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arXiv:2608.23380

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

Sonu Verma, Sangmo Cheon, Myung-Joong Hwang, Moon Jip Park

quant-phcond-mat.mes-hallSubmitted Aug 24, 2026• Announced Aug 25, 2026• 0 citations

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.

Comments: 43 pages, 6 figures

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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.

We formulate the geometric structure of non-compact bosonic quantum systems in regimes where vacuum instability renders the relevant quantum states non-normalizable, causing conventional state-space quantum geometry -- described by the Berry connection, curvature, and quantum metric -- to become ill-defined. To overcome this breakdown, we develop a formulation of quantum geometry at the level of canonical operator frames, allowing for complexified Bogoliubov-Valatin transformations that lift the requirement that creation operators be Hermitian conjugates of annihilation operators. Canonical operator frames are defined as choices of bosonic creation and annihilation operators realizing the canonical commutation relations. A natural equivalence relation among such frames generalizes the phase ambiguity of quantum states and determines a parameter space that analytically extends the stable-regime parameter space. The space of canonical operator frames forms a principal bundle over parameter space -- the operator-frame bundle -- equipped with a natural Ehresmann connection that defines parallel transport while preserving the canonical commutation relations. In the stable regime, this construction reduces to the Berry connection, while more generally it yields a well-defined operator-space quantum geometric tensor (QGT) that remains valid across vacuum instabilities. Using the framework of rigged Hilbert spaces, we define a notion of quantum frame vacuum and obtain a consistent state-space QGT. Focusing on a single bosonic mode, we demonstrate analyticity of the QGT across the quantum frame-vacuum phase transition and present the corresponding phase diagram on the complexified squeezing-parameter plane. We further introduce a realistic physical setting that allows continuous paths connecting stable and unstable regimes, along which the QGT evolves smoothly across Stokes lines.

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

Ω^2 \to Ω^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

Ω

and

U

. In particular,

\operatorname{rank}(β_\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.

The spatial confinement of defect-induced bound states is conventionally governed by the effective mass in dispersive bands. More recently, Compact Localized States (CLSs) arising from exact destructive interference have been utilized to achieve confinement in flat bands. However, CLSs rely on pristine lattice symmetries and fine-tuned defect profiles. The introduction of a generic local impurity inevitably breaks these strict phase-matching conditions, resulting in extensive bound states whose fundamental length scale has remained an open question. Here, we establish a third regime of confinement: the quantum metric bound state. We provide a rigorous mathematical proof demonstrating that in the absence of kinetic energy and CLS protection, the exponential decay length of these states is lower-bounded by the quantum metric of the unperturbed flat band. We demonstrate the tightness of this geometric limit by constructing a family of highly tunable flat-band generators, and we verify its universality across diverse realistic architectures. Ultimately, this classification establishes the independently measurable quantum metric as a predictive design principle for engineering confined modes in synthetic wave platforms.

We investigate squeezed states of composite bosons (cobosons) formed by pairs of spin-

1/2

fermions, with emphasis on Frenkel-like cobosons. While squeezing for standard bosonic modes is well established, its extension to cobosons requires accounting for Pauli blocking and the resulting non-canonical commutation algebra. Building on earlier constructions of coboson coherent states, we define squeezed cobosons as eigenstates of a Bogoliubov transformed coboson operator and derive explicit expressions for the associated quadrature variances. We show that the underlying fermionic structure leads to state-dependent modifications of the Heisenberg--Robertson uncertainty bound, which may fall below the canonical bosonic limit without implying any violation of uncertainty principles. Numerical results based on finite-dimensional matrix representations illustrate how these effects constrain the attainable squeezing. Our framework is relevant to composite boson systems such as tightly bound electron-hole pairs and provides a physically transparent setting to probe compositeness through observable quadrature fluctuations.

Bosonic Bogoliubov-de Gennes (BBdG) Hamiltonians describe the excitations of weakly interacting Bose condensates as well as photonic systems under parametric driving. Their topological features have been studied mainly by utilizing a generalized symplectic version of the Berry curvature and related Chern numbers. However, a full characterization of geometrical features in BBdG systems is still lacking. Here, we propose a symplectic quantum geometric tensor (SQGT), whose imaginary part leads to the previously studied symplectic Berry curvature, while the real part gives rise to a symplectic quantum metric, providing a natural distance measure in the space of bosonic Bogoliubov modes. The SQGT is directly related to observable properties of BBdG systems. We show how to measure all components of the SQGT by extracting excitation rates in response to periodic modulations of the system’s parameters. Moreover, we connect the symplectic Berry curvature to a generalized symplectic anomalous velocity term for Bogoliubov-Bloch wave packets. We test our results for a bosonic Bogoliubov-Haldane model.

A quantum oscillator always carries vacuum fluctuations, and modulating its parameters converts them into correlated pairs of real quanta. This vacuum amplification underlies squeezed light and Hawking radiation. In a closed quadratic system, pair creation competes with the energy splitting of the modes, and growth turns exponential only once pairing overcomes the splitting. Here we realize this competition in two motional modes of a trapped-ion chain. A two-mode-squeezing drive creates the pairs, and a beam-splitter coupling splits the motion into two coupled modes at different frequencies. The two rates are tuned and calibrated independently. Starting near the motional ground state, we cross the stability boundary in two ways. First, with the drive at the sum of the two bare-mode frequencies, the phonon number stays bounded until the pairing rate exceeds the splitting and grows exponentially beyond it. Second, tuning the drive into resonance with one coupled mode lowers the threshold to zero, and only that mode is amplified into a squeezed vacuum. Separate control of the two interactions makes the threshold for vacuum amplification programmable. This control can be used to simulate particle creation and quantum phase transitions, and to operate parametrically driven bosonic devices close to their stability limit.