 Solution

The stated $M$ is specifically the transfer from the external input to the internal field $b_1$. Combining $b_1=G{b}_0$ with $b_0=\beta b_{min}-\alpha b_1$ givesFor $\gamma >0$ and a nondegenerate beam splitter $|\alpha |<1$, the closed-loop pole is $-{\gamma }_{meff}/2<0$. The return ratio for the Nyquist stability criterion is $L=\alpha G$. Its frequency locus is a circle of radius $|\alpha |<1$, so it cannot reach or encircle $-1$. The open cavity has no unstable poles; hence the closed loop has none. This agrees with the directly derived state damping rate.

The externally accessible output is different. From $b_2=\alpha b_{min}+\beta b_1$,It is again an all-pass cavity response. Confusing $b_1$ and $b_2$ would give the wrong output steady state.

For a constant coherent input mean $B=⟨b_{min}⟩$, the steady mean cavity amplitude follows by setting its derivative to zero:The input-output values also follow from $G(0)=-1$ and $M(0)=-\beta /(1-\alpha )$. For any initial mean amplitude, its difference from the stationary mean decays as $e^{-{\gamma }_{meff}t/2}$.

The stability statement needs endpoint qualifications. At $\alpha =1$, normalization forces $\beta =0$, and the cavity obeys $\dot{a}=0$: it is isolated and undamped, not asymptotically stable. Its external output is nevertheless the decoupled input. At $\alpha =-1$, the algebraic elimination divides by zero; the instantaneous feedback connection is singular and is not a well-posed instance of these reduced equations. Thus strict cavity stability holds for $|\alpha |<1$, not for every real pair satisfying only ${\alpha }^2+{\beta }^2=1$.

Finally, $b_{min}(t)$ is a quantum stochastic field, so arbitrary inputs do not approach a literal time-independent operator amplitude. The stationary expressions above describe a constant coherent drive's means, or the formal zero-frequency response. The full solution contains the filtered input-noise convolutionThe stationary cavity density operator also depends on the input noise statistics. For this ideal passive cavity with a coherent drive and vacuum noise, it is the coherent state with the amplitude just calculated; without such input statistics, the gain function alone does not specify a complete stationary quantum state.