  The fundamental concept underlying topological phenomena posits the geometric
phase associated with eigenstates. In contrast to this prevailing notion,
theoretical studies on time-varying Hamiltonians allow for a new type of
topological phenomenon, known as topological dynamics, where the evolution
process allows a hidden topological invariant associated with continuous flows.
To validate this conjecture, we study topological chiral and nonreciprocal
dynamics by encircling the exceptional points (EPs) of non-Hermitian
Hamiltonians in a trapped ion system. These dynamics are topologically robust
against external perturbations even in the presence dissipation-induced
nonadiabatic processes. Our findings indicate that they are protected by
dynamical vorticity -- an emerging topological invariant associated with the
energy dispersion of non-Hermitian band structures in a parallel transported
eigenbasis. The symmetry breaking and other key features of topological
dynamics are directly observed through quantum state tomography. Our results
mark a significant step towards exploring topological properties of open
quantum systems.
