While coherently-driven Kerr microcavities have rapidly matured as a platform for frequency comb formation, such microresonators generally possess weak Kerr coefficients; consequently, triggering comb generation requires millions of photons to be circulating inside the cavity. This suppresses the role of quantum fluctuations in the combs dynamics. In this paper, we realize a minimal version of coherently-driven Kerr-mediated microwave frequency combs in the circuit QED architecture, where the quantum vacuums fluctuations are the primary limitation on comb coherence. We achieve a comb phase coherence of up to 35~$mu$s, approaching the theoretical device quantum limit of 55~$mu$s, and vastly longer than the modes inherent lifetimes of 13~ns. The ability within cQED to engineer stronger nonlinearities than optical microresonators, together with operation at cryogenic temperatures, and excellent agreement of comb dynamics with quantum theory indicates a promising platform for the study of complex dynamics of quantum nonlinear systems
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