Particle number from Bogoliubov coefficients
= Particle number from Bogoliubov coefficients
If
$$
u_i^{\rm out}=\sum_j(\alpha_{ij}u_j^{\rm in}+\beta_{ij}u_j^{{\rm in}*}),
$$
then the expected out-particle number in mode $i$ in the in-vacuum is
$$
\langle0_{\rm in}|N_i^{\rm out}|0_{\rm in}\rangle
=\sum_j|\beta_{ij}|^2.
$$