Measurement of topological invariants in a 2D photonic system

A hallmark feature of topological physics is the presence of one-way propagating chiral modes at the system boundary1,2. The chirality of edge modes is a consequence of the topological character of the bulk. For example, in a non-interacting quantum Hall model, edge modes manifest as mid-gap states between two topologically distinct bulk bands. The bulk– boundary correspondence dictates that the number of chiral edge modes, a topological invariant called the winding number, is completely determined by the bulk topological invariant, the Chern number3. Here, for the first time, we measure the winding number in a 2D photonic system. By inserting a unit flux quantum at the edge, we show that the edge spectrum resonances shift by the winding number. This experiment provides a new approach for unambiguous measurement of topological invariants, independent of the microscopic details, and could possibly be extended to probe strongly correlated topological orders. Recently, there has been a surge of interest in investigating topological states with synthetic gauge fields. Synthetic gauge fields have been realized in various atomic4–7 and photonic systems8,9. In particular, topological photonic edge states have been imaged in two recent concurrent experiments10,11 and the robustness of their transport has been quantitatively confirmed both in the microwave12 and telecom domains13. Several other interesting studies have investigated topological states in 1D14–16, 2D17–21 and also 3D22 synthetic structures. Topological states are characterized by topologically invariant integers2. In fermionic systems, conductance measurements reveal these integer invariants. However, direct measurement of these integers is non-trivial in bosonic systems, mainly because the concept of conductance is not well defined23,24. Whereas these integers have been measured in 1D bosonic systems15,25,26, the 2D bosonic case has been limited to atomic lattices7. Here, we experimentally demonstrate that selective manipulation of the edge can be exploited to measure topological invariants, that is, the winding number of the edge states. We implement an integer quantum Hall system using a fixed, uniform synthetic gauge field in the bulk and couple an extra, tunable gauge field only to the edge. The edge state energy spectrum flows as a function of this tunable flux. With the insertion of a unit quantum of flux, the edge state resonances move by ±1, which is the winding number of edge states in our system. This spectral flow can be directly observed in an experiment as the flow of transmission resonances, and thus provides a direct measurement of the winding number. For this demonstration, we employ the unique ability of our photonic system to selectively manipulate edge states—a feature that is challenging to achieve in current electronic and atomic systems. To model the spectral flow of a quantum Hall edge, with winding number k = 1, we consider a linear edge dispersion Ep = vp where Ep is the energy, v is the group velocity, and p is the momentum along the edge. When a gauge flux (θ) is coupled to the edge, the momentum is replaced by the covariant momentum: