Quantum behavior of a superconducting Duffing oscillator at the dissipative phase transition

  1. Qi-Ming Chen,
  2. Michael Fischer,
  3. Yuki Nojiri,
  4. Michael Renger,
  5. Edwar Xie,
  6. Matti Partanen,
  7. Stefan Pogorzalek,
  8. Kirill G. Fedorov,
  9. Achim Marx,
  10. Frank Deppe,
  11. and Rudolf Gross
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.

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