Direct observation of Pancharatnam geometric phase in a quenched topological system

Berry phase, the geometric phase accumulated in cyclic adiabatic evolution, has been commonly used to define topological invariants for equilibrium states. Pancharatnam geometric phase, a purely geometric phase accumulated in generic time-evolution, extends the Berry phase to non-adiabatic and non-cyclic dynamics. Theoretically, the Pancharatnam geometric phase can perfectly identify dynamical phase transitions in quenched systems, which are analog to equilibrium phase transitions in the Ginzburg-Landau paradigm. However, due to the great challenge in eliminating the dynamical phase during a non-cyclic evolution, it is hard to observe the Pancharatnam geometric phase in dynamical phase transitions. Here, we directly observe the Pancharatnam geometric phase after sudden quenches from a topological edge state in the Su-Schrieffer-Heeger model, which is realized by a reconfigurable array of nanomechanical oscillators. Due to the chiral symmetry in our system, the initial edge state equally populates all symmetrical pairs of final eigenstates and so that the dynamical phase is naturally eliminated. We found that, the Pancharatnam geometric phase jumps $pi$ at each critical time when dynamical phase transition takes place, otherwise the Pancharatnam geometric phase keeps unchanged. This work not only provides a quantitative method to identify dynamical phase transitions, but also paves the way to reveal the bulk-edge correspondence for dynamical phase transitions in topological systems.