Gaussian quantum noise Ito table ($dBd{B}^{†}=(N+1)dt,d{B}^{†}dB=Ndt,d{B}^2=Mdt$)

For a zero-mean Gaussian bosonic input, $d{B}^{†}d{B}^{†}=M^{\ast }dt$ completes the displayed table, with $N\ge 0$ and $|M{|}^2\le N(N+1)$. The vacuum has $N=M=0$; an unsqueezed thermal input has $M=0$. Products with $dt$ and higher differentials vanish in the first-order Ito equation. The difference between the first two products preserves the bosonic commutator.