Uhlmann geometric phase of mixed states measured directly in a photonic quantum walk experiment

In a paper in Science Advances, the authors report an experimentally accessible protocol for directly measuring a topological invariant of mixed quantum states, implemented in a photonic quantum walk science.orgScience Advancesscience.orgOpen source ↗. The invariant is based on the Uhlmann geometric phase, a generalization of the pure-state geometric phase to mixed states.

The geometric phase of pure states is well established and used across essentially all branches of physics, while its mixed-state counterpart remains largely unexplored experimentally. According to the authors, two obstacles stood in the way: Uhlmann parallel transport is incompatible with Hamiltonian dynamics, and topologically nontrivial mixed states are difficult to prepare.

To get around this, the authors engineered controlled nonunitary dynamics in the photonic quantum walk. This let them prepare topologically nontrivial mixed states starting from a trivial initial state. They then reconstructed the full density matrix in momentum space and directly extracted the quantized geometric phase of the nontrivial mixed states.

The authors also observed that the transition of the quantized invariant from a trivial to a nontrivial value is closely associated with the emergence of a topological kink structure science.orgScience Advancesscience.orgOpen source ↗. They state that the results show a geometric-phase framework that naturally extends to open quantum systems, both in and out of thermal equilibrium.