Experimental measurement of non-Hermitian left eigenvectors

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. In contrast, non-Hermitian eigenstates 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 objects 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 right eigenvectors (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(Figure Presented).