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Use units $\hslash =1$. A Hermitian quadrature coupling of a cavity mode to a continuum of external modes can be writtenwith real coupling amplitude $\kappa$. The free Quantum Hamiltonian is $H_0={\omega }_c{a}^{†}a+\int \omega b_{\omega }^{†}{b}_{\omega }d\omega$. In the interaction picture, the two photon-exchange products $a{b}_{\omega }^{†}$ and $a^{†}{b}_{\omega }$ acquire phases with frequency $\omega -{\omega }_c$, whereas $a^{†}{b}_{\omega }^{†}$ and $a{b}_{\omega }$ oscillate at $\omega +{\omega }_c$.

For weak coupling near resonance, the rotating-wave approximation drops those rapidly oscillating pair-creation and pair-annihilation terms and retainsFor a broadband flat coupling $\kappa =\sqrt{\gamma /(2\pi )}$, define the envelope annihilation operator $b(t)=(2\pi {)}^{-1/2}\int b_{\omega }{e}^{-i(\omega -{\omega }_c)t}d\omega$. The Markov approximation extends the detuning integral over the full real line and gives $[b(t),b^{†}(s)]=\delta (t-s)$. Then the cavity-reservoir interaction isThe phases and normalization of the external field fix the sign convention. The creation operators and annihilation operators exchange one excitation between cavity and field, rather than creating or destroying a pair.

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.

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.