Quantum-Geometric Length Scale for Long-distance Squeezing in Bosonic Bogoliubov Systems

Abstract

Multimode squeezing is a key resource for continuous-variable quantum technologies, but its spatial range in bosonic lattices is usually tied to dispersive propagation. Here we show that parametric pairing can create an exactly flat Bogoliubov band spanned by compact Bogoliubov generators, while producing phase-sensitive anomalous correlations and sub-vacuum collective-mode squeezing that extend far beyond their finite support. Each compact generator mixes annihilation and creation operators, thereby encoding the Bogoliubov squeezing structure, and neighboring translated generators can have nonzero commutator overlap. Enforcing canonical bosonic commutation relations therefore requires spatially extended linear combinations of these translated compact generators, which define the canonical Bogoliubov modes. The squeezing transformation of these modes varies with momentum and is quantified by the squeezing-sector symplectic quantum metric. A complex-momentum singularity of the analytically continued canonical Bogoliubov modes sets both the anomalous-correlation decay length and the momentum-space width of this metric, defining an intrinsic quantum-geometric length scale. This singularity can be tuned continuously while preserving exact flatness, thereby controlling the spatial range of correlations and squeezing. Away from exact flatness, long-distance correlations persist through multiple decay channels, while modes localized by defects and dimerized boundaries provide complementary probes of the underlying quantum geometry. In the weak-damping limit, the same quantum-geometric length scale can be extracted from frequency-filtered two-port correlations. Our results establish the quantum geometry of canonical Bogoliubov modes as a mechanism for spatially extended quantum resources arising from compact Bogoliubov generators.