[Paper Review] Direct measurement of the quantum geometric tensor in a two-dimensional continuous medium

Antonio Gianfrate, O. Bleu+9 more|arXiv (Cornell University)|Jan 10, 2019

Topological Materials and Phenomena25 references41 citations

TL;DR

The paper directly measures both Berry curvature and quantum metric in a 2D continuous photonic system (exciton-polaritons) using polarization-resolved photoluminescence in a high-Q planar microcavity, revealing monopole and half-skyrmion pseudospin textures and validating the quantum geometric tensor formalism.

ABSTRACT

Topological Physics relies on the specific structure of the eigenstates of Hamiltonians. Their geometry is encoded in the quantum geometric tensor containing both the celebrated Berry curvature, crucial for topological matter, and the quantum metric. The latter is at the heart of a growing number of physical phenomena such as superfluidity in flat bands, orbital magnetic susceptibility, exciton Lamb shift, and non-adiabatic corrections to the anomalous Hall effect. Here, we report the first direct measurement of both Berry curvature and quantum metric in a two-dimensional continuous medium. The studied platform is a planar microcavity of extremely high finesse, in the strong coupling regime. It hosts mixed exciton-photon modes (exciton-polaritons) subject to photonic spin-orbit-coupling which makes emerge Dirac cones and exciton Zeeman splitting breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured by polarisation-resolved photoluminescence. The associated quantum geometry of the bands is straightforwardly extracted from these measurements. Our results unveil the intrinsic chirality of photonic modes which is at the basis of topological photonics. This technique can be extended to measure Bloch band geometries in artificial lattices. The use of exciton-polaritons (interacting photons) opens wide perspectives for future studies of quantum fluid physics in topological systems.

Motivation & Objective

• Motivate the measurement of local geometric properties of bands beyond energy spectra in 2D continuous media.
• Demonstrate a direct optical protocol to extract the quantum geometric tensor (Berry curvature and quantum metric) from pseudospin textures.
• Characterize how TE-TM spin-orbit coupling and Zeeman splitting shape the quantum geometry of exciton-polariton bands.

Proposed method

• Use a high-Q planar GaAs/AlGaAs microcavity hosting strongly coupled exciton-polaritons with TE-TM splitting.
• Model the system with a two-band effective Hamiltonian H_k = (ħ^2k^2/2m*)I + Ω(k)·σ and extract Ω(k) from polarization-resolved PL.
• Compute the quantum geometric tensor components g_ij and Berry curvature B_z from the pseudospin texture via θ(k) and φ(k) as θ=arccos(S_3) and φ=arctan(S_2/S_1) and equations g_ij and B_z.
• Acquire polarization tomography data in six basis polarizations (H,V,D,A,R,L) across k-space and energy, then reconstruct E_±(k) and S_i(k) per k.
• Extract experimental parameters α, β, Δ_z from dispersion and polarization maps, and compare measured g_ij and B_z with analytical expressions derived from the effective Hamiltonian.

Experimental results

Research questions

• RQ1 Can the full quantum geometric tensor (Berry curvature and quantum metric) be directly mapped in a homogeneous 2D photonic system?
• RQ2 How do TE-TM spin-orbit coupling and Zeeman splitting shape the Berry curvature and quantum metric textures in momentum space?
• RQ3 Do emergent features like monopoles and half-skyrmions appear in the pseudospin textures of exciton-polariton bands under accessible experimental conditions?
• RQ4 Is there quantitative agreement between directly measured geometry and analytical predictions for a two-band polaritonic system?

Key findings

• Direct experimental access to both Berry curvature and quantum metric in a 2D continuous medium through polarization tomography.
• Observation of pseudospin monopoles and their associated strong peaks in the quantum metric near degeneracy points.
• Emergence of half-skyrmion pseudospin textures under applied magnetic field, evidenced by elliptical polarization and anti-crossings.
• Quantitative maps of B_z and g_ij in k-space that agree with analytic expressions derived from the effective two-band Hamiltonian (α, β, Δ_z) with measured dispersions.
• Demonstration that TE-TM spin-orbit coupling is essential for nontrivial Berry curvature and metric, while Zeeman splitting alone does not generate them.
• Discussion of implications for topological photonics and potential to extend the protocol to Bloch bands in photonic lattices and polariton-based quantum fluids.