  Understanding the non-deterministic behavior of deterministic nonlinear
systems has been an implicit dream since Lorenz named it the "butterfly
effect". A prominent example is the hysteresis and bistability of the Duffing
oscillator, which in the classical description is attributed to the coexistence
of two steady states in a double-well potential. However, this interpretation
fails in the quantum-mechanical perspective, where a single unique steady state
is allowed in the whole parameter space. Here, we measure the non-equilibrium
dynamics of a superconducting Duffing oscillator and reconcile the classical
and quantum descriptions in a unified picture of quantum metastability. We
demonstrate that the two classically regarded steady states are in fact
metastable states. They have a remarkably long lifetime in the classical
hysteresis regime but must eventually relax into a single unique steady state
allowed by quantum mechanics. By engineering the lifetime of the metastable
states sufficiently large, we observe a first-order dissipative phase
transition, which mimics a sudden change of the mean field in a 11-site
Bose-Hubbard lattice. We also reveal the two distinct phases of the transition
by quantum state tomography, namely a coherent-state phase and a squeezed-state
phase separated by a critical point. Our results reveal a smooth quantum state
evolution behind a sudden dissipative phase transition, and they form an
essential step towards understanding hysteresis and instability in
non-equilibrium systems.
