Generalized Quantum Geometric Tensor in a Non-Hermitian Exciton-Polariton System

In this work, we review different generalizations of the quantum geometric tensor (QGT) in two-band non-Hermitian systems and propose a protocol for measuring them in experiments. We present the generalized QGT components, i.e. the quantum metric and Berry curvature, for a non-Hermitian hybrid photonic (exciton-polariton) system and show that the generalized non-Hermitian QGT can be constructed from experimental observables. In particular, we extend the existing method of measuring the QGT that uses the pseudospins in photonic and exciton-polariton systems by suggesting a method to construct the left eigenstates from experiments. We also show that the QGT components have clear signatures in wave-packet dynamics, where the anomalous Hall drift arises from both the non-Hermitian Berry curvature and Berry connection, suggesting that both left and right eigenstates are necessary for defining non-Hermitian band geometries and topologies.