Dynamically encircling an exceptional point through phase-tracked closed-loop control

The intricate complex eigenvalues of non-Hermitian Hamiltonians manifest as Riemann surfaces in control parameter spaces. At the exceptional points (EPs), the degeneracy of both eigenvalues and eigenvectors introduces noteworthy topological features, particularly during the encirclement of the EPs. Traditional methods for probing the state information on the Riemann surfaces involve static measurements; however, realizing continuous encircling remains a formidable challenge due to non-adiabatic transitions that disrupt the transport paths. Here, we propose an approach leveraging the phase-locked loop (PLL) technique to facilitate smooth, dynamic encircling of EPs while maintaining resonance. Our methodology strategically ties the excitation frequencies of steady states to their response phases, enabling controlled traversal along the Riemann surfaces of real eigenvalues. This study advances the concept of phase-tracked dynamical encircling and explores its practical implementation within a fully electrically controlled non-Hermitian microelectromechanical system, highlighting robust in-situ tunability and providing methods for exploring non-Hermitian topologies. Encircling exceptional points (EPs) is of great significance in studying topological properties in non-Hermitian systems; however, non-adiabatic transitions pose a major challenge to the continuous dynamically encircling of EPs. This work presents a scheme for the dynamically encircling of EPs through phase-tracked closed-loop control, enabling the smooth evolution along the eigenfrequency Riemann surface.