Gaussian quantum noise Ito table
= Gaussian quantum noise Ito table
{c}
{title2=$dB\,dB^\dagger=(N+1)dt,\quad dB^\dagger dB=Ndt,\quad dB^2=Mdt$}
For a zero-mean Gaussian bosonic input, $dB^\dagger dB^\dagger=M^*dt$ completes the displayed table, with $N\geq0$ and $|M|^2\leq N(N+1)$. 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.