= Solution
The Heisenberg update is $ds=U^\dagger sU-s$; the left side of the printed first update relation must be this increment, rather than $s(t)$ itself. For $s=a$, the linear noise term in the supplied stochastic equation is
$$
\sqrt\gamma[a^\dagger\otimes dB-a\otimes dB^\dagger,a]=-\sqrt\gamma\,dB,
$$
by the bosonic <canonical commutation relation> $[a,a^\dagger]=I$.
The two thermal drift expressions simplify separately to
$$
2a^\dagger a a-aa^\dagger a-a^\dagger a a=-a,\qquad
2aaa^\dagger-aaa^\dagger-aa^\dagger a=a.
$$
The anomalous terms vanish because $[a^\dagger,[a^\dagger,a]]=0$ and $[a,[a,a]]=0$. Hence the factor inside the drift braces is $-(N+1)a+Na=-a$, independent of $N$ and $M$. Therefore
$$
\boxed{da=-\frac\gamma2a\,dt-\sqrt\gamma\,dB_{\mathrm{in}}.}
$$
Equivalently, in the formal white-noise notation of the <cavity-reservoir interaction>,
$$
\dot a(t)=-\frac\gamma2a(t)-\sqrt\gamma\,b_{\mathrm{in}}(t).
$$
The dot is a derivative notation for this stochastic differential equation, not a claim that the white-noise field is an ordinary differentiable function.
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