthesis improve Fourier coverage, but each channel must use its own baseline in wavelengths, primary beam and SZ spectrum. Use the longer baselines and spectra to identify variable point sources, flag bad data, and test the decrement with time, antenna and frequency splits. Primary-beam and absolute-flux calibration must be included. Large-scale microwave-background fluctuations and residual ground or atmospheric signals can remain even below the thermal-noise estimate. The narrow low-frequency band alone gives limited separation of SZ and primary microwave-background temperature structure. There is no zero-spacing interferometric datum: integrated flux is model dependent unless sufficiently short baselines or additional total-power information are available. Exact cluster detection significance cannot be obtained from the point-source noise alone without the actual projected baseline distribution and weighting.

The beta barium borate crystal supplies spontaneous parametric down-conversion: a weak ultraviolet pump can produce a pair of lower-frequency photons, with their frequencies and wavevectors selected by conservation of energy and phase matching. A suitably aligned and compensated source makes the two polarization alternatives coherent, giving the stated Bell state. The crystal alone does not guarantee that every collected pair has that state.

The two polarizing beam splitters route horizontal polarization and vertical polarization to different ports. Take $H$ from arm $a$ to the trigger $T$ and $V$ to arm $c$. The ordinary beam splitter in arm $b$ is a lossless, polarization-independent 50:50 splitter, with the unused input in the vacuum state. It splits probability amplitudes, rather than measuring which output a photon takes. The half-wave plate changes the relative phase of its principal polarizations by $\pi$; orient it so that a $V$ photon in $c$ becomes $(H+V)/\sqrt{2}$ in $c^{\prime }$, up to a common phase. Finally, photodetectors absorb arriving photons and supply electrical clicks. They need not resolve photon number for the ideal four-photon, fourfold-coincidence calculation: a click at every detector already requires one photon at each.

Choose the relative wave phases so that the final polarizing beam splitter sends $H_{c^{\prime }}$ to $H_2$, $V_{c^{\prime }}$ to $V_1$, $H_d$ to $H_1$ and $V_d$ to $V_2$. Such phase conventions must be used consistently: an uncompensated relative phase would give a phase-shifted GHZ state.

Put $b=\mathcal{E}{e}^{i(kr-\omega t)}$, allowing $\mathcal{E}$ to be complex. The Hermitian electric field is $\hat{E}=b\hat{a}+b^{\ast }{\hat{a}}^{†}$. The annihilation operator and creation operator change photon number, so their diagonal matrix elements in a Fock state vanish. Since $\hat{a}{\hat{a}}^{†}=\hat{n}+1$ and ${\hat{a}}^{†}\hat{a}=\hat{n}$,The terms containing ${\hat{a}}^2$ or ${\hat{a}}^{†2}$ connect orthogonal number states and contribute zero. Even as $n$ grows, there is no definite oscillating mean electric field; its quadrature fluctuations grow instead. A Fock state can represent fixed field energy, but not a classical sinusoidal wave of prescribed wave phase. This is distinct from representing a classical field with a randomly averaged phase.

The photon number probabilities of a coherent state form a Poisson distribution with mean $\mu =|\alpha {|}^2$. Alternatively normal ordering gives $⟨\hat{n}(\hat{n}-1)⟩=⟨{\hat{a}}^{†2}{\hat{a}}^2⟩=|\alpha {|}^4$. ThusThe last ratio is defined only for nonzero $\alpha$. At large $|\alpha |$ the absolute fluctuation grows but its size relative to the mean becomes negligible. This is the deterministic intensity limit within quantum optics; it does not assert the disappearance of vacuum-scale quadrature noise.

Choose the normalization constant real and positive; its otherwise arbitrary phase is physically immaterial. Orthogonality of the Fock states givesActing with the annihilation operator and shifting the summation index givesThus a coherent state is an annihilation-operator eigenket. It is not also an eigenket of the creation operator: a common eigenket would make $[\hat{a},{\hat{a}}^{†}]$ annihilate it, contradicting $[\hat{a},{\hat{a}}^{†}]=I$. Taking the adjoint of the annihilation-operator equation instead gives the distinct, valid bra relationWith $\hat{n}={\hat{a}}^{†}\hat{a}$, the photon number expected value is $\mu =|\alpha {|}^2$. Normal ordering ${\hat{n}}^2=({\hat{a}}^{†}{)}^2{\hat{a}}^2+{\hat{a}}^{†}\hat{a}$ gives $⟨n^2⟩={\mu }^2+\mu$. ThereforeAt large mean photon number the relative fluctuations vanish, giving a classical-field limit even though quantum fluctuations remain. For the vacuum $\mu =0$, the relative ratio itself is undefined.

The Born rule applied to the number-state coefficient gives the Poisson distributionIt depends on the single parameter $\mu =⟨n⟩$, and has equal expected value and variance, consistent with the preceding moments.

For a Fock state $|n⟩$, raising or lowering the number makes an orthogonal state, so the optical quadrature has $⟨\hat{X}⟩=0$. Expanding its square givesThe first two terms have zero diagonal matrix elements, while the last two contribute $n+1$ and $n$. In particular, the vacuum retains quadrature variance $1/2$.

A laser relaxation oscillation is the damped exchange between stored upper-level excitation and intracavity photon number after a small perturbation of a lasing steady state. Extra photons increase stimulated depletion of the upper level; the resulting smaller gain then reduces the photon population. Pumping subsequently replenishes the inversion, so the two populations overshoot their equilibrium values before damping settles them.

Laser spiking is the corresponding large-signal sequence of intense output pulses, often during switch-on. Excitation first accumulates while the optical field is small. Once gain exceeds loss, rapid stimulated emission releases this stored energy in a spike, depleting the inversion below threshold. Recovery can produce further spikes. This is distinct from an externally imposed Q-switching cycle, although both draw on stored excitation.

Many solid-state gain media have upper-state lifetimes far longer than the cavity photon lifetime. The optical polarization relaxes rapidly, leaving the two slow rate-equation variables of a class-B laser. A long-lived population reservoir and a much faster photon response make underdamped recovery and pronounced startup spiking common. The criterion and frequency follow from the linearization in part c; they are not universal consequences of the word “solid-state”.

Take ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. In the resonant Jaynes-Cummings model, the interaction connects only $|e,n⟩$ and $|g,n+1⟩$. Both have free energy $\hslash \omega (n+1/2)$, and their interaction matrix is $\hslash g\sqrt{n+1}{\sigma }_x$. ThereforeExponentiating this two-state block and applying it to $|e,n⟩$ yieldsThe populations undergo Rabi oscillations at angular frequency $2g\sqrt{n+1}$. The common phase in this fixed-$n$ block does not affect its probabilities, but it must be retained when blocks with different photon numbers are superposed.

Let the initial field be the normalized coherent state $|z⟩={\sum }_{n=0}^{\infty }{c}_n|n⟩$, where $c_n=e^{-|z{|}^2/2}{z}^n/\sqrt{n!}$. The atom is again initially excited. Linearity of the Schrödinger equation givesThe different photon numbers have different Rabi frequencies, so the state generally becomes entangled between atom and field. It cannot in general be replaced by one Rabi-oscillating atomic state multiplied by an unchanged coherent state.