Classifying reconfigurable photonic topology using matrix homotopy

One drawback of linear photonic topological systems is that the locations of their protected boundary-localized states are generally fixed during the fabrication of a device, as it is challenging to subsequently alter the system’s geometry. Instead, ideally, it would be desirable to realize reconfigurable topological interfaces, such that the propagation path of the boundary-localized states could be altered to yield a variety of devices, such as routers. Here, I will show how a local theory of material topology rooted in ideas from matrix homotopy – not band theory – enables the classification of topology in dynamic nonlinear and non-Hermitian systems. This framework, rooted in the spectral localizer, a suite of local markers and enables the rigorous definition of topological dynamics, i.e., changes in the system’s local topology as a function of time. Applying this framework to exciton-polariton lattices, we show that a non-resonant pump can be used to realize picosecond-scale reconfigurability.