Theorem: Joint count generating function and Wick identities

For $0\le s_1,s_2\le 1$, with $S=\mathrm{diag}(s_1,s_2)$,

More explicitly,

Its Taylor coefficients at $(0,0)$ give the complete joint count law.

Each marginal is geometric with mean ${\nu }_i$, and

Moreover,

The packet factors are statistically independent if and only if $A_{12}=0$.

Proof. Conditional on a coherent amplitude $\alpha$, the two counts are independent Poisson variables of means $|{\alpha }_i{|}^2$. The coherent-state mixture in Theorem thm:gaussian gives

The circular Gaussian integral equals

This also covers singular $A$ by its diagonal Gaussian construction. Setting either $s_i=1$ gives its geometric marginal.

Gaussian contraction gives (eq:wick). For different modes, $N_1{N}_2={b_1}^{†}{b_2}^{†}{b}_1{b}_2$; hence $⟨N_1{N}_2⟩={\nu }_1{\nu }_2+|A_{12}{|}^2$. The same-mode calculation gives the stated variance. All polynomial moments are finite.

If $A_{12}=0$, the characteristic function factorizes, so the state is the product of its marginals. Conversely, a product centered state has zero cross covariance.