Measuring the Adiabatic Non-Hermitian Berry Phase in Synthetically Coupled Mechanical Oscillators.*

The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the quantum state. Here we measure the adiabatic non-Hermitian Berry phase by using a two-site PT-symmetric quantum Hamiltonian, inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further exploration of non-Hermitian and topological physics in classical synthetic metamaterials.