= Solution
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 $dB_{\mathrm{in}}(t)=\int_t^{t+dt}b(\tau)d\tau$; 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 propagator
$$
\boxed{U_I(t+dt,t)=\exp\!\left[\sqrt\gamma\{a(t)\otimes dB_{\mathrm{in}}^\dagger(t)-a^\dagger(t)\otimes dB_{\mathrm{in}}(t)\}\right].}
$$
The 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.
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