Balanced-splitter twin-Fock output distribution ($P(2j,2N-2j)=4^{-N}\left(\genfrac{}{}{0ex}{}{2j}{j}\right)\left(\genfrac{}{}{0ex}{}{2N-2j}{N-j}\right)$)

For two identical bosonic input populations of $N$ in a balanced two-mode mixer, only even occupancies occur at either output. Expanding $[(b_1^{†}{)}^2-(b_2^{†}{)}^2{]}^N$ and retaining normalized number-state factorials gives the displayed law. Its sum is one by the generating function $(1-4z{)}^{-1/2}(1-4z{)}^{-1/2}=(1-4z{)}^{-1}$.