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Under the Markov approximation, a short incoming time bin is independent of the earlier input and commutes with adapted cavity operators. Write its quantum noise increment as $d{B}_{\mathrm{in}}(t)={\int }_t^{t+dt}b(\tau )d\tau$; this is the increment denoted $B_{\mathrm{in}}(t)$ in the short-time definition, not an accumulated integral from zero.

During this bin, use the cavity operator at its left endpoint. Integrating the cavity-reservoir interaction and applying $U=e^{-i\int H_{I}d\tau }$ yields the bin propagatorThe exponent is anti-Hermitian, so the bin evolution is unitary. This is the short-time stochastic form: the time-bin limit, or equivalently the time-ordered product of these adapted propagators, gives continuous evolution. The quantum noise increments scale as $\sqrt{dt}$; therefore one must retain the square of this exponent, rather than treating its integral as an ordinary order-$dt$ bounded Hamiltonian.