Balanced-splitter twin-Fock output distribution
= Balanced-splitter twin-Fock output distribution
{title2=$P(2j,2N-2j)=4^{-N}\binom{2j}{j}\binom{2N-2j}{N-j}$}
For two identical bosonic input populations of $N$ in a balanced two-mode mixer, only even occupancies occur at either output. Expanding $[(b_1^\dagger)^2-(b_2^\dagger)^2]^N$ and retaining normalized number-state factorials gives the displayed law. Its sum is one by the <generating function> $(1-4z)^{-1/2}(1-4z)^{-1/2}=(1-4z)^{-1}$.