Quantum depletion of a Bose condensate ($n-n_0={\mathcal{V}}^{-1}{\sum }_{k\ne 0}{v}_k^2$)

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.