Direct Measurement of Non-Hermitian Left Eigenvectors via Non-Hermitian Green's Functions

Non-Hermitian systems are fundamentally distinct from Hermitian systems in several important ways. One of them is the left and right eigenvectors are different in non-Hermitian systems, and together, they underpin the bi-orthogonality, which is crucial for the correct description of the system's behaviors. Yet, it remains unclear whether left eigenvectors are directly observable and, if so, how to observe them. Here, we demonstrate the first direct experimental measurement and extraction of both left and right eigenvectors in a non-Hermitian acoustic crystal, in which the nearest-neighbor coupling is non-reciprocal. The method to obtain the left eigenvectors roots in the fact that they are associated with the real-space position of the source in Green's function. Therefore, the left eigenvector can be experimentally obtained by moving the source across the acoustic crystal while keeping the probe fixed. Our findings deepen the understanding of fundamental aspects of non-Hermitian systems and provide a new experimental framework for investigating non-Hermitian phenomena, such as the non-Hermitian skin effects topological phase transitions.