Measuring Electromagnetic and Gravitational Responses of Photonic Landau Levels

Topology describes global properties insensitive to local perturbation or manipulation. Mathematical examples include knots in strings, where no manipulation of a closed loop besides cutting it can change its knottedness, and the genus (number of handles) of a closed surface, where no smooth deformation can change its number of handles. Topological materials have recently become a distinct focus in condensed matter physics, appearing famously in the quantum Hall effect and topological insulators.

Synthetic materials in which the constituent particles are photons trapped in an optical resonator offer an exciting platform on which to study topological materials. Recent efforts have realized broad control over the single-photon Hamiltonian, including a strong synthetic magnetic field for photons, and strong photon-photon interactions. In this talk, I will present how a nonplanar resonator can harbor a quantum Hall system in curved space. I will then discuss measurements of three distinct topological indices, offering insight onto their physical meaning and application. We measure the Chern number via real-space local projectors: non-reciprocal products of transmission amplitudes reveal an Aharanov-Bohm phase associated with a non-zero Chern number. Two additional topological invariants, the mean orbital spin and chiral central charge, are encoded in the variation of the local density of states near a singularity of spatial curvature, revealing a complex interplay between geometry and topology. I will conclude with a view towards the experimental introduction of interactions and the role these invariants play in characterizing topological phases of matter.