= Quantum depletion of a Bose condensate
{title2=$n-n_0=\mathcal V^{-1}\sum_{k\ne0}v_k^2$}
The <quasiparticle> vacuum contains physical particles at nonzero momentum because the <Bogoliubov transformation> mixes creation and <annihilation operators>. Their occupation is $v_k^2=(A_k/E_k-1)/2$, and the depletion density is its momentum sum divided by the volume. At nonzero temperature an additional $(A_k/E_k)n_B(E_k)$ contribution is required; quantum depletion alone is a zero-temperature expression.
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