Arbitrary photonic linear transformations using synthetic frequency dimensions

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.