Spectroscopic observation of the crossover from a classical Duffing oscillator to a Kerr parametric oscillator

We study microwave response of a Josephson parametric oscillator consisting of a superconducting transmission-line resonator with an embedded DC-superconducting quantum interference device (-SQUID). The DC-SQUID allows to control the magnitude of a Kerr nonlinearity over the ranges where it is smaller or larger than the photon loss rate. Spectroscopy measurements reveal the change in the microwave response from a classical Duffing oscillator to a Kerr parametric oscillator in a single device. In the single-photon Kerr regime, we observe parametric oscillations with a well-defined phase of either 0 or pi, whose probability can be controlled by an externally injected signal.