Experimental observation of energy-band Riemann surface

Non-Hermiticity naturally arises in physical systems that exchange energy with their environment. The presence of non-Hermiticity leads to many topological physics phenomena and device applications. In the non-Hermitian energy band theory, the foundation of these physics and applications, both energies and wave vectors take complex values. The energy bands thus become a Riemann surface, and such an energy-band Riemann surface underlies all important signatures of non-Hermitian topology. Despite a long history and recent theoretical interests, the energy-band Riemann surface has not been experimentally studied. Here, we provide a photonic observation of the energy-band Riemann surface of a non-Hermitian system. This is achieved by a tunable imaginary gauge transformation in photonic synthetic frequency dimensions. From measured topologies of the Riemann surface, we reveal the complex-energy winding, the open-boundary-condition spectrum, the generalized Brillouin zone, and the branch points. Our findings demonstrate a unified framework in the studies of diverse effects in non-Hermitian topological physics through an experimental observation of energy-band Riemann surfaces.