When a two-dimensional (2D) electron gas is placed in a perpendicular magnetic field, its in-plane transverse conductance becomes quantized; this is known as the quantum Hall effect1. It arises from the non-trivial topology of the electronic band structure of the system, where an integer topological invariant (the first Chern number) leads to quantized Hall conductance. It has been shown theoretically that the quantum Hall effect can be generalized to four spatial dimensions2,3,4, but so far this has not been realized experimentally because experimental systems are limited to three spatial dimensions. Here we use tunable 2D arrays of photonic waveguides to realize a dynamically generated four-dimensional (4D) quantum Hall system experimentally. The inter-waveguide separation in the array is constructed in such a way that the propagation of light through the device samples over momenta in two additional synthetic dimensions, thus realizing a 2D topological pump5,6,7,8. As a result, the band structure has 4D topological invariants (known as second Chern numbers) that support a quantized bulk Hall response with 4D symmetry7. In a finite-sized system, the 4D topological bulk response is carried by localized edge modes that cross the sample when the synthetic momenta are modulated. We observe this crossing directly through photon pumping of our system from edge to edge and corner to corner. These crossings are equivalent to charge pumping across a 4D system from one three-dimensional hypersurface to the spatially opposite one and from one 2D hyperedge to another. Our results provide a platform for the study of higher-dimensional topological physics.
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We thank H. M. Price, M. Lohse, C.-X. Liu, W. Benalcazar, E. Prodan and T. Ozawa for their comments and feedback on the manuscript. O.Z. thanks the Swiss National Science Foundation for financial support. M.C.R. acknowledges the National Science Foundation under award number ECCS-1509546, the Charles E. Kaufman Foundation, a supporting organization of the Pittsburgh Foundation, and the Alfred P. Sloan Foundation under fellowship number FG-2016-6418. K.P.C. acknowledges the National Science Foundation under award numbers ECCS-1509199 and DMS-1620218.
The authors declare no competing financial interests.
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Extended data figures and tables
a, The overall scale A (dashed red line) and exponential decay prefactor γ (solid orange line) that describe the inter-waveguide coupling as a function of their separation s: t(s) = Aexp(−γs). The parameters were obtained using a thorough calibration procedure (see Methods) and are plotted as a function of wavelength. b, An additional illustration of the waveguide spacing used to implement our topological pump. To simplify the diagram, we show a 1D waveguide array, which corresponds to an implementation of a 1D pump. This configuration can be thought of as resulting from a constant y slice through the full 2D waveguide array.
See equation (2). a, Finite-sample band structure (energy E versus pump parameter) for a single Harper model aligned along the x direction. Boundary modes highlighted in orange (red) are localized on the left (right) end of the 1D sample. The first Chern number associated with each bulk band is also shown. b, Band structure for the fully separable 2D pump taken along the path ϕx = ϕy for a system that decomposes into two independent Harper models. Each band in b is obtained by summing a pair of bands from a. The resulting bands can be classified by the types of state that appear in the sum: bulk–bulk (2D bulk), bulk–boundary (2D edge) or boundary–boundary (2D corner). These types are respectively coloured grey, red or orange, and black. As a function of ϕi, the edge modes form ‘dispersive’ bands that thread through the 2D bulk gaps. The corner modes thread between the edge bands and are therefore forced to cross 2D bulk bands along their ϕi trajectory.
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Zilberberg, O., Huang, S., Guglielmon, J. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018). https://doi.org/10.1038/nature25011
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