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For one bosonic mode, define the unnormalized bosonic coherent state $|z⟩=e^{z{a}^{†}}|0⟩$. The commutator $[a,a^{†}]=1$ gives $a|z⟩=z|z⟩$ and $⟨z^{\prime }|z⟩=e^{{\bar{z}}^{\prime }z}$. The normalized state is $e^{-|z{|}^2/2}|z⟩$; it is not orthogonal to a different coherent state. The coherent-state resolution of identity follows from Gaussian moments,For many modes take products of these states and measures. The complex amplitudes have the natural scale of square root occupation; coherent states turn normal-ordered matrix elements into classical symbols without replacing the quantum trace by a single classical configuration.

For fermions, $\{c_i,c_j^{†}\}={\delta }_{ij}$ and a nilpotent annihilation operator cannot have an arbitrary nonzero ordinary complex eigenvalue. Introduce independent Grassmann variables ${\eta }_i,{\bar{\eta }}_i$, anticommuting among themselves and with the fermionic operators, and define $|\eta ⟩=\exp (-{\sum }_i{\eta }_i{c}_i^{†})|0⟩$. Then $c_i|\eta ⟩={\eta }_i|\eta ⟩$ and $⟨{\bar{\eta }}^{\prime }|\eta ⟩=e^{{\sum }_i{\bar{\eta }}_i^{\prime }{\eta }_i}$. With an ordered Berezin integral measure satisfying $\int d\bar{\eta }d\eta \bar{\eta }\eta =-1$ for one pair,for an even operator $O$; this is the fermionic coherent-state trace. In one mode the trace integrand is $e^{-\bar{\eta }\eta }(O_{00}-\bar{\eta }\eta O_{11})$, whose integral is $O_{00}+O_{11}$. Without the sign reversal it would be the supertrace, not the thermal trace. These formal Grassmann-valued coherent states do not represent ordinary complex-valued fermion amplitudes.

To construct the many-body thermal path integral, write $\mathcal{Z}=\mathrm{Tr}{e}^{-\beta K}$ with $K=H-\mu \hat{N}$ and divide the imaginary-time interval $\mathcal{T}=\beta \hslash$ into slices of width $\delta \tau$. Insert the appropriate coherent-state resolution of identity between factors. For a normal-ordered Hamiltonian, the adjacent-slice kernel iscoherent-state time slicing makes the overlap and Gaussian measure combine into the first-order time derivative. The grand-canonical coherent-state action in the regulated continuum limit isFor bosons the integration variables are complex, with the conjugate labels on the original integration contour; for fermions the barred and unbarred Grassmann variables are independent. The continuum notation retains the time-sliced ordering prescription. For example, $n(n-1)=a^{†2}{a}^2$ has normal symbol ${\bar{z}}^2{z}^2$, whereas $n^2$ has that symbol plus $\bar{z}z$. Dropping that distinction changes a theory. The coherent-state thermal boundary conditions follow because the bosonic trace closes paths periodically and the fermionic sign in the trace closes them antiperiodically:The barred fields obey the same respective boundary conditions. Hence Matsubara frequencies are $2\pi n/(\beta \hslash )$ and $(2n+1)\pi /(\beta \hslash )$. The derivative term encodes the canonical structure; omitting it would remove the quantum dynamics.^[ref]

As a check, a free diagonal mode with $K=\xi a^{†}a$ gives $Z_B=(1-e^{-\beta \xi }{)}^{-1}$ for $\xi >0$, whereas $K=\xi c^{†}c$ gives $Z_F=1+e^{-\beta \xi }$. Gaussian integration produces an inverse determinant for bosons and a determinant for independent complex fermions, with their respective temporal boundary conditions. Differentiating $\log Z$ recovers Bose-Einstein and Fermi-Dirac occupations. Interactions become higher powers in the normal-ordered symbols, and sources generate time-ordered correlations. The coherent-state construction preserves statistics, ordering and boundary conditions together; a formal continuum expression without its measure and temporal prescription is incomplete.

A bosonic coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$ and has expansion $e^{-|\alpha {|}^2/2}{\sum }_n{\alpha }^n|n⟩/\sqrt{n!}$. Its occupation is Poisson distributed, with mean and variance $|\alpha {|}^2$. Its nonzero field amplitude makes phase coherence natural, although physically observable phase comparisons require a reference.

For paired fermions, the corresponding Cooper-pair pseudospin coherent state isIt coherently superposes empty and occupied pair states and has an anomalous average $⟨c_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }⟩\ne 0$. It is not an ordinary complex-eigenvalue coherent state of a single fermion annihilation operator. The broken-symmetry state has indefinite particle number; number projection can restore a fixed-number description without discarding the underlying pairing correlations.

Hartree-Fock-Bogoliubov theory approximates a two-body interaction using both the normal density ${\rho }_{ij}=⟨c_j^{†}{c}_i⟩$ and anomalous pairing tensor ${\kappa }_{ij}=⟨c_j{c}_i⟩$. Direct and exchange contractions determine a normal one-particle field, while anomalous contractions determine the pair potential. The resulting quadratic Hamiltonian is diagonalized by a fermionic Bogoliubov transformation. In a homogeneous singlet case its Nambu block isThe fields are determined self-consistently, rather than selected independently of the quasiparticle state. For example,together with the particle-number equation. The pairing field extends ordinary Hartree-Fock theory to states with pair correlations. This mean-field treatment omits fluctuations beyond its self-consistent quadratic description.

For bosons, write the field as a coherent condensate plus fluctuations, $\hat{\psi }=\varphi +\delta \hat{\psi }$. The same normal/anomalous strategy retains $\tilde{n}=⟨\delta {\hat{\psi }}^{†}\delta \hat{\psi }⟩$ and $\tilde{m}=⟨\delta \hat{\psi }\delta \hat{\psi }⟩$, with a bosonic, commutation-preserving Bogoliubov transformation. In the leading weakly depleted contact-gas limit, the condensate density $n_0$ and effective coupling $g>0$ give $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g{n}_0)}$, with ${ϵ}_k={\hslash }^2{k}^2/(2m)$. This Bogoliubov theory connects low-momentum sound to high-momentum particle excitations. More elaborate self-consistent truncations must respect the symmetry constraints needed for a gapless condensate mode.

Ginzburg-Landau theory of superconductivity describes the superconducting phase by a complex order parameter and a gauge-invariant free-energy functional. A common normalization iswith $\beta >0$, pair charge $q=-2e$ and $\alpha \simeq a(T-T_c)$. In zero field the uniform minimum has $|\Psi {|}^2=-\alpha /\beta$ below $T_c$. Variation gives the order-parameter equation and supercurrent. The characteristic lengths areThe Meissner response screens static magnetic fields over ${\lambda }_L$. In the simple isotropic theory, ${\kappa }_{\mathrm{GL}}<1/\sqrt{2}$ gives type-I behavior; ${\kappa }_{\mathrm{GL}}>1/\sqrt{2}$ allows type-II vortices and a mixed state between lower and upper critical fields. Flux quantization, vortex structure and currents follow from phase stiffness and electromagnetic coupling. This is a phenomenological expansion most directly controlled near $T_c$; static coefficients alone do not determine every dynamical mode or dissipation rate.

BCS theory supplies a microscopic pairing description. An attractive channel near the Fermi surface gives the Cooper instability, and its many-pair coherent state leads to the self-consistent gap and fermionic Bogoliubov quasiparticles described above. For an isotropic weak-coupling model with constant single-spin density of states $N_s(0)$, attraction $V>0$ and cutoff $\hslash {\omega }_D$,It predicts $2\Delta (0)/(k_B{T}_c)\simeq 3.53$ in that specified model. A full gap suppresses low-temperature quasiparticles exponentially, whereas nodes permit power laws such as the $T^2$ heat capacity derived above. The weak-coupling ratio is not a universal value for all superconductors. Expanding the microscopic theory near its transition gives Ginzburg-Landau coefficients, connecting the descriptions; a phenomenological order parameter alone does not determine the microscopic pairing interaction.

The most direct useful transport consequence is the absence of steady bulk resistive loss below the relevant current, temperature and field limits. Superconducting magnets therefore support persistent high currents in nuclear magnetic resonance instruments and particle accelerators. Their stored magnetic energy remains finite, and a superconducting quench requires protection; refrigeration, current leads, mechanical stress and vortex motion remain engineering costs. Zero dc resistance does not eliminate all ac loss or justify unlimited current.

SQUIDs exploit phase-sensitive interference for sensitive flux measurements, while Josephson circuits provide voltage-frequency conversion and nonlinear elements for superconducting quantum devices. These applications use coherence, not just low resistance. Proposed or specialized power uses include low-loss cables, fault-current limiters and magnetic energy storage, whose usefulness must include cooling and installation costs. Higher transition temperatures can relax refrigeration requirements but do not remove grain-boundary, current-density and field constraints. Neither BCS nor Ginzburg-Landau theory alone supplies a complete device design or guarantees that a particular material is economical.

A bosonic coherent state is an annihilation-operator eigenstate,It has $⟨N⟩=\mathrm{Var}N=|\alpha {|}^2$, a selected complex amplitude, and Poisson-distributed particle number. A coherent-field condensate variational state extends this construction to a spatial field with $⟨\hat{\psi }(\mathbf{r})⟩=\Psi (\mathbf{r})$. Evaluating a normal-ordered interacting Bose Hamiltonian in that state gives a classical-field energy functional; its variational equation is the Gross–Pitaevskii equation. This connects a macroscopic phase to interference, quantized circulation and neutral Josephson effects. A fixed-number condensate instead has vanishing one-point field expectation; off-diagonal long-range order identifies the condensate without requiring exact number fluctuations.

The BCS ground state has the pair-coherent formIt is a product of Cooper-pair pseudospin coherent states: each paired momentum sector mixes its empty and doubly occupied configurations. The anomalous pair expectation is $⟨c_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }⟩=u_{\mathbf{k}}^{\ast }{v}_{\mathbf{k}}$, while an ordinary single-electron expectation vanishes by fermion parity. Pair operators are not unrestricted canonical boson operators, so this state must not be identified with a naive oscillator coherent state of individual electrons. Number projection gives a fixed-even-number description with pair correlations. Phase differences between condensates enter the Josephson effect; in a charged condensate that phase difference must be gauge invariant.

A bosonic coherent state is an eigenstate of its annihilation operator,Its mean occupation is $|\alpha {|}^2$ and its number variance is also $|\alpha {|}^2$, giving a Poisson number distribution. It has a well-defined complex mean field but fluctuating particle number. Its overlap with another coherent state is $\exp [-(|\alpha {|}^2+|\beta {|}^2)/2+{\alpha }^{\ast }\beta ]$, so distinct amplitudes need not be orthogonal. For large occupation, a phase-space wave packet has small relative number fluctuations; this does not require an exact self-adjoint phase operator conjugate to a number bounded below.

For a bosonic field, a useful coherent-field condensate variational state isIt obeys $⟨\hat{\Psi }(\mathbf{r})⟩=\psi (\mathbf{r})$. For a contact-interacting dilute Bose gas, its normal-ordered energy expectation isMinimizing $E-\mu \bar{N}$ gives the stationary Gross–Pitaevskii equation, $[-{\hslash }^2{\nabla }^2/(2m)+V+g|\psi {|}^2]\psi =\mu \psi$. Its phase gradient determines superfluid flow, while weak-link phase differences give Josephson currents. The complex microscopic expectation supplies the order field used by a mean-field description of a bosonic superfluid.

An exactly fixed-number condensate can instead use the Hartree product state $(a_{\phi }^{†}{)}^N|0⟩/\sqrt{N!}$. Its one-point field expectation vanishes by number conservation, but its one-body density matrix has a macroscopic eigenvalue. Off-diagonal long-range order expresses that number-conserving criterion. Selecting a phase is a useful thermodynamic symmetry-broken description; it is not a claim that every finite isolated condensate literally has indefinite total number. A plain coherent product also misses interaction-induced quantum depletion, which needs correlations such as those in the Bogoliubov quasiparticle vacuum.

In a spin-singlet superconductor, the corresponding variational state is the BCS ground stateEach block superposes an empty and an occupied pair state. Where $u_{\mathbf{k}}\ne 0$, it can be rewritten as $(\prod u_{\mathbf{k}})\exp [\sum (v_{\mathbf{k}}/u_{\mathbf{k}})a_{\mathbf{k}\uparrow }^{†}{a}_{-\mathbf{k}\downarrow }^{†}]|0⟩$, treating any fully occupied blocks separately. The exponential truncates within each block because a fermionic pair cannot be created twice in the same two modes. This is a Cooper-pair pseudospin coherent state, rather than a coherent state of unrestricted ideal bosons.

The one-electron expectation remains zero in a fermion-parity-even BCS state. Its nonzero microscopic order is the BCS anomalous average $F_{\mathbf{k}}=⟨a_{-\mathbf{k}\downarrow }{a}_{\mathbf{k}\uparrow }⟩=u_{\mathbf{k}}^{\ast }{v}_{\mathbf{k}}$. For an attractive interaction written with matrix elements $V_{\mathbf{k}{\mathbf{k}}^{\prime }}$, the gap is ${\Delta }_{\mathbf{k}}=-{\sum }_{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{F}_{{\mathbf{k}}^{\prime }}$. Minimization yields the coherence factors and gap equation. Fermionic coherent states using Grassmann variables are valuable integration tools, but they are not ordinary complex-eigenvalue electron states replacing this physical paired wavefunction.^[ref]

Near the transition, a slowly varying pair field connects the microscopic anomalous average to Ginzburg-Landau theory of superconductivity. The phenomenological complex field has pair charge $-2e$, a phase that controls supercurrent and a magnitude whose normalization is model dependent. It is not the expectation of a single-electron field. Microscopic derivation of Ginzburg-Landau theory supplies this connection under its stated near-critical assumptions. Pair phase coherence, Josephson interference and flux quantization then become different manifestations of the same macroscopic paired order. Paired-fermion superfluids likewise use a pair order field, while bosonic atomic superfluids can use a one-particle condensate field.

Put $C=\cos (\chi /2)$ and $S=\sin (\chi /2)$. Commutativity of the two creation operators and the binomial theorem giveThusThe state is a fixed-number two-mode condensate with binomial mode occupations; it is not a product of two independent Poisson-distributed Glauber coherent states.

The exponential input has norm squared $\exp (|{\alpha }_1{|}^2+|{\alpha }_2{|}^2)$. Its normalized form is a product of coherent states. Substitution of the inverse transformation givesSince the output creation operators commute, the normalized output factorizes as $|{\beta }_1⟩|{\beta }_2⟩$. For all nonnegative integers $n_1,n_2$,These are independent Poisson distributions, not a fixed-total-particle-number distribution. unitarity gives $|{\beta }_1{|}^2+|{\beta }_2{|}^2=|{\alpha }_1{|}^2+|{\alpha }_2{|}^2$. Equal complex inputs make ${\beta }_2=0$, so that port is certainly empty; opposite inputs empty port 1. This is amplitude interference of coherent inputs, distinct from the fixed-number two-particle interference in (c).

A normalized oscillator coherent state is an eigenstate of the annihilation operator, $a|\alpha ⟩=\alpha |\alpha ⟩$. Expanding it in Fock states givesSince $x=\sqrt{\hslash /(2M\omega )}(a+a^{†})$, $⟨a^2⟩={\alpha }^2$, and $⟨a{a}^{†}⟩=|\alpha {|}^2+1$, the variance is $⟨x^2⟩-⟨x{⟩}^2=\hslash /(2M\omega )$. Henceindependently of the complex amplitude $\alpha$.

For $A=\alpha a^{†}$ and $B=-{\alpha }^{\ast }a$, the commutator is the scalar $[A,B]=|\alpha {|}^{2}I$. The Baker--Campbell--Hausdorff formula therefore terminates and gives the displacement operatorBecause $a|0⟩=0$, applying it to the Fock vacuum yields exactly the Fock state expansion above:For a general complex $\alpha$, its complex conjugate is required in the annihilation term. The expression with $-\alpha a$ printed in the paper is valid when $\alpha$ is real.

The annihilation operator associated with the new coordinate $X=x-x_0$ isAs the initial state is the old vacuum,Equivalently, applying $a^{\prime }$ to the number-state expansion in part (c) and shifting its summation index returns $-\lambda$ times the same state. It is therefore a coherent state of the displaced oscillator.

The new Hamiltonian is $H=U_{\min }+\hslash \omega (a^{\prime †}{a}^{\prime }+1/2)$. In the Schrödinger picture its evolution givesConsequentlyThe displayed prefactor is only a global phase. Equivalently, in the Heisenberg picture $a^{\prime }(t)=a^{\prime }(0)e^{-i\omega t}$. The mean occupation $|\alpha (t){|}^2={\lambda }^2$ is constant, and $\hslash \omega ({\lambda }^2+1/2)$ reproduces the energy above the new minimum found in part (b). Its mean position and momentum likewise reproduce that part's oscillating wavepacket centre.

The normalized coherent state has number amplitude $e^{-|\alpha {|}^2/2}{\alpha }^n/\sqrt{n!}$, hence the displayed Poisson law with mean and variance $|\alpha {|}^2$. A product of coherent modes has factorized number probabilities.

The position uncertainty of a quantum state is the standard deviation of its measured position, provided the second moment is finite. It is computed from the position operator with the Born rule. An oscillator coherent state has $\Delta x=\sqrt{\hslash /(2m\omega )}$ independently of its mean position; translating the state changes the mean without changing this variance.

For a single-mode Gaussian state with no quadrature covariance, the covariance contour has semiaxes $\sqrt{\mathrm{Var}X}$ and $\sqrt{\mathrm{Var}P}$. A coherent state has both equal to $1/2$ in the convention $[X,P]=i/2$. A pure squeezed vacuum state has semiaxes $e^{-r}/2$ and $e^r/2$; the uncertainty area remains fixed while one noise width decreases. An ellipse of means alone does not specify a general non-Gaussian state.^[ref]

A classical oscillator of amplitude $A$ has energy $m{\omega }^2{A}^2/2$. Comparing with the oscillator spacing $\hslash \omega$ gives the large-occupation scale displayed. A coherent state with this classical amplitude has exactly this mean excitation occupation and a Poisson number distribution; its total mean energy also includes the zero-point term. An observed macroscopic amplitude does not, by itself, establish a pure coherent state.

Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

Fix a detected position and polarization of waves. Write the electric field operator as $\hat{E}(t)={\hat{E}}^{(+)}(t)+{\hat{E}}^{(-)}(t)$, where the positive-frequency part contains annihilation operators and ${\hat{E}}^{(-)}=({\hat{E}}^{(+)}{)}^{†}$. An ideal absorbing photodetector responds towith density matrix $\rho$. The diagonal gives the single-count rate up to the detector coupling. Multiple absorptions require higher, normal-ordered field products. Roy J. Glauber used these correlations to develop the quantum account of optical coherence.^[ref]

The ordering follows from a detection amplitude. If absorption takes an initial pure state $|i⟩$ to an unobserved final state $|f⟩$, its field factor is $⟨f|{\hat{E}}^{(+)}|i⟩$. Squaring its complex modulus and summing over a complete orthonormal basis of final states gives $⟨i|{\hat{E}}^{(-)}{\hat{E}}^{(+)}|i⟩$. Averaging over initial preparations replaces this expression by a trace. Thus a detector that absorbs cannot count the Fock vacuum, although the vacuum has quantum fluctuations.

For one selected normal mode, ${\hat{E}}^{(+)}(t)=\mathcal{E}u(t)a$. All Fock states with nonzero occupation, coherent states, and thermal light in this same mode havewhere both denominators are nonzero. Its magnitude is one even when the expected value of the field vanishes. Perfect first-order optical coherence therefore does not identify a coherent state. Pair detection supplies additional information.

Displacing a one-dimensional quantum harmonic oscillator produces a linear force term and a scalar shift relative to the unshifted Hamiltonian. The scalar adds only a common phase. The linear term couples number states separated by one excitation. Arbitrary linear driving of the ground state creates a coherent state; a first-order expansion of that state contains only the vacuum and first excited state.^[ref]

For one selected polarization of waves, write the detected complex fields as $E_1(t),E_2(t+\tau )$ and their light intensities as $I_j=|E_j{|}^2$, absorbing a common conversion constant. With stationary ensemble averages and nonzero ${\bar{I}}_j=⟨I_j⟩$, defineHere second-order optical coherence concerns distinct detector channels, with the corresponding normal ordering in the quantum formulation. Independent detection has $g^{(2)}=1$. Subtracting the means gives the normalized covarianceThus multiplying raw photocurrents contains a large mean-product contribution; multiplying their fluctuations isolates the small excess. Independent detector shot noise does not supply a mean cross-channel excess, although it increases measurement uncertainty.

The displayed Siegert relation requires a zero-mean, jointly circular complex Gaussian random field, the usual chaotic thermal light model. Circular means invariance under a common arbitrary complex phase rotation; in particular $⟨E_i{E}_j⟩=0$. Apply the Gaussian moment pairing theorem to the real and imaginary parts:The remaining pairing vanishes by circularity. Division proves the Siegert relation. Gaussianity is an assumption about fourth moments, not a consequence of merely measuring first-order optical coherence; a coherent state or photon antibunching cannot be interpreted by this thermal formula.

For equal-intensity beams combined in a common optical mode with matched wave polarizations, interferometric visibility is $V=|g_{12}^{(1)}|$. Consequently ideal thermal stellar intensity interferometry measures $g^{(2)}-1=V^2$, while the full $g^{(2)}$ includes the independent-count reference one. Unpolarized thermal light with two independent equally populated wave polarizations has half this excess: adding their covariances doubles the numerator while multiplying total means quadruples the denominator. Finite detector response further suppresses the peak and must be calibrated.^[ref]

For a single mode in thermal equilibrium, its quantum harmonic oscillator energy ladder gives a geometric distributionIts probability generating function is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F^{\prime \prime }(1)=2q^2/(1-q{)}^2=2{\mu }^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu =2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\mathrm{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of coherent states.^[ref]

Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N={\sum }_{j=1}^M{N}_j$ with $⟨N_j⟩=\mu /M$. Adding their variances gives $\mathrm{Var}N=\mu +{\mu }^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.

A product of number states with equal fixed populations in two modes. Its total particle number is $2N$. It differs from a pair of coherent states, whose number distributions fluctuate.

An unnormalized bosonic coherent state is $|\psi ⟩=e^{\psi a^{†}}|0⟩$ for a complex label $\psi$. The canonical commutation relation gives $a|\psi ⟩=\psi |\psi ⟩$ and overlap $⟨\psi |{\psi }^{\prime }⟩=e^{\bar{\psi }{\psi }^{\prime }}$. Multiplication by $e^{-|\psi {|}^2/2}$ produces a normalized coherent state. The unnormalized form makes the coherent-state resolution of identity and thermal time slicing particularly simple.