Maximal-Rank Squeezing from Rank-One Quenches: An Exact Spectral-Reachability Theorem for Gaussian Bosonic Systems

Abstract

We prove an exact spectral-reachability theorem for multimode squeezing generated by finite-rank quadratic quenches in Gaussian bosonic systems. For a stable sign-definite stiffness quench $\Omega^2 \to \Omega^2+\varepsilon U U^T$, the kernel, range, and rank of the anomalous Bogoliubov block are determined exactly by the block-Krylov subspace generated by $\Omega$ and $U$. In particular, $\operatorname{rank}(\beta_\varepsilon)=\sum_a \operatorname{rank}(P_a U)$, where $P_a$ projects onto each distinct free-frequency eigenspace. Consequently, the microscopic rank of a perturbation does not bound the number of squeezing channels it can activate: for a simple spectrum, a single rank-one element with nonzero overlap with every mode produces a full-rank anomalous Bogoliubov response at every nonzero stable coupling. A signed Lyapunov-Gramian path shows that spectrally reachable directions cannot disappear through finite-coupling cancellation. The theorem also gives exact dark sectors, degeneracy bottlenecks, and actuator-position rules without computing the full Bogoliubov transformation. These results provide a direct hardware-screening principle for multimode Gaussian systems: one or a few physical elements can provide simultaneous support across many canonical squeezing channels, although algebraic rank alone does not imply strong squeezing or independent tunability.