Quantum geometry of bosonic Bogoliubov quasiparticles

TL;DR

This paper introduces a symplectic quantum geometric tensor for bosonic Bogoliubov systems, unifying Berry curvature and quantum metric, and proposes experimental methods to measure these geometric features.

Contribution

It develops a comprehensive symplectic quantum geometric tensor for BBdG systems, linking geometry to observable properties and extending topological characterization.

Findings

Defined the symplectic quantum geometric tensor (SQGT) for BBdG systems.

Connected SQGT components to measurable excitation responses.

Applied the framework to a bosonic Bogoliubov-Haldane model.

Abstract

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 systems' 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.