We present a new approach for deriving exact closed-form solutions for the steady state of a wide class of driven-dissipative nonlinear resonators that is distinct from more common complex-P-function methods. Our method generalizes the coherent quantum-absorber approach of Stannigel et al. [New J. Phys. 14, 063014 (2012)] to include nonlinear driving and dissipation and relies crucially on exploiting the Segal-Bargmann representation of Fock space. Our solutions and method reveal a wealth of previously unexplored observable phenomena in these systems, including new generalized photon-blockade and antiblockade effects and an infinite number of new parameter choices that yield quantum bistability.
Multi-tone external driving offers a route to parametric physics without directly modulating the device. However, the validity of the parametric response in the few-photon regime remains underexplored. Here, we apply two coherent microwave tones to a Josephson-junction Kerr oscillator and stimulate degenerate parametric downconversion via four-wave mixing. Using transmission spectroscopy, we observe that the response retains the qualitative semiclassical Kerr parametric oscillator structure, including its instability lobe and bistable phase-space topology. Interestingly, we demonstrate that a conventional single-mode reduction fails to capture the system quantitatively: the predicted AC Stark shift is severely underestimated, and the reported distributions might not be fully physical when the single-photon Kerr shift
K exceeds the cavity linewidth
κ. Instead, we show that a full three-tone quantum description accurately reproduces the experimental observables. There, quantum fluctuations of the drive tones become dynamically dominant over dissipation, and all three interacting tones operate in a deep few-photon limit where the expected semiclassical macroscopic lobes undergo fundamental renormalization due to profound mixing with quantum variance. Our results establish two-tone-driven Kerr oscillators as potential parametric amplifiers and open new horizons to explore the quantum-to-classical crossover in driven-dissipative circuits.