Pair-number statistics of polarization-entangled down-conversion ($P_n=(n+1)(1-\lambda {)}^2{\lambda }^n$)

For an ideal undepleted pump and one collected pair of spatial modes, the two orthogonal polarization-pair creation terms generate two commuting two-mode squeezed vacua. If $\lambda ={\tanh }^{2}r$, either squeezed pair has geometric occupation probability $(1-\lambda ){\lambda }^k$. Adding their pair counts gives $n+1$ equal-weight possibilities, hence $P_n=(n+1)(1-\lambda {)}^2{\lambda }^n$. Thus $P_1=2(1-\lambda {)}^2\lambda$ and $P_2=3(1-\lambda {)}^2{\lambda }^2\simeq 3P_1^2/4$ at weak gain. Different unresolved temporal or spectral modes need not obey this single-mode formula.