= Pair-number statistics of polarization-entangled down-conversion
{title2=$P_n=(n+1)(1-\lambda)^2\lambda^n$}
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 $\lambda=\tanh^2r$, 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 $P_2=3(1-\lambda)^2\lambda^2\simeq3P_1^2/4$ at weak gain. Different unresolved temporal or spectral modes need not obey this single-mode formula.
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