Fixed-number two-mode condensate ($|\chi ⟩=(C{a}_0^{†}+S{a}_1^{†}{)}^N|0⟩/\sqrt{N!}$)

All $N$ bosons occupy the same normalized one-particle superposition with $|C{|}^2+|S{|}^2=1$. Its occupation distribution in the original two modes is binomial, not two independent Poisson distributions. In particular, $⟨N_1⟩=N|S{|}^2$ and $⟨N_0{N}_1⟩=N(N-1)|C{|}^2|S{|}^2$. These factorial moments give finite-particle corrections to interaction energies. The state is also the two-mode realization of a spin coherent state with spin $N/2$.