Frequency Comb Formation in Driven High Impedance Josephson Junction Arrays*

Optical frequency combs (OFCs) have a broad array of applications in metrology, spectroscopy, communications and have been realized across a range of platforms including mode-locked lasers, nonlinear microresonators, microcavity exciton-polaritons and superconducting circuits. The latter has enabled a platform for generation of OFCs in a fundamentally quantum setting, mediated by a single nonlinear Josephson Junction (JJ) coupled to a single LC mode. In this work, we study the multimode extension of such a system, analyzing the driven-dissipative dynamics of a nonlinear JJ coupled to an open high-impedance JJ array. Starting from a circuit-level description, we analyze the linearized system exactly using a non-Hermitian modal description. By then including the Josephson nonlinearity via a perturbative approach, we explore instability thresholds for the formation of frequency combs and analyze the dependence of comb properties on the circuit and input parameters. Our results are pertinent to recent experiments with artificial atoms coupled to high impedance JJ arrays that exhibit strong light-matter couplings.