Fig. 1: Hysteresis and its quantum features.

a Schematic of the experimental setup for pulsed heterodyne measurement. The Duffing oscillator is initially prepared in one of the two potential wells (blue and red) or in the steady state (purple). Then, we switch the driving strength to ξ, and trigger a short measurement after a waiting time of τ. The direct-current (DC) port is used to control the nonlinearity of the resonator, and we drive the resonator through one of the two radio-frequency (RF) ports for transmission- or reflection-type measurements (See Methods section). b The absolute mean field, ∣〈a〉∣, and photon number, 〈a^†a〉, measured for ω_A/2π = 7.00 GHz show a clear dependence on different initial states in the classical hysteretic regime, which is enclosed by the dashed curves calculated without any fitting parameter (See Supplementary Note 1). A drastic change happens at either of the two boundaries if the system is initially prepared in one potential well. c At a fixed detuning frequency marked by the arrows in (b), the ∣〈a〉∣ vs. ξ curves show a dip around the transition point (vertical dashed lines), while 〈a^†a〉 is a monotonic function of ξ. The error bars represent the standard deviation over 8 independent experiments. The second-order correlation function g⁽²⁾(0) is strongly peaked around the transition point, which decays towards unity for large ξ. The solid curves are Lorentzian functions serving as guides to the eyes. Source data are provided as a Source Data file.