Strict stationary squeezing limit for one-mode thermal diffusion
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Problem
Does every stable one-mode thermal Gaussian diffusion obey the sharp stationary squeezing bound \(\lambda_{\min}(V_\infty)>\nu/2\)? Use \(R=(q,p)^{\mathsf T}\), \([q,p]=i\), and covariance \(V_{jk}=\langle\{R_j-\langle R_j\rangle,R_k-\langle R_k\rangle\}\rangle\). Let \(G\) be any real symmetric \(2\times2\) matrix, \(\kappa>0\), and \(\nu=2\bar n+1\) with \(\bar n\geq0\). Define the drift and diffusion in Eq. (1).
Assume \(A\) is Hurwitz stable: both eigenvalues have negative real part. The stationary covariance \(V_\infty\) solves Eq. (2).
Sharpness means that the infimum of \(\lambda_{\min}(V_\infty)\) over stable choices of \(G\) is \(\nu/2\).
Source
This is the single-mode isotropic-diffusion form of the stationary squeezing limit in Harwood and Serafini, Eq. (15) [HS20]. Their physical setting is passive interferometric coherent feedback with rotating-wave coupling to equal-temperature Markov baths.
Progress
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Stability implies the unique solution \(V_\infty=\int_0^\infty e^{At}D e^{A^{\mathsf T}t}\,dt\) is positive definite. Multiplying Eq. (2) by \(V_\infty^{-1}\) and taking the trace gives Eq. (3), since \(\operatorname{tr}A=-\kappa\).
\begin{equation} \operatorname{tr}(V_\infty^{-1})=\frac2\nu,\qquad \frac1{\lambda_{\min}(V_\infty)}<\operatorname{tr}(V_\infty^{-1}). \tag{3} \end{equation}Equation (3) proves the strict bound directly and agrees with the published result [HS20].
For \(0\leq s<\kappa\), choose \(G_s=\begin{pmatrix}0&-s/2\\-s/2&0\end{pmatrix}\). The stable drift and covariance in Eq. (4) show sharpness.
\begin{equation} \begin{aligned} A_s&=\frac12\operatorname{diag}(-\kappa-s,-\kappa+s),\\ V_\infty(s)&=\nu\operatorname{diag}\!\left(\frac\kappa{\kappa+s},\frac\kappa{\kappa-s}\right),\\ \lim_{s\uparrow\kappa}\lambda_{\min}(V_\infty(s))&=\frac\nu2. \end{aligned} \tag{4} \end{equation}
Comment
The question is completely resolved, with a peer-reviewed source and a direct verification above. The limit is an infimum approached at instability, not a stable minimum. Isotropic noise and one mode are essential hypotheses; the statement does not cover arbitrary multimode feedback or anisotropic baths.