Parametric instability and effective non-Hermitian dynamics in a bosonic Kitaev model with Kerr nonlinearity

A multimode superconducting parametric cavity can efficiently simulate bosonic lattice models, whose lattice sites in synthetic dimensions are resonant modes subject to flux-induced parametric couplings. In particular, the combination of beam-splitter and downconversion couplings allows to realize a bosonic Kitaev chain, known to exhibit phase-dependent chiral transport and an effective non-Hermitian skin-effect. In this work, we theoretically study in the semiclassical framework the interplay of the intrinsic Kerr nonlinearity and the non-Hermitian dynamics in a short synthetic Kitaev chain. We focus on the coupling strength threshold for parametric instability and the above-threshold parametric oscillations, and confirm their sensitive dependence on the amplitudes and phases of the couplings at the edge. This provides an example for the non-Hermitian skin-effect in an experimentally realistic nonlinear system.