Observation of energy-band Riemann surfaces in a non-Hermitian photonic system

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