Quantum geometry of bosonic Bogoliubov quasiparticles

TL;DR

This work extends the geometry of bosonic Bogoliubov quasiparticles by introducing the symplectic quantum geometric tensor (SQGT), whose imaginary part yields the symplectic Berry curvature and whose real part defines a symplectic quantum metric, thereby providing a natural distance in Bogoliubov mode space. The SQGT is gauge-invariant, respects the indefinite Bogoliubov inner product, and has a spectral representation that couples particle and hole sectors; crucially, all components are experimentally accessible via periodic driving and measurement of Bogoliubov excitation rates. The authors develop the paraunitary Bogoliubov framework, establish local conservation laws for the symplectic curvature, and connect these geometric quantities to anomalous (transverse) velocity in Bogoliubov Bloch waves. As a concrete testbed, they study a Bogoliubov-Haldane model on a hexagonal lattice, providing a concrete algorithm to construct paraunitary transformations and demonstrating how integrated excitation rates reveal the SQGT components, with implications for ultracold-atom experiments and potential extensions to mixed or driven-dissipative BBdG systems.

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.

Figures (5)

• Figure 1: (a) Illustration of the symplectic quantum metric providing a distance measure $ds$ between infinitesimally close Bogoliubov modes $\boldsymbol{w}(\boldsymbol{\lambda}_0)$ and $\boldsymbol{w}(\boldsymbol{\lambda}_0+d\boldsymbol{\lambda})$ [cf. \ref{['eq:distance-def-sympl-final-SQGT']}]. The integrated excitation rate $\Gamma^n_{\text{int}}$ of a Bogoliubov system in response to (weak) periodic perturbations is governed by the symplectic quantum geometry [cf. \ref{['eq:ETDPT-integrated-rate-SQG', 'eq:ETDPT-integrated-rate-offdiag']}]. (b) Illustration of the transverse velocity a Bogoliubov Bloch wave packet experiences in response to an external force, which is proportional to the symplectic Berry curvature [cf. \ref{['eq:ADPT-anomal-corretion-velocity-final']}].
• Figure 2: (a) Sketch of the measurement scheme: Selecting transitions by the frequency of the drive, first the Bogoliubov mode $n$ is populated (red), second, a transition to another mode $m$ is induced (green). (b) Illustration of the Bogoliubov-Haldane model. (c) Bogoliubov energy spectrum along high-symmetry path in the first Brillouin zone (inset) for $J_2=0.1J_1$, $\theta=\pi/2$, $\Delta=3\sqrt{3}J_2/2$ and different $Un/J_1$.
• Figure 3: Symplectic quantum metric component $g_{xx}(\boldsymbol{\mathbf{q}})$ and Berry curvature $B_{xy}(\boldsymbol{\mathbf{q}})$ for the upper particle band of the Bogoliubov-Haldane model Note5 obtained via integrated excitation rates \ref{['eq:ETDPT-integrated-rate-SQG']}-\ref{['eq:ETDPT-integrated-rate-offdiag']} and via exact diagonalization \ref{['eq:SQGT-def']}. (a) Symplectic quantum metric component $g_{xx}(\boldsymbol{\mathbf{q}})$ for $Un/J_1=3$ shown in 2D quasimomentum space from numerical integration of excitation rates (left) and exact diagonalization (right). (b) Same as in (a) but for the symplectic Berry curvature $B_{xy}(\boldsymbol{\mathbf{q}})$. (c) Exact (solid line) and integrated (crosses) results for the symplectic quantum metric component $g_{xx}(\boldsymbol{\mathbf{q}})$ along a high-symmetry path (cf. inset in \ref{['fig:Ebog-Sp']}(c)) for two interactions strengths $Un/J_1=0$ (non-interacting Haldane model) and $Un/J_1=3$. (d) Same as in (c) but for the symplectic Berry curvature $B_{xy}(\boldsymbol{\mathbf{q}})$. Simulation parameters: $A/J_1=0.01$, $J_1 t/\hbar=10$ (final integration time), $\hbar\omega/J_1\in [0.1 (0.2),5.5]$ for $Un/J_1=3\,(0)$ with a frequency spacing of $\delta \omega=0.05$. Other parameters are given in \ref{['fig:Ebog-Sp']}.
• Figure S1: (a) Illustration of the Bogoliubov-Haldane model \ref{['eq:supp:Bogoliubov-Hamiltonian']} on a hexagonal lattice with two sublattice sites $A$ ($B$) with potential energy offset $\Delta$ ($-\Delta$), real (complex-valued) tunneling matrix element $J_1$ ($J_2e^{i\theta}$) for the nearest (next-) nearest neighbors, and on-site interaction strength $U$. The translation vectors connecting the (next-) nearest neighbors ($\boldsymbol{\mathbf{d}}_i$) $\boldsymbol{\mathbf{a}}_i$\ref{['eq:supp:translation-vectors']} are shown. (b) Illustration of reciprocal lattice of the hexagonal lattice structure of (a) including the reciprocal lattice vectors $\boldsymbol{\mathbf{b}}_i$\ref{['eq:supp:reciprocal-lattice-vectors']}. In the 1st Brillouin zone, we depict a high-symmetry path along the high-symmetry points $\Gamma$, $K$, $M$, and $K'$.
• Figure S2: Integrated and exact symplectic quantum metric component $g_{yy}(\boldsymbol{\mathbf{q}})$ and $g_{xy}(\boldsymbol{\mathbf{q}})$ for the upper particle band of the Bogoliubov-Haldane model \ref{['eq:supp:Bogoliubov-Hamiltonian']}. (a) Integrated (left) and exact (right) symplectic quantum metric component $g_{yy}(\boldsymbol{\mathbf{q}})$ for $Un/J_1=3$ shown in 2D quasimomentum space. (b) Same as in (a) but for the off-diagonal component $g_{xy}(\boldsymbol{\mathbf{q}})$. (c) Exact (solid line) and integrated (crosses) symplectic quantum metric component $g_{yy}(\boldsymbol{\mathbf{q}})$ along a high-symmetry path in quasimomentum space for two interactions strengths $Un/J_1=0$ (non-interacting Haldane model) and $Un/J_1=3$. (d) Same as in (c) but for the off-diagonal component $g_{xy}(\boldsymbol{\mathbf{q}})$. All the simulations have been performed for driving strengths $A/J_1=0.01$ up to a final time of $J_1 t/\hbar=10$ for a frequency range of $\hbar\omega/J_1\in [0.1 (0.2),5.5]$, for $Un/J_1=3\,(0)$ with a spacing of $\delta \omega=0.05$ over which the excitation rates have been integrated. The red-colored hexagon in (a) and (b) indicates the first Brillouin zone. Other parameters of the underlying Haldane model \ref{['eq:supp:Haldane-Model']} are the same as in \ref{['fig:Int-Rate-GxxBxy']} in the main text, i.e., $J_2=0.1J_1$, $\theta=0.5\pi$, $\Delta/(3\sqrt{3}J_2)=0.5$.

Theorems & Definitions (2)

• Definition 1: Krein space
• Definition 2: Real Krein space