![]() ![]() Our experimental results settle the seeming controversy between the classical and quantum properties of the Duffing oscillator and provide support to the recent results of the Liouvillian spectral theory 18, 19, 20, 21. Our experimental setup allows for a proper control of the initial state at different parameter settings as well as a high time resolution readout. Besides the wide tunability range of sample parameters in one device, the pulsed heterodyne measurement distinguishes our experiment from the experiments already reported in the literature. Here, we use an in-situ tunable superconducting nonlinear resonator to simulate the non-equilibrium quantum dynamics of the Duffing oscillator. These experiments are performed around a fixed parameter setting in a continuous-wave measurement setup. Recently, signatures of dissipative phase transition (DPT) have been observed in the scattering coefficient 14, 15, decay rate 15, 16, and second-order correlation function 17 of the Duffing oscillator, which indicate a prominent role of the quantum fluctuation in the SS. However, the seeming two classical SSs are still observed even in a typical quantum experiment setup 5, 6. These two perspectives indicate fundamentally different behaviors of the Duffing oscillator. However, it has been revealed by Drummond and Walls already in the 1980s that a fully-quantum treatment of the Duffing oscillator yields a single unique SS over the entire parameter space, such that it does not exhibit bistability or hysteresis 13. The underlying double-well potential model has been used to explain a variety of physical processes, such as optical bistability 7, 8, parametric amplification 9, 10, and self-oscillation 11, 12. This classical behavior of the Duffing oscillator has been observed in a considerable number of experiments, for example, in superconducting quantum circuits 4, 5, 6. Thermal fluctuations may induce unpredictable jumps between the two potential wells and lead to the bistability of the oscillation amplitude 3. Depending on whether the system is initially at rest or in strong oscillation, the oscillator spontaneously chooses one of the amplitudes when adiabatically tuning the parameters into the hysteretic regime. It gives rise to a hysteretic behavior where two different amplitudes of the forced oscillation are possible. In a certain parameter regime, classical mechanics predicts a double-well potential that allows two steady states (SSs) at the same parameter setting 2. The Duffing oscillator is a simple but prototypical model in nonlinear physics, which describes a forced oscillation with cubic (Kerr) nonlinearity and linear viscous damping 1. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. ![]()
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