= Solution
A bosonic <coherent state> obeys $a|\alpha\rangle=\alpha|\alpha\rangle$ and has expansion $e^{-|\alpha|^2/2}\sum_n\alpha^n|n\rangle/\sqrt{n!}$. 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> is
$$
|\mathrm{BCS}\rangle=\prod_{\mathbf k}(u_{\mathbf k}+v_{\mathbf k}c^\dagger_{\mathbf k\uparrow}c^\dagger_{-\mathbf k\downarrow})|0\rangle,
\qquad |u_{\mathbf k}|^2+|v_{\mathbf k}|^2=1.
$$
It coherently superposes empty and occupied pair states and has an anomalous average $\langle c_{-\mathbf k\downarrow}c_{\mathbf k\uparrow}\rangle\ne0$. 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 $\rho_{ij}=\langle c_j^\dagger c_i\rangle$ and anomalous pairing tensor $\kappa_{ij}=\langle c_jc_i\rangle$. 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 is
$$
\begin{pmatrix}\xi_{\mathbf k}&\Delta_{\mathbf k}\\\Delta_{\mathbf k}^*&-\xi_{\mathbf k}\end{pmatrix},\qquad
E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+|\Delta_{\mathbf k}|^2}.
$$
The fields are determined self-consistently, rather than selected independently of the <quasiparticle> state. For example,
$$
\Delta_{\mathbf k}=-\sum_{\mathbf k'}V_{\mathbf k\mathbf k'}\frac{\Delta_{\mathbf k'}}{2E_{\mathbf k'}}\tanh\frac{E_{\mathbf k'}}{2k_BT},
$$
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, $\widehat\psi=\phi+\delta\widehat\psi$. The same normal/anomalous strategy retains $\widetilde n=\langle\delta\widehat\psi^\dagger\delta\widehat\psi\rangle$ and $\widetilde m=\langle\delta\widehat\psi\delta\widehat\psi\rangle$, 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 $E_k=\sqrt{\epsilon_k(\epsilon_k+2gn_0)}$, with $\epsilon_k=\hbar^2k^2/(2m)$. 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.
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