Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system

When a two-dimensional electron gas is exposed to a perpendicular magnetic field and an in-plane electric field, its conductance becomes quantized in the transverse in-plane direction: this is known as the quantum Hall (QH) effect. This effect is a result of the nontrivial topology of the system's electronic band structure, where an integer topological invariant known as the first Chern number leads to the quantization of the Hall conductance. Interestingly, it was shown that the QH effect can be generalized mathematically to four spatial dimensions (4D), but this effect has never been realized for the obvious reason that experimental systems are bound to three spatial dimensions. In this work, we harness the high tunability and control offered by photonic waveguide arrays to experimentally realize a dynamically-generated 4D QH system using a 2D array of coupled optical waveguides. The inter-waveguide separation is constructed such that the propagation of light along the device samples over higher-dimensional momenta in the directions orthogonal to the two physical dimensions, thus realizing a 2D topological pump. As a result, the device's band structure is associated with 4D topological invariants known as second Chern numbers which support a quantized bulk Hall response with a 4D symmetry. In a finite-sized system, the 4D topological bulk response is carried by localized edges modes that cross the sample as a function of of the modulated auxiliary momenta. We directly observe this crossing through photon pumping from edge-to-edge and corner-to-corner of our system. These are equivalent to the pumping of charge across a 4D system from one 3D hypersurface to the opposite one and from one 2D hyperedge to another, and serve as first experimental realization of higher-dimensional topological physics.