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For $\bar{n}=|z{|}^2\gg 1$, the Poisson distribution of photon number has width $\sqrt{\bar{n}}\ll \bar{n}$. Initially the atom therefore undergoes nearly classical Rabi oscillations at angular frequency approximately $2g\sqrt{\bar{n}}$. However, the atomic inversion is a sum over slightly different frequencies:The components dephase, causing collapse of the visible Rabi oscillations on a time scale of order $1/g$; a Gaussian photon-number approximation gives an initial envelope approximately $e^{-g^2{t}^2/2}$. Their discrete phases later rephase, giving revivals nearThe revival estimate follows by requiring the phase difference of neighbouring photon-number components to advance by $2\pi$. These Jaynes-Cummings collapse and revival effects arise from atom-field entanglement and photon-number discreteness. Large amplitude makes the initial classical description accurate over many Rabi periods, but does not eliminate quantum collapse and revival at all later times.

Four complementary classical features of a large-amplitude coherent state are:

These properties identify a particular classical oscillation, rather than a state of definite photon number. Absolute quantum fluctuations remain even when their relative size is small.

From b, the excited-state probability isFor a single short measurement interval $\Delta t$ beginning in $|e,0⟩$, requiring $P_e>0.9$ before the first departure from unity givesA zero-photon outcome resets the conditional state to $|e,0⟩$. After $n=T/\Delta t$ such tests, the probability of the all-zero-photon record isThus a 90 percent survival requirement over a specified total duration $T$ is stronger than the single-interval condition: approximately $\Delta t<-\ln (0.9)/(g^{2}T)$ in the short-interval limit. The limiting preservation as $\Delta t\to 0$ is the quantum Zeno effect.

The measurement instrument matters. If photons are absorbed and their first detection removes the excitation, this survival probability also equals the unconditional excited population. For ideal nondestructive photon-number measurements, excitation can transfer back after a one-photon result; the unconditional result after $n$ equal intervals is instead $P_e^{(n)}=[1+{\cos }^n(2g\Delta t)]/2$. It also tends to one for fixed $T$ as the intervals shrink. Neither interpretation guarantees indefinite 90 percent survival merely by imposing a 90 percent bound on each separate interval.

Discarding the photon-number result gives a classical mixture of the two outcome branches. The atomic state, in the basis $(|e⟩,|g⟩)$, isIt is also the reduced state before measuring the field; a nonselective operation on the field does not change the atomic reduced density matrix. The atom is not in a coherent superposition with these populations.

For an atomic energy gap $\hslash \omega$, the thermal population ratio is $p_e/p_g=e^{-\hslash \omega /(k_{B}T)}$. Matching it givesA finite positive temperature exists when $0<p_e<p_g$, with $T\to 0^{+}$ at $p_e=0$ and $T\to +\infty$ as $p_e\to 1/2$ from below. Equal populations correspond to zero inverse temperature. If $p_e>1/2$, the expression is a negative effective temperature for a bounded two-level system, not an equilibrium state at any ordinary positive temperature. For a concrete counterexample to an unrestricted positive-temperature interpretation, at $t=0$ the atom is certainly excited, whereas every positive-temperature Gibbs distribution has $p_e\le 1/2$. The fully excited endpoint is the formal limit $T\to 0^{-}$. Matching one density matrix does not establish genuine thermalization with a bath.

An ideal projective measurement of the field's photon number selects the coefficient of $|1⟩$. ThereforeWhenever this probability is nonzero, normalization of the projected joint quantum state gives the atomic Bell stateup to a common global phase. The maximum heralding probability is $1/3$, attained at $\sqrt{6}gt=\pi /2+n\pi$. At a time with $P_1=0$, that conditioning event is impossible, so a conditional state is not defined there. The interpretation is a photon number measurement at the selected time; continuous monitoring of escaping photons would instead change the dynamics and require a different model.

An ideal zero-photon number result projects onto the first term in part (b). ConsequentlyThis is a product state, not an entangled success. The remaining ideal outcome hasand leaves $|gg⟩$. In particular, simply discarding the field outcome does not reproduce the pure heralded Bell state.

The practical requirements are strong coherent coupling of both atoms to the same optical normal mode with stable relative coupling phases, preparation of $|ee;0⟩$, and interaction timing on the scale $g^{-1}$ before spontaneous emission or cavity loss destroys quantum coherence. The field must then be measured or extracted after halting the coherent exchange, with enough photon number resolution to distinguish one from two. Otherwise an inefficient detector can mistake a two-photon event for the desired herald. Even ideally the one-photon success probability is at most $1/3$, so trials must be repeated.

“No photons detected” implies $|ee⟩$ only for the ideal zero-number projection specified above. With loss or finite detector efficiency, no click also includes missed one- and two-photon events and generally leaves a conditional mixed state. Detecting a leaking photon during the interaction is also a different measurement protocol from the end-time projective measurement analyzed here.

In Q-switching, hold the optical resonator's quality factor low while pumping builds population inversion, then rapidly reduce the loss. The stored excitation produces a large optical gain, the intracavity light intensity grows, and stimulated emission depletes the population inversion until the optical gain falls below the laser threshold. The resulting optical pulse is a transient discharge of stored excitation.

Its duration depends on the high-quality factor photon lifetime, initial population inversion, saturation dynamics and switching time. For example, a simple laser rate equation model iswhere $S$ is intracavity photon number, $N$ is the available inversion in compatible units, $R_p$ is the pump rate and ${\tau }_c$ is the photon lifetime. During the short discharge, pumping and upper-state decay can often be neglected. The build-up and depletion rates determine the width; there is no universal equality between the width and ${\tau }_c$. The initially stored energy and the output coupling also matter. Typical Q-switching produces nanosecond-scale optical pulses.

Set $A=\alpha a^{†}$ and $B=-{\alpha }^{\ast }a$. Their commutator isIt commutes with both operators, so the stated Baker--Campbell--Hausdorff formula terminates:In particular $e^{-{\alpha }^{\ast }a}|0⟩=|0⟩$. Expanding the remaining exponential and using $(a^{†}{)}^n|0⟩=\sqrt{n!}|n⟩$ gives the number state expansionThe orthonormality of the number states givesThis is a Poisson distribution with parameter $\mu =|\alpha {|}^2$. Its probabilities sum to $e^{-\mu }{e}^{\mu }=1$. The phase of $\alpha$ affects the coherent state's relative probability amplitudes but not its photon number probabilities. At $\alpha =0$ the distribution is concentrated at $n=0$.

The number operator is $\hat{N}=a^{†}a$. The number state expansion gives $a|\alpha ⟩=\alpha |\alpha ⟩$ by shifting the summation index; taking the adjoint gives $⟨\alpha |a^{†}={\alpha }^{\ast }⟨\alpha |$. Thus the expectation value isFor the second moment, use $a{a}^{†}=a^{†}a+I$:Its expectation value in the coherent state is $⟨{\hat{N}}^2⟩=|\alpha {|}^4+|\alpha {|}^2$. ThereforeThe variance equals the mean, exactly as for the Poisson distribution found in b. For $\mu >0$ the fractional standard deviation is $1/\sqrt{\mu }$, the usual photon shot noise scaling. For the vacuum state, both the mean photon number and its variance vanish; this does not imply zero optical quadrature fluctuations. At $\mu =0$ the fractional fluctuation is undefined.

Zero-point fields are the irreducible electromagnetic field fluctuations in the vacuum state. Each normal mode has ground energy $\hslash \omega /2$. Although $⟨\mathcal{E}⟩=⟨\mathcal{B}⟩=0$ in the vacuum state, d gives $⟨{\mathcal{E}}^2⟩=⟨{\mathcal{B}}^2⟩=1$. These are quantum fluctuations, not a classical random coherent field with a known instantaneous value.

Two consequences are the Lamb shift, where coupling to the quantized electromagnetic field changes atomic energy levels, and spontaneous emission, where an excited chemical atom can emit a photon into an initially empty normal mode. In the latter process the creation operator has a nonzero matrix element $⟨1|a^{†}|0⟩=1$; stimulated-plus-spontaneous factors have the form $n+1$, which does not vanish at $n=0$. Descriptions splitting spontaneous emission into vacuum fluctuations and radiation reaction depend on operator-ordering conventions; the measurable total rate does not. Zero-point fields do not mean that the vacuum state contains a nonzero mean photon number.

A degenerate optical parametric amplifier sends a strong pump at $2\omega$ through a phase-matched second-order nonlinear medium, creating pairs in a selected mode near $\omega$. With the pump treated classically and undepleted, the effective interaction is quadratic in the mode's creation and annihilation operators. Vacuum input evolves into an optical squeezed vacuum; the pump phase fixes the quiet optical quadrature. A cavity enhances the interaction below oscillation threshold. balanced homodyne detection compares this optical quadrature's noise with the blocked-signal vacuum reference; phase locking, detector balance, electronic-noise measurement and loss calibration distinguish squeezing from an instrumental noise reduction. A primary cavity experiment observed below-vacuum noise by degenerate down conversion and balanced homodyne readout.^[ref]

Use $S(r)=\exp [\frac{r}{2}(a^2-a^{†2})]$, so $S^{†}aS=a\cosh r-a^{†}\sinh r$. Write $s={\sinh }^{2}r$, $c=\cosh r$ and $N=a^{†}a$. Vacuum expectations then givePair creation makes the mean field zero while the mean photon number is positive. Expanding the transformed ladder operators in the vacuum gives the squeezed-vacuum photon-number moments: $⟨N(N-1)⟩=3s^2+s$ and $⟨N^2⟩=3s^2+2s$; equivalently $\mathrm{Var}N=2s(s+1)$.

The vectors produced by photon addition and photon subtraction in the paper are not normalized. Their squared norms are $s+1$ and $s$, respectively. A particularly direct evaluation isBoth successful operations therefore prepare the same squeezed one-photon state, up to an overall phase, whenever subtraction has nonzero probability. Since $⟨1|S^{†}NS|1⟩=c^2+2s=1+3s$, the unnormalized moments areThe printed photon-added formula equals $M_{+}$, not a normalized mean. With its literal unnormalized convention, the corresponding subtraction result is $M_{-}$ andThis difference tends to one as $r\to 0$ and grows as $\frac{3}{4}{e}^{2r}$ for large positive $r$. Keeping this convention recovers the paper's requested expression without misidentifying it as a physical conditional mean.

For the normalization of photon-added and photon-subtracted states, divide by the squared norms to obtain actual conditional means:These equalities for subtraction require $s>0$. As $r\to 0$ the normalized successful subtraction tends to $|1⟩$, but its success probability vanishes; exactly at $r=0$, $a|0⟩=0$ gives no conditional state. At large $r$, both normalized means grow as $3e^{2r}/4$, while their difference remains zero.

For the photon statistics of a squeezed vacuum, the even-number probabilities areAfter subtraction the normalized distribution is $p_n^{-}=(n+1)p_{n+1}/s$. It has only odd occupation and no vacuum component. photon addition gives $p_n^{+}=n{p}_{n-1}/(s+1)$, identical here to $p_n^{-}$. The diagram uses normalized distributions with an explicitly chosen $r$, rather than treating the unnormalized moments as means.

A subtraction event samples initial occupation with weight $n$, preferentially selecting the larger components before removing one photon. In general ${\bar{n}}_{-}=⟨N(N-1)⟩/⟨N⟩=⟨N⟩-1+\mathrm{Var}N/⟨N⟩$. Here the large variance gives ${\bar{n}}_{-}-⟨N⟩=1+2s>0$. This conditional photon-number bias does not create energy by removing a photon: the unconditional ensemble also includes the runs without a heralded event.

An ideal cross-Kerr interaction between two single modes is $H_{\mathrm{int}}=\hslash \chi n_1{n}_2$, with $n_j=a_j^{†}{a}_j$. The number operators commute with the interaction, so this coupling changes phases without transferring photons between the modes. To implement the specified positive conditional phase, usewhere $n_{2V}=|V⟩⟨V|$ within the one-photon polarization subspace. The horizontal branch has zero interacting occupation. Since $e^{i\Theta n}|\alpha ⟩=|e^{i\Theta }\alpha ⟩$, the specified conditional rotation follows. This is an ideal lossless single-mode model; an actual travelling-wave medium's bandwidth, loss and noise are not established by writing this Hamiltonian.

Use $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$. A coherent state has centre $(\mathrm{Re}\alpha ,\mathrm{Im}\alpha )$ and variance $1/4$ in either optical quadrature. Multiplication of $\alpha$ by $e^{i\Theta }$ rotates that centre anticlockwise through $\Theta$, keeping the coherent state noise disk unchanged. For real positive $\alpha$ the two centres are $(\alpha ,0)$ and $(\alpha \cos \Theta ,\alpha \sin \Theta )$.

Normalize the initial polarization of an electromagnetic wave qubit. The output of the medium isThe joint state is generally an entangled state. The overlap magnitude is $|⟨\alpha |\beta ⟩|=e^{-2|\alpha {|}^2{\sin }^2(\Theta /2)}$. The state factorizes when the amplitudes coincide, while nearly distinguishable amplitudes make the polarization nearly maximally entangled with the optical mode. Before any polarization conditioning, tracing out mode 2 gives the mixture $(|\alpha ⟩⟨\alpha |+|\beta ⟩⟨\beta |)/2$, with no interference between branches.

Include the normalization factors in the half-wave plate map: $|H⟩\mapsto (|H⟩+|V⟩)/\sqrt{2}$ and $|V⟩\mapsto (-|H⟩+|V⟩)/\sqrt{2}$. The transformed joint state isConditioning on a vertical result and setting $\Theta =\pi$ therefore preparesThis pure optical cat state is the conditional output; it is not the unconditioned reduced state immediately after the Kerr medium. The measurement erases the polarization label, allowing the two amplitudes to interfere. The calculated optical Wigner functions below display the resulting fringes; their negative regions are quasiprobability, not negative detection probabilities. These illustrative plots choose $\alpha =1.15$: the rotation panel uses $\Theta =2\pi /3$, while the cat panels use the conditional Kerr rotation $\Theta =\pi$ and three different relative phases $\phi$.

The coherent-state overlap follows by expanding in the Fock basis:Thus $s=⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$ andThe postselection probability for Kerr-cat preparation includes both the normalized input qubit and the wave plate. At $\alpha =0,\varphi =\pi$ the vertical branch has zero probability and no normalized conditional state. For the remaining calculation take $\alpha$ real, as required.

Using $a|\pm \alpha ⟩=\pm \alpha |\pm \alpha ⟩$, put $D=1+s\cos \varphi$. Directly evaluating diagonal and cross terms givesThe last quantity is the mean photon number in the conditional cat, which differs from the input coherent state mean ${\alpha }^2$. Since $a{a}^{†}=n+1$,However, its first moment iswhich is generally nonzero. The original PDF's assertion of a zero mean for arbitrary $\varphi$ is inconsistent with its stated pure state. Consequently its displayed expression is the second moment, not the general variance. The phase-dependent cat-state quadrature variance isThe printed result does become the variance for $\sin \varphi =0$, notably the even and odd cat states. It could also describe an ensemble mixing equal weights of $+\varphi$ and $-\varphi$: that different state's mean vanishes while its second moment is unchanged. Such phase averaging is not part of the specified pure-state preparation.

For nonzero finite $\alpha$, $0<s<1$ and $D>0$. The correct quadrature squeezing condition in this quadrature is thereforeFor a representative phase in $-\pi <\varphi \le \pi$, this is $|\varphi |<\arccos (-e^{-2{\alpha }^2})$. If instead one applies the printed second moment as a variance, the resulting condition is $\cos \varphi >0$, or $|\varphi |<\pi /2$ modulo $2\pi$. That is sufficient, but not necessary, for the actual pure state's squeezing. For example, at $\varphi =\pi /2$, $⟨X_2^2⟩=1/4$ while $\mathrm{Var}{X}_2=1/4-{\alpha }^2{e}^{-4{\alpha }^2}<1/4$. This explicitly tests the difference.

For a fixed phase other than $\pi$ modulo $2\pi$, expandAs the input mean number ${\alpha }^2\to 0$, the normalized state tends to vacuum, $⟨X_2⟩\sim -\alpha \tan (\varphi /2)$ andThis expansion is nonuniform near the odd-cat phase. At exactly $\varphi =\pi$ the vacuum terms cancel and the normalized odd coherent-state superposition tends to $|1⟩$, giving $\mathrm{Var}{X}_2\to 3/4$, while its postselection probability tends to zero. The even coherent-state superposition at $\varphi =0$ instead contains only even photon numbers and tends smoothly to vacuum.

At large input mean number, $s\to 0$, the two coherent state branches become nearly orthogonal, and $\mathrm{Var}{X}_2\to 1/4$ for every fixed phase. The mean tends to zero as well. This does not make the state an incoherent mixture: phase-space interference still distinguishes the ideal quantum superposition, although its contribution to this low-order moment is exponentially small. The loss of coherence in an optical cat state arises when ordinary loss reveals the branch to the environment and suppresses that interference, with factor $e^{-2(1-\eta ){\alpha }^2}$ for transmissivity $\eta$.

For the especially simple even cat,It starts at $1/4$, falls to about $0.1108$ at $|\alpha |\simeq 0.7995$, and returns towards $1/4$ from below. The odd-cat curve decreases from $3/4$ towards $1/4$ from above. Other fixed-phase curves have the small- and large-amplitude limits derived above; for negative $\cos \varphi$, squeezing ends when $e^{-2{\alpha }^2}=-\cos \varphi$. In balanced homodyne detection, centre each quadrature distribution on its measured mean before inferring its variance.

Assume the quoted intensities represent relative emitted photon numbers, with efficiency corrections implicit, as required for population-flow bookkeeping. The prominent branch obeys $I_1=I_2+I_4$, and $I_2=I_3$ identifies a serial two-photon branch. The energies obey $E_2+E_3\simeq E_4$ and $E_1+E_4\simeq E_5$ within their uncertainties. The weak fifth line directly spans D to A. Reversing which end contains the line-1 step and interchanging the serial line-2/line-3 order produces the four possibilities.

Take A as zero. A gamma-ray energy-network fit to the five energy differences gives the following approximate excitation energies in keV:The line connections, in the order $1,2,3,4,5$, are respectively: I: $(B\to A,C\to B,D\to C,D\to B,D\to A)$; II: $(B\to A,D\to C,C\to B,D\to B,D\to A)$; III: $(D\to C,B\to A,C\to B,C\to A,D\to A)$; IV: $(D\to C,C\to B,B\to A,C\to A,D\to A)$.

For I and II, the B population entering through lines 2 or 3 and line 4 exits through line 1. For III and IV, line 1 feeds C and its intensity splits into the direct line 4 and serial lines 2/3. The small direct D-to-A branch bypasses these intermediate balances. The central measured energies do not obey every sum exactly; all four fitted arrangements have ${\chi }^2\simeq 0.81$ for two energy constraints. Assigning the weakest line's central energy as an exact excitation energy would discard the more precise cascade measurements.

Each electromagnetic normal mode is a quantum harmonic oscillator with $E_n=(n+1/2)h\nu$. For $x=h\nu /(k_{B}T)$ its canonical partition function and occupation probabilities areThe geometric distribution is normalized and gives thermal mode energy $h\nu /(e^x-1)$. The zero-point term $h\nu /2$ is not part of the thermally emitted black-body intensity. The equilibrium photon chemical potential is zero because photon number is not conserved.

Counting two transverse polarizations gives $8\pi {\nu }^2/c^3$ modes per volume per frequency interval. Thus the spectral energy density is $u_{\nu }=8\pi h{\nu }^3/[c^3(e^x-1)]$. In an isotropic radiation field, spectral radiance is $B_{\nu }=c{u}_{\nu }/(4\pi )$. Transforming the spectral measure, $B_{\lambda }=B_{\nu }|d\nu /d\lambda |$, givesHere intensity means radiance per wavelength and solid angle; the hemispheric spectral exitance of a black surface is $M_{\lambda }=\pi B_{\lambda }$. The extra Jacobian is essential: the wavelength peak is not found by simply converting the frequency peak using $\lambda =c/\nu$.

At short wavelength the exponential suppresses radiation. At long wavelength the Rayleigh-Jeans limit is $B_{\lambda }\simeq 2c{k}_{B}T/{\lambda }^4$; using classical equipartition at every frequency instead would produce the ultraviolet catastrophe. The wavelength maximum solves $5(1-e^{-x})=x$, with nonzero root $x=4.9651$, yielding Wien's displacement law. For the specified temperature,Integrating the hemispheric flux gives the Stefan-Boltzmann law, $M=\sigma T^4\simeq 3.01\times 10^{-6}\mathrm{W}{\mathrm{m}}^{-2}$. The plot uses the specified ideal temperature, rather than substituting a later measured value. The physical black-body description of the cosmic microwave background is tested by comparing the sky spectrum with an absolute reference blackbody and separating instrumental and Galactic contributions.^[ref]

Equilibrium radiation and cavity frequencies. A blackbody absorbs all incident radiation at every frequency and direction. In thermal equilibrium its emission spectrum depends only on its temperature. Radiation inside an isothermal absorbing cavity, sampled through a small aperture that hardly disturbs equilibrium, realizes black-body radiation to a good approximation. Microscopic absorption and emission establish the equilibrium distribution; a perfectly reflecting wall is the convenient idealization used to count the modes once that distribution is specified.

In a rectangular electromagnetic cavity, the wave equation and perfect conductor wall boundary conditions select component wavenumbers $k_x=\pi l/L_x$, $k_y=\pi m/L_y$, $k_z=\pi n/L_z$. Combining the vacuum dispersion relation $\omega =c|\mathbf{k}|$ and $\nu =\omega /(2\pi )$ givesFor example, electric-field components can have the spatial dependenceFor three positive indices, Gauss's law requires $k_x{A}_x+k_y{A}_y+k_z{A}_z=0$, leaving two independent polarizations of an electromagnetic wave.

Index qualification. The original paper describes the indices as positive integers. That describes the three-nonzero-index family but omits valid electromagnetic modes with exactly one zero index, which have one polarization of an electromagnetic wave. For instance, with $n=0$, $l,m\ge 1$, only $E_z$ above remains; it is normal to the top and bottom faces and vanishes at the side walls. Two zero indices give no nontrivial cavity field. The omitted one-zero-index family does not affect the leading three-dimensional mode count, but is essential to the thin-cavity limit below. Retaining the paper's strict positivity restriction for every mode would instead give a thickness gap and would not yield its requested two-dimensional power law.

Three-dimensional mode count. A lattice cell in positive $\mathbf{k}$ space has volume ${\pi }^3/V$, where $V=L_x{L}_y{L}_z$. In a continuum approximation, a shell of radius $k$ and thickness $dk$ contains its positive octant, multiplied by two polarizations:Using $k=2\pi \nu /c$ givesFor a finite cavity the exact density of states is discrete; this formula is its bulk continuum limit, appropriate when many modes contribute and wall corrections are small.

Mean photon number. A mode is a quantum harmonic oscillator with thermal excitation energies $sh\nu$, $s=0,1,2,\dots$. The zero-point energy does not enter its thermal occupation. Put $q=e^{-h\nu /(k_{B}T)}$. Its partition function and mean occupation areThis is the Bose-Einstein distribution with zero photon chemical potential: equilibrium absorption and emission do not conserve photon number.

Thermal energy. Multiply the density of states by the mean excitation energy $h\nu ⟨s⟩$ and integrate:The substitution $x=h\nu /(k_{B}T)$ isolates all the dependence on cavity dimensions and temperature:Expanding $1/(e^x-1)={\sum }_{r=1}^{\infty }{e}^{-rx}$ gives an integral ${\sum }_{r}6/r^4=6\zeta (4)={\pi }^4/15$. HenceThis is the thermal energy only, excluding the zero-point contribution. The three-dimensional continuum regime requires $k_{B}T$ large compared with the geometric level spacings in all three directions.

Spectral sketch. The spectral energy density per unit ordinary frequency is Planck's law,At small frequency it rises as $8\pi k_{B}T{\nu }^2/c^3$, while at large frequency it decays exponentially. Raising temperature increases the spectrum at every fixed positive frequency, moves the peak to larger frequency in proportion to $T$, and increases the integrated area in proportion to $T^4$. A plot per unit wavelength has a different peak position because the integration variable changes.

Thin-cavity radiation. Let $A=L_x{L}_y$. The relevant low-temperature window isNonzero thickness indices are then thermally suppressed, while many in-plane modes are populated. The $n=0$ family has one polarization of an electromagnetic wave. Its allowed in-plane wavenumbers form a lattice of cell area ${\pi }^2/A$; a quarter of a circular shell therefore containsThus the dimensional crossover of thermal cavity radiation givesThe integral is $2\zeta (3)$ by the same geometric-series argument. A model with a different number of independent polarizations of an electromagnetic wave changes the prefactor but not the $T^3$ scaling. For a cavity with fixed finite $L_x,L_y$, cooling still further eventually freezes its lowest in-plane mode as well: the strict $T\to 0$ thermal energy is exponentially suppressed. The requested area law is the intermediate quasi-two-dimensional regime, not an arbitrarily low-temperature continuum limit.

A photon is one quantum of the electromagnetic field. In vacuum it has zero rest mass, energy $E=h\nu =\hslash \omega$, and momentum magnitude $p=E/c$. Its two physical polarization degrees of freedom are transverse to its propagation direction. A monochromatic field mode can contain zero, one or several photons; its photon number distinguishes these quantum states.

The photoelectric effect illustrates a photon's discrete energy exchange: increasing incident frequency increases the available electron kinetic energy, whereas increasing light intensity primarily increases the rate of events in the weak-illumination regime. Compton scattering additionally tests the associated momentum transfer. Interference still depends on probability amplitudes, so describing light in terms of photons does not discard its wave behavior.

An explicit original ideal meter model explains how photon number survives while field quantum coherence changes. Let $\hat{N}$ be the number operator and prepare an atomic qubit in $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. The dispersive unitary operatorcommutes with $\hat{N}\otimes I$. For a field quantum superposition ${\sum }_n{c}_n|n⟩$, it produces ${\sum }_n{c}_n|n⟩|a_n⟩$, where $|a_n⟩=(e^{-in\chi /2}|e⟩+e^{in\chi /2}|g⟩)/\sqrt{2}$. The atomic meter states encode $n$ through relative quantum phase and can become entangled with the field. The second Ramsey interferometry pulse and detector turn those correlations into the outcome probabilities above when $C=1$.

Tracing out the atom multiplies a field density matrix entry ${\rho }_{nm}$ by $⟨a_m|a_n⟩=\cos ((n-m)\chi /2)$. Hence the nonselective quantum measurement preserves every photon number population ${\rho }_{nn}$ while changing off-diagonal quantum coherence. For $\chi =\pi /2$, coherence between $|0⟩$ and $|1⟩$ is multiplied by $1/\sqrt{2}$ per fresh meter, even with no photons absorbed. Field quantum phase information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.

One possible finite-contrast quantum instrument makes the distinction sharper. Define diagonal Kraus operators $K_e=\cos ((\varphi +\chi \hat{N})/2)$ and $K_g=\sin ((\varphi +\chi \hat{N})/2)$. After their ideal readout, flip the reported label with probability $r=(1-C)/2$. The unnormalized selective maps areTheir traces reproduce $L_e,L_g$ on a Fock state, and their sum reproduces the displayed dephasing quantum channel, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause measurement backaction. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal likelihoods alone do not determine the disturbance.

The single-mode squeeze operator changes occupation in pairs. Consequently an optical squeezed vacuum has only even photon numbers, with mean $s={\sinh }^{2}r$ and variance $2s(s+1)$. Its large number fluctuations are compatible with quadrature squeezing: they concern different observables.

Ideal conditional application of an annihilation operator, defined only for positive initial mean photon number. Number-dependent weighting means that the conditional mean need not fall. Vacuum input gives the zero vector and therefore no successful conditional state.

An ideal measurement preserving the measured eigenspaces. In the projective setting, the selective instrument $\rho \mapsto P_j\rho P_j$ preserves every vector within outcome $j$, including coherence inside a degenerate eigenspace. Repeated measurements agree in the absence of intervening dynamics changing the measured quantity. Measuring a finer observable can destroy degeneracy coherence and need not realize the same Lüders rule instrument.

A physical measurement interaction must correlate the system with a meter; in Haroche cavity photon nondemolition experiment that meter is a chemical atom. For a measured observable $A$, a useful sufficient ideal condition is $[A,H_{\mathrm{int}}]=0$ together with $[A,H_0]=0$, where $H_{\mathrm{int}}$ is the interaction Hamiltonian operator and $H_0$ the free Hamiltonian operator. These conditions protect its eigenvalues during probing and between probes. They do not protect every other observable, nor prevent conditional quantum state changes. In particular, preserving photon number can disturb the field's quantum phase.