 Solution

The Heisenberg update is $ds=U^{†}sU-s$; 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 isby the bosonic canonical commutation relation $[a,a^{†}]=I$.

The two thermal drift expressions simplify separately toThe anomalous terms vanish because $[a^{†},[a^{†},a]]=0$ and $[a,[a,a]]=0$. Hence the factor inside the drift braces is $-(N+1)a+Na=-a$, independent of $N$ and $M$. ThereforeEquivalently, in the formal white-noise notation of the cavity-reservoir interaction,The dot is a derivative notation for this stochastic differential equation, not a claim that the white-noise field is an ordinary differentiable function.