 Two-species uniform boson number state

Two independent boson species occupy uniform single-particle modes on $L$ sites, with fixed occupation numbers $n_A,n_B$. This differs from a fixed-number two-mode condensate whose particles all occupy one superposition of the two modes. Here $⟨a_i^{†}{a}_j⟩=n_A/L$, $⟨b_i^{†}{b}_j⟩=n_B/L$, and the local total number operator has variance $(n_A+n_B)L^{-1}(1-L^{-1})$. The latter follows from two independent binomial distributions with success probability $1/L$. This state satisfies a prescribed Schwinger boson constraint only in the mean unless the local variance vanishes.