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For a product of bosonic number states, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $⟨a^{†2}{a}^2⟩=N(N-1)$, while Wick theorem would give $2N^2$. Subtracting the excess yields the displayed exact two-point correction. At higher order the occupation decrements carry falling factorials. This distinction is essential when comparing a fixed-number condensate with independent thermal modes.

For a coherent state, write $a|\alpha ⟩=\alpha |\alpha ⟩$ and $|\alpha ⟩={\sum }_n{c}_n|n⟩$ in number states. The annihilation operator gives $c_{n+1}\sqrt{n+1}=\alpha c_n$, so $c_n=c_0{\alpha }^n/\sqrt{n!}$. Normalization yields $|c_0{|}^2=e^{-|\alpha {|}^2}$. Thus photon statistics follows a Poisson distribution with $\mu =|\alpha {|}^2$, and the normal ordering of every moment gives $⟨(a^{†}{)}^k{a}^k⟩={\mu }^k$. This factorization is the defining virtue of coherent states for optical calculations.^[ref]

As an original comparison, choose $\mu =2$. Then $p_0=e^{-2}$, $p_1=p_2=2e^{-2}$, $\mathrm{Var}N=2$, and $⟨N(N-1)⟩=4$. A two-photon Fock state also has mean two but has $\mathrm{Var}N=0$ and pair moment two. ConsequentlyIn particular, $|1⟩$ cannot supply a pair. Nevertheless, photon statistics alone does not determine a quantum state: averaging $|\alpha ⟩⟨\alpha |$ over a uniform quantum phase removes its off-diagonal entries while retaining exactly the same $p_n$ and these equal-time moments.

A single-component chaotic Gaussian bosonic field has normalized coincident normally ordered correlation $n!$: every pairing contributes the same product of densities. A same-component fermionic field instead has zero coincident correlations for $n\ge 2$ by exclusion. These are state-dependent statements: a coherent bosonic field has factor one, and a single-mode number state has $N^{\underline{n}}/N^n$. Ratios to distant unnormalized correlations additionally require the compared densities to agree and distant correlations to factorize.

A concrete quantum quench experiment loaded a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ chemical atoms from a magnetic field trap into an optical lattice: three orthogonal $838\mathrm{nm}$ laser standing waves, with approximately $125\mu \mathrm{m}$ beam waists. An $80\mathrm{ms}$ loading ramp preceded a $50\mu \mathrm{s}$ increase in lattice depth. This suppressed quantum tunnelling while retaining each well's vibrational ground state. The switch preserved atom-number populations rather than freezing every motional degree of freedom.^[ref]

After variable hold times, releasing the traps and taking photon absorption images after $16\mathrm{ms}$ of flight revealed disappearance and return of interference peaks. Background-subtracted peak counts supplied a coherent fraction. Varying final depth tested the predicted revival period $h/U$; varying initial depth tested the collapse's sensitivity to atom-number statistics. Peak broadening outside counting windows and spatial variation of $U$ limited later signals.^[ref]

Here $U$ is the on-site interaction energy. The approximate isolated-well Hamiltonian operator $H=U\hat{n}(\hat{n}-1)/2$, with number operator $\hat{n}$, gives different quantum phases to its number states, which later rephase. This is a many-level realization of the distinction between fixed populations and changing quantum phases.^[ref]

Electron shelving distinguishes an internal quantum state outside a cycling optical transition from one that scatters many photons. In the mercury-ion example in David J. Wineland's lecture, $S\leftrightarrow P$ excitation produces $194\mathrm{nm}$ fluorescence; transfer to a metastable atomic level $D$ suppresses it. A photomultiplier records bright/dark intervals. Coherent $282\mathrm{nm}$ interrogation precedes detection to avoid perturbation by the detection beam.^[ref]

“Dark” means weak scattering under a particular detection setting. The motional ground state is dark to an ideal red motional sideband for a different reason: there is no lower number state. It can still fluoresce strongly on a cycling detection transition. Shelving's population readout also need not preserve a quantum superposition's quantum coherence; scattered photons carry information about the internal state, and their recoil can disturb motion.

Retain the first $N\ge 2$ number states of a quantum harmonic oscillator. Let ${\Pi }_N$ be the orthogonal projection onto their span and compress $X_N={\Pi }_{N}X{\Pi }_N$, $P_N={\Pi }_{N}P{\Pi }_N$, restricted to this span. The compressed annihilation operator $a_N$ satisfiesFor $n<N-1$, the two products give $n+1$ and $n$; on the top state they give $0$ and $N-1$. Thus, with $\ell =\sqrt{\hslash /(m\omega )}$,On every quantum superposition of $|0⟩,\dots ,|N-2⟩$, this commutator acts exactly as $i\hslash I_N$. For a normalized retained state with coefficients $c_n$,Small probability at the cutoff must therefore be judged relative to $1/N$ for this expectation. For $N=8$, the diagonal of $[X_N,P_N]/(i\hslash )$ is $(1,1,1,1,1,1,1,-7)$, with zero trace.

This is unavoidable for finite matrices: trace of a matrix commutator gives $\mathrm{tr}(XP-PX)=0$, whereas $\mathrm{tr}(i\hslash I_N)=i\hslash N\ne 0$. Compression does not preserve products: ${\Pi }_{N}XP{\Pi }_N$ can use intermediate states outside the retained span, while $X_N{P}_N$ cannot. Accordingly, compressing the full oscillator Hamiltonian operator gives ${\Pi }_{N}H{\Pi }_N=\hslash \omega (a_N^{†}{a}_N+I_N/2)$, but building it from the compressed squares givesThe two prescriptions agree on lower number states and disagree at the boundary. In the infinite oscillator, the canonical commutation relation holds on the common invariant dense subspace of finite number state combinations. Its unbounded position and momentum operators do not admit the finite trace argument; writing the formal identity does not assert that both products act on every vector of the Hilbert space.

In 1925, Werner Heisenberg proposed replacing classical orbital quantities with arrays of transition amplitudes and frequencies. His paper's opening demanded relations between quantities observable in principle; section 1 built products by summing over an intermediate state and noted their dependence on order. The formulas above express that construction in modern linear operator notation.^[ref]

Pumping and a heated bath prepared Mercury vapor around a hot platinum wire, mesh and collector in the original Franck-Hertz experiment. A galvanometer read collector electric current while accelerating voltage varied at fixed retarding voltage. Retarding scans distinguished outgoing electron energies.^[ref] The losses constrain excitation of chemical atoms; they do not determine every optical coupling. The Franck-Hertz experiment explains the original mistaken ionization interpretation.

Such excitation and spectral line evidence motivated a mechanics of discrete energy levels and transitions. Replacing one representation by another was a theoretical construction; the Franck-Hertz experiment does not measure a matrix product or establish a unique representation of an observable.

Later in 1925, Max Born and Pascual Jordan developed the array calculus systematically. Their section 3 states $pq-qp=hI/(2\pi i)$, equivalently the modern canonical commutation relationThis relates position and momentum through an intrinsically noncommuting product.^[ref]

For the quantum harmonic oscillator, the canonical commutation relation and creation and annihilation operators give $H=\hslash \omega (a^{†}a+I/2)$, recovering both the spacing and zero-point energy algebraically. One must retain infinitely many number states for the exact relation: the finite cutoff below reveals the obstruction explicitly.

The Lamb-Dicke regime means the wave-induced phase change across the occupied motional wavepacket is small. For one normal mode, let $x_0=\sqrt{\hslash /(2m\nu )}$ be the ground state's position standard deviation, and write $x=x_0(a+a^{†})$ using creation and annihilation operators. Here $m$ is mass and $\hslash$ the Planck constant divided by $2\pi$. The effective wave vector component $k_{\mathrm{eff}}$ is the single beam's component for a direct transition, or the difference of two beam components for a Raman transition. With dimensionless $\eta =|k_{\mathrm{eff}}|x_0$, the useful condition is $\eta \sqrt{2n+1}\ll 1$ over the populated number states, rather than merely $\eta \ll 1$.

Absorb the sign of $k_{\mathrm{eff}}$ into the pulse phase. Expanding the coupling factor givesThe constant term preserves $n$; the annihilation operator gives $a|n⟩=\sqrt{n}|n-1⟩$; the creation operator gives $a^{†}|n⟩=\sqrt{n+1}|n+1⟩$. The different square roots explain why removing a motional quantum becomes impossible at $n=0$, while adding one remains possible. Smallness of $\eta$ controls this spatial expansion; resolving the resulting transitions is a separate frequency requirement.

A state of light is nonclassical when it is not a classical mixture of coherent states. The Glauber-Sudarshan representation states this criterion in phase space. Examples include nonvacuum number states and quadrature squeezing states. A coherent state is on the classical boundary despite its vacuum fluctuations; thermal light is a classical mixture despite its non-Poisson fluctuations. None of these nonclassicality witnesses is a necessary condition for nonclassicality.

Let $x_{\mathrm{zp}}=\sqrt{\hslash /(2m\omega )}$ and $\eta =x_0/(2x_{\mathrm{zp}})$. The normalized first-order truncation is $(|0⟩+\eta |1⟩)/\sqrt{1+{\eta }^2}$. Its mean displacement is $2\eta x_{\mathrm{zp}}/(1+{\eta }^2)$, agreeing with $x_0$ only at small $|\eta |$. The exact translated ground state is a coherent state whose expansion contains all number states and has mean occupation ${\eta }^2$. Correct linear response does not justify extrapolating a truncated state to large fields.

Here number-conserving means invariant under a global particle-number quantum phase rotation: the field mean and anomalous averages $⟨\psi \psi ⟩$ vanish. It does not mean a sharp total particle number. All normal correlations are obtained by Wick pairings of the one-body density matrix. Independent thermal bosonic modes satisfy this rule; generic nonvacuum bosonic number states do not. A Slater determinant is the analogous fermionic pairing state.

For real coefficients in $a_{\pm }=c_{\pm }X+i{d}_{\pm }P$, imposing $a_{-}^{†}=a_{+}$, $[a_{-},a_{+}]=I$ and $H=\hslash \omega (a_{+}{a}_{-}+\lambda I)$ givesIndeed the adjoint operator condition makes the two $c$ coefficients equal and the two $d$ coefficients opposite. The canonical commutation relation gives $2\hslash c_{-}{d}_{-}=1$, and matching the coefficients of $X^2$ and $P^2$ fixes their magnitudes. Both signs change together. The conventional creation and annihilation operators choose $\epsilon =1$; the other sign is an equivalent phase convention for the number states.

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.

Normalize the specified state by $⟨0|0⟩=1$ and write $⟨A⟩=⟨0|A|0⟩$. Its zero mode is a coherent state, so any normally ordered product of zero-mode creation operators and annihilation operators can be evaluated using ${\psi }_0\mapsto \sqrt{N_0}$ and ${\psi }_0^{†}\mapsto \sqrt{N_0}$. In particular $⟨({\psi }_0^{†}{)}^2{\psi }_0^2⟩=N_0^2$, rather than $N_0(N_0-1)$, which would apply to a zero-mode number state.

Sort the momentum-space contact-interaction Hamiltonian by its number of zero-mode factors. Four such factors give $u{N}_0^2/(2V)$. Three zero-mode factors and one nonzero momentum violate momentum conservation, so no such term survives. With two zero-mode factors there are three possibilities:

The sums over $\mathbf{k}\ne 0$ include both members of each opposite-momentum pair, consistently with the overall factor $1/2$. Terms with one zero-mode factor contain three nonzero-mode factors; their expected values vanish in the centered multimode squeezed vacuum. Indeed each $\psi$ is linear in $\phi ,{\phi }^{†}$, and an odd product has zero vacuum expectation value. Terms with no zero-mode factors are independent of $N_0$ at fixed ${\theta }_k$. ThereforeTo evaluate these vacuum expectation values, use the inverse Bogoliubov transformation and its Hermitian adjoint:Since ${\phi }_{\mathbf{k}}|0⟩=0$, the only nonzero two-factor vacuum expectation value is $⟨{\phi }_{\mathbf{k}}{\phi }_{\mathbf{q}}^{†}⟩={\delta }_{\mathbf{k},\mathbf{q}}$. The bosonic squeezed-vacuum contractions are consequentlyIn particular the three requested quantities are $-\sinh {\theta }_k\cosh {\theta }_k$, $-\sinh {\theta }_k\cosh {\theta }_k$ and ${\sinh }^2{\theta }_k$, respectively. The minus sign in the pair vacuum expectation values comes from the inverse Bogoliubov transformation and is essential to lowering the interaction energy. Substitution yields

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.

Orthogonality of the number states makes the coherent state norm an exponential series:Using $\hat{a}|n⟩=\sqrt{n}|n-1⟩$ and reindexing $m=n-1$,It is not an eigenstate of the creation operator. Indeed its supposed eigenvalue would have to be ${\alpha }^{\ast }$, from its expectation, whereasThis nonzero residual holds even at $\alpha =0$.

Project the field onto the five-photon number state. The excited-atom contribution comes from the initial $n=5$ block, while the ground-atom contribution comes from the initial $n=4$ block. The unnormalized conditional atomic state isThus the five-count probability and normalized post-measurement state areThe relative free-evolution phase is physically retained; it is absent only in an explicitly rotating atomic frame. The conditional state is defined when $P_5>0$. In particular, an initial vacuum cannot produce five photons in this one-atom model, so that conditioning event has probability zero when $z=0$.

Track the $-i$ factors only when the corresponding transition is addressed. They are common within the addressed two-state subspace, but become physically relevant relative phases between computational basis states. Using a semicolon to separate the shared motional number state, the complete paths areThe middle arrow comprises both Y pulses. In the last row those individual steps are $-i|ge;1⟩\to -|gf;0⟩\to +i|ge;1⟩$. The two X pulses contribute $(-i{)}^2=-1$ for either initially excited control; the Y pair supplies a second minus sign only when the second ion is also excited.

The shared motion returns to its ground state for all four inputs, and the auxiliary internal state is empty. Quantum superposition therefore gives the two-qubit gateDiscarding the X-pulse factors would incorrectly identify $G$ with a Controlled-Z gate itself.

For one electromagnetic normal mode, $a$ and $a^{†}$ are the annihilation operator and creation operator, with commutator $[a,a^{†}]=I$. Their action on normalized number states isThey remove or add one photon, whose energy is $\hslash \omega$. The vacuum state $|0⟩$ is normalized and obeys $a|0⟩=0$. It has no photons, but retains zero-point energy $\hslash \omega /2$ for the quantum harmonic oscillator Hamiltonian $H=\hslash \omega (a^{†}a+1/2)$. The coherent state $|\alpha ⟩=D(\alpha )|0⟩$ is a displaced vacuum state.

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 $n=0$.