 Quantum noise increment

For a broadband bosonic input with $[b(t),b^{†}(s)]=\delta (t-s)$, disjoint field increments commute and $[d{B}_t,d{B}_t^{†}]=dt$. An adapted system operator commutes with the next incoming increment. Noise increments scale as $\sqrt{dt}$, so quadratic products contribute to first-order stochastic evolution.