Experimental Measurement of Non-Hermitian Left Eigenvectors

Abstract: The duality of left and right eigenvectors underpins the comprehensive understanding of many physical phenomena. In Hermitian systems, left and right eigenvectors are simply Hermitian-conjugate pairs. Non-Hermitian eigenstates in contrast, have left and right eigenvectors that are distinct from each other. However, despite the tremendous interest in non-Hermitian physics in recent years, the roles of non-Hermitian left eigenvectors (LEVs) are still inadequately explored-their physical consequences and observable effects remain elusive, so much so that LEVs seem largely like an object of primarily mathematical purpose. In this study, we present a method based on the non-Hermitian Green's function for directly retrieving both LEVs and REVs from experimentally measured steady-state responses. We validate the effectiveness of this approach in two separate acoustic experiments: one characterizes the non-Hermitian Berry phase, and the other measures extended topological modes. Our results not only unambiguously demonstrate observable effects related to non-Hermitian LEVs, but also highlight the under-appreciated role of LEVs in non-Hermitian phenomena.