Recent technological advances in integrated photonics have spurred on thestudy of topological phenomena in engineered bosonic systems. Indeed, thecontrollability of silicon ring-resonator arrays has opened up new perspectivesfor building lattices for photons with topologically nontrivial bands andintegrating them into photonic devices for practical applications. Here, wepush these developments even further by exploiting the different modes of asilicon ring resonator as an extra dimension for photons. Tunneling along thissynthetic dimension is implemented via an external time-dependent modulationthat allows for the generation of engineered gauge fields. We show how thisapproach can be used to generate a variety of exciting topological phenomena inintegrated photonics, ranging from a topologically-robust optical isolator in aspatially one-dimensional (1D) ring-resonator chain to a driven-dissipativeanalog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Ourproposal paves the way towards the use of topological effects in the design ofnovel photonic lattices supporting many frequency channels and displayinghigher connectivities.
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