= Fixed-mean-number Bogoliubov variational energy
{title2=$\mathcal E=\frac{gN}{2}+\sum_{k\ne0}[(\epsilon_k+g)\sinh^2\theta_k-g\sinh\theta_k\cosh\theta_k]$}
For a weakly depleted homogeneous <Bose gas> with repulsive <contact interaction>, set $g=uN/V$ and $\epsilon_k=\hbar^2k^2/(2m)$. A zero-mode <coherent state> times a <two-mode squeezed vacuum> for each opposite-<momentum> pair has mean <particle number> $N=N_0+\sum_{k\ne0}\sinh^2\theta_k$. 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 $A_k=\epsilon_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh(2\theta_k^*)=g/A_k$. The mode contribution equals $\tfrac12E_k\cosh[2(\theta_k-\theta_k^*)]-A_k/2$, 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>.
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