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A normalized nonnegative probability measure over coherent states produces a classical mixture. The measure may contain Dirac delta distributions, so a pure coherent state is included. Conditioning on $\alpha$ gives Poisson distribution photon statistics; hence $\mathrm{Var}n=⟨n⟩+\mathrm{Var}(|\alpha {|}^2)\ge ⟨n⟩$. Each optical quadrature has variance at least its vacuum value because mixing adds nonnegative variance of conditional means. These inequalities provide simple nonclassicality witnesses.

Suddenly shifting a quantum harmonic oscillator equilibrium from zero to a signed position $x_0$ leaves the old ground state unchanged at the instant of the quantum quench. The new annihilation operator is $a^{\prime }=a-\lambda$, with $\lambda =\sqrt{m\omega /(2\hslash )}{x}_0$, so $a^{\prime }|0⟩=-\lambda |0⟩$. In the translated energy eigenstates $|n^{\prime }⟩=e^{-ip{x}_0/\hslash }|n⟩$, the expansion isThe sign follows by translating the old vacuum by $-x_0$ before expanding it in the old Fock states. A Baker--Campbell--Hausdorff formula with central commutator separates the resulting exponential into creation and annihilation parts. Evolution with the shifted Hamiltonian preserves this coherent state, with $\alpha (t)={\alpha }_0{e}^{-i\omega t}$. Its mean occupation is $|{\alpha }_0{|}^2$ and its mean energy above the new potential minimum is $\hslash \omega (|{\alpha }_0{|}^2+1/2)$. Omitting $(-\lambda {)}^n$ in favour of $|\lambda {|}^n$ changes the state when $x_0>0$.

For one optical mode, $E=E_0(a+a^{†})$ and $[a,a^{†}]=1$. A coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so normal ordering gives $⟨E^2⟩=E_0^2[(\alpha +{\alpha }^{\ast }{)}^2+1]$. Subtracting the squared mean leaves $E_0^2$, independent of the coherent displacement. The extra one is the canonical commutation relation, not noise from a random coherent amplitude. A rotated optical quadrature has the same coherent-state variance with the same amplitude normalization.

A normalized oscillator coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so its mean occupation number is $N=|\alpha {|}^2$. Under the quantum harmonic oscillator Hamiltonian operator $\hslash \omega (a^{†}a+1/2)$, its evolution is $|\alpha (t)⟩=e^{-i\omega t/2}|\alpha e^{-i\omega t}⟩$. Inserting $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ and $p=-i\sqrt{m\hslash \omega /2}(a-a^{†})$ givesThese mean values obey the classical harmonic oscillator equations. The displacement amplitude is $\sqrt{2\hslash N/(m\omega )}$, whose classical orbit has energy $\hslash \omega N$. The full quantum mean energy has the additional zero-point energy $\hslash \omega /2$. Indeed the variances are $\hslash /(2m\omega )$ and $m\hslash \omega /2$, and their kinetic and potential contributions sum to $\hslash \omega /2$.

Expand both states in the Fock basis and sum the exponential series to obtain the displayed overlap. In particular $⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$. Distinct coherent states are nonorthogonal at finite separation; dropping this overlap can give incorrect normalization and small-amplitude limits.

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.

Here $\alpha >0$ and $q,p,\gamma$ are real functions of time. Differentiation in position gives ${\psi }_x=[-\alpha (x-q)+ip/\hslash ]\psi$. The standard probability current is thereforeFor a coherent state moving in a quadratic Hamiltonian, the centre follows the classical equation $\dot{q}=p/m$, so the current is the normalized Gaussian probability density transported at its centre velocity. The factor $\sqrt{\alpha /\pi }$ is essential: omitting it changes the current by a dimensional constant and violates the continuity equation for the normalized density. A spatially constant phase does not affect the current.

The displacement operator translates oscillator phase space. It is unitary and obeys $D(\alpha {)}^{†}aD(\alpha )=a+\alpha$ and $D(\alpha )|0⟩=|\alpha ⟩$, a coherent state. The Baker--Campbell--Hausdorff formula with a scalar commutator givesThe conjugation on ${\alpha }^{\ast }$ ensures that the exponent is anti-Hermitian for arbitrary complex displacement. For real $\alpha =\sqrt{m\omega /(2\hslash )}{x}_0$, this becomes the spatial translation $e^{-i{x}_{0}p/\hslash }$.

For $H(t)=\hslash \omega (a^{†}a+1/2)-Fx{e}^{t/\tau }$ on $t\le 0$, start in the unforced ground state at $t=-\infty$. Define $x_{01}=\sqrt{\hslash /(2m\omega )}$ and $\eta =F{x}_{01}/(\hslash \omega )$. Time-dependent perturbation theory givesThus the first excited-state probability in the original oscillator basis tends to zero for a rapid turn-on and to ${\eta }^2$ for slow turn-on, within the weak-force approximation $|\eta |\ll 1$.

This finite adiabatic limit is not a failure of the adiabatic theorem: the instantaneous ground state is displaced and has a nonzero projection onto the original excited states. In fact a linear force preserves a coherent state, whose exact original-basis displacement at $t=0$ is $\alpha =i\eta \omega \tau /(1+i\omega \tau )$. It follows either from $\dot{\alpha }=-i\omega \alpha +iF{x}_{01}{e}^{t/\tau }/\hslash$ or from the forced oscillator equations. Consequently the exact original-basis probability is $P_1^{(0)}=|\alpha {|}^2{e}^{-|\alpha {|}^2}$. The instantaneous oscillator basis has displacement $\eta$, so its mean excitation is instead $|\alpha -\eta {|}^2={\eta }^2/[1+(\omega \tau {)}^2]$, which vanishes adiabatically.

Hermitian combinations of pair creation and annihilation operators. With $K_3=(N+1/2)/2$, their commutators are $[K_1,K_2]=-4i{K}_3$, $[K_2,K_3]=i{K}_1$ and $[K_3,K_1]=i{K}_2$. A coherent state has both variances $⟨N⟩+1/2$, saturating the corresponding uncertainty product.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

All $N$ bosons occupy the same normalized one-particle superposition with $|C{|}^2+|S{|}^2=1$. Its occupation distribution in the original two modes is binomial, not two independent Poisson distributions. In particular, $⟨N_1⟩=N|S{|}^2$ and $⟨N_0{N}_1⟩=N(N-1)|C{|}^2|S{|}^2$. These factorial moments give finite-particle corrections to interaction energies. The state is also the two-mode realization of a spin coherent state with spin $N/2$.

An orthonormal basis labelled by the occupations of specified single-particle modes. For one bosonic mode, $|n⟩=(a^{†}{)}^n|0⟩/\sqrt{n!}$ and the number operator has eigenvalue $n$. A coherent state's expansion in this basis gives its overlap and photon-number statistics.

Under a quantum harmonic oscillator Hamiltonian, the Heisenberg equations of motion giveSubtract the expectation values, square the first equation, and use the symmetrized covariance $C_0=⟨\{x-⟨x⟩,p-⟨p⟩\}⟩/2$. This yieldsThese moment equations hold for any state with finite second moments; a quadratic Hamiltonian additionally preserves Gaussian wave packets. A coherent state has ${\sigma }_x(0{)}^2=\hslash /(2m\omega )$, ${\sigma }_p(0{)}^2=m\hslash \omega /2$, and $C_0=0$, so its width stays fixed while its center follows the classical orbit. A squeezed coherent state generally has a width that oscillates at $2\omega$.

A bosonic state with Gaussian phase-space characteristic function, specified by its first and second optical quadrature moments. Vacuum, coherent states, optical squeezed vacuum and thermal states are examples. Gaussian here describes phase-space statistics, not a Gaussian optical pulse envelope.

The diagonal coherent state representation writes a single-mode density matrix in the form shown. Its weight may be a generalized distribution rather than an ordinary nonnegative probability density. A nonclassical state of light has no nonnegative measure of this form. This use of $P$ differs from the Wigner distribution; a nonnegative Wigner function does not by itself certify classicality.

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.

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.

For $X_1=x$, $X_2=y$ with $[X_1,X_2]=i/2$, normalize the phase-space function by $\int W(x,y)dxdy=1$. A coherent state of amplitude $\alpha$ has $W=2e^{-2|x+iy-\alpha {|}^2}/\pi$. Quadrature marginals are physical probability distributions, but the full function can be negative. Cross terms of an optical cat state give interference fringes, even when low-order moments approach those of a mixture.