 Boson-pair realization of SU(1,1)

Pair creation and annihilation in two independent oscillator modes obey $[K_{-},K_{+}]=2K_3$ and $[K_3,K_{\pm }]=\pm K_{\pm }$. The Hermitian components $K_1=(K_{+}+K_{-})/2$ and $K_2=(K_{+}-K_{-})/(2i)$ satisfy $[K_1,K_2]=-i{K}_3$. Multiplying all three Hermitian components by $i$ gives the real Lie bracket signs of SU(1,1). Both occupations change equally, so their difference is conserved. This realizes the algebra used in two-mode squeezed states.