  Phase transitions are a fundamental concept in science describing diverse
phenomena ranging from, e.g., the freezing of water to Bose-Einstein
condensation. While the concept is well-established in equilibrium, similarly
fundamental concepts for systems far from equilibrium are just being explored,
such as the recently introduced dynamical phase transition (DPT). Here we
report on the first observation of a DPT in the dynamics of a fermionic
many-body state after a quench between two lattice Hamiltonians. With
time-resolved state tomography in a system of ultracold atoms in optical
lattices, we obtain full access to the evolution of the wave function. We
observe the appearance, movement, and annihilation of vortices in reciprocal
space. We identify their number as a dynamical topological order parameter,
which suddenly changes its value at the critical times of the DPT. Our
observation of a DPT is an important step towards a more comprehensive
understanding of non-equilibrium dynamics in general.
