= Solution
A bosonic <coherent state> is an eigenstate of its annihilation operator,
$$
|\alpha\rangle=e^{-|\alpha|^2/2}e^{\alpha a^\dagger}|0\rangle
=e^{-|\alpha|^2/2}\sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle,
\qquad a|\alpha\rangle=\alpha|\alpha\rangle.
$$
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 $\exp[-(|\alpha|^2+|\beta|^2)/2+\alpha^*\beta]$, 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> is
$$
|\psi\rangle=e^{-\bar N/2}\exp\left[\int d^3r\,\psi(\mathbf r)\widehat\Psi^\dagger(\mathbf r)\right]|0\rangle,
\qquad \bar N=\int d^3r\,|\psi|^2.
$$
It obeys $\langle\widehat\Psi(\mathbf r)\rangle=\psi(\mathbf r)$. For a contact-interacting dilute Bose gas, its normal-ordered <energy> expectation is
$$
E[\psi]=\int d^3r\left[\frac{\hbar^2}{2m}|\nabla\psi|^2+V|\psi|^2+\frac g2|\psi|^4\right].
$$
Minimizing $E-\mu\bar N$ gives the stationary <Gross–Pitaevskii equation>, $[-\hbar^2\nabla^2/(2m)+V+g|\psi|^2]\psi=\mu\psi$. 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> $(a^\dagger_\varphi)^N|0\rangle/\sqrt{N!}$. 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 state>
$$
|\mathrm{BCS}\rangle=\prod_{\mathbf k}\left(u_{\mathbf k}
+v_{\mathbf k}a^\dagger_{\mathbf k\uparrow}a^\dagger_{-\mathbf k\downarrow}\right)|0\rangle,
\qquad |u_{\mathbf k}|^2+|v_{\mathbf k}|^2=1.
$$
Each block superposes an empty and an occupied pair state. Where $u_{\mathbf k}\ne0$, it can be rewritten as $(\prod u_{\mathbf k})\exp[\sum(v_{\mathbf k}/u_{\mathbf k})a^\dagger_{\mathbf k\uparrow}a^\dagger_{-\mathbf k\downarrow}]|0\rangle$, 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> $F_{\mathbf k}=\langle a_{-\mathbf k\downarrow}a_{\mathbf k\uparrow}\rangle=u_{\mathbf k}^*v_{\mathbf k}$. For an attractive interaction written with matrix elements $V_{\mathbf k\mathbf k'}$, the gap is $\Delta_{\mathbf k}=-\sum_{\mathbf k'}V_{\mathbf k\mathbf k'}F_{\mathbf k'}$. 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>.<Theory of Superconductivity by John Bardeen and others>[1957, variational paired ground state and elementary excitations]{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.
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