Tomographic Characterization of Non-Hermitian Hamiltonians in Reciprocal Space.

Non-Hermitian Hamiltonians enrich quantum physics by extending conventional phase diagrams, enabling novel topological phenomena, and realizing exceptional points with applications in quantum sensing. Here, we present a programmable liquid-crystal-based photonic platform for implementing nonunitary quantum walks with topological band structures. The novelty of this platform lies in its direct access to the reciprocal space, which enables us to perform a momentum-resolved characterization of the underlying Hamiltonian. We report a complete process tomography of a single step of several walks, from which we retrieve complex-valued energy bands, resolve exceptional points, and detect the associated parity-time symmetry breaking through eigenvector coalescence. Our Letter offers a powerful complementary route to retrieving topology beyond dynamical measurements.