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Topological insulators possess protected boundary states which are robustagainst disorders and have immense implications in both fermionic and bosonicsystems. Harnessing these topological effects in non-equilibrium scenarios ishighly desirable and has led to the development of topological lasers. Thetopologically protected boundary states usually lie within the bulk bandgap,and selectively exciting them without inducing instability in the bulk modes ofbosonic systems is challenging. Here, we consider topological parametricallydriven nonlinear resonator arrays that possess complex eigenvalues only in theedge modes in spite of the uniform pumping. We show parametric oscillationoccurs in the topological boundary modes of one and two-dimensional systems aswell as in the corner modes of a higher-order topological insulator system.Furthermore, we demonstrate squeezing dynamics below the oscillation threshold,where the quantum properties of the topological edge modes are robust againstcertain disorders. Our work sheds light on the dynamics of weakly nonlineartopological systems driven out of equilibrium and reveals their intriguingbehavior in the quantum regime.

Topological Optical Parametric Oscillation