Implementing frequency-encoded photonic linear transformations can be of significant interest not only for quantum information processing and machine learning hardware accelerators, but also for optical signal processing, communications, and spectrotemporal shaping of light. We present a flexible, reconfigurable architecture to implement such arbitrary linear transformations for photons using the synthetic frequency dimension of dynamically modulated microring resonators. Inverse design of the coupling between the frequency modes enables arbitrary scattering matrices to be scalably implemented with high fidelity, allowing for nonreciprocal frequency translation, unitary and nonunitary transformations. Our results introduce new functionalities for linear transformations beyond those possible with real-space architectures that are typically time-invariant.
Higher-order topological insulators (HOTIs), originating from quantized quadrupole and octupole moments, have attracted significant interest since they support boundary states that are two or more dimensions lower than their bulk. However, previous HOTIs have been restricted to real-space dimensions. Here we construct photonic HOTIs using synthetic dimensions, comprising frequency modes of dynamically modulated rings. We show how quadrupole and octupole HOTIs supporting topologically protected corner modes emerge in a lattice of modulated photonic molecules and predict a dynamical topological phase transition in this system. Additionally, we propose a quantized hexadecapole (16-pole) insulator by leveraging synthetic dimensions to create a 4D hypercubic lattice that cannot be realized in real space.
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