Quantum depletion of the condensate

Another interesting result that can be obtained from Bogoliubov theory is the momentum distribution of the particles. The number of particles with momentum ${\bf p}$ is given by $N_{\bf p} = {\hat a_{\bf p}}^\dagger \hat a_{\bf p}$ or using the transformation (1.32) it is given by

$$N_{\bf p} = \frac{n_{\bf p}+L_p^2(n_{\bf p}+1)}{1-L_p^2},$$ (1.41)

here $n_{\bf p}= {\hat b_{\bf p}}^\dagger\hat b_{\bf p}$ is the number of elementary excitations, which satisfy the usual Bose distribution $n_p = (\exp\{E(p)/k_B T\}-1)^{-1}$. At zero temperature such excitations are absent and (1.41) simplifies to

$$N_p = \frac{(mc^2)^2}{2E(p)\left[E(p)+\frac{p^2}{2m} + mc^2\right]}$$ (1.42)

As $p \to 0$ the momentum distribution diverges as $N_p \to mc/2p$. The number of atoms in the condensate can be obtained by taking the difference between $N$ and the number of non-condensed atoms.

$$N_0 = N - \sum\limits_{\bf p \ne 0}N_p = N -\frac{V}{(2\pi\hbar)^3}\int N_p {\bf dp}$$ (1.43)

The integration can be carried out and gives the result

$$N_0 = N \left(1 - \frac{8}{3\sqrt{\pi}} (na^3)^{1/2}\right)$$ (1.44)

Due to interactions particles are pushed out of the condensate and a fraction of particles with nonzero momenta is present even at zero temperature. This phenomenon is called quantum depletion of the condensate Also result (1.44) is valid in the dilute regime $na^3 \ll 1$ in which the quantum depletion is small and Bogoliubov theory applies.