In open quantum systems, dissipative phase transitions (DPTs) emerge from the competition between unitary evolution, driving terms, and dissipation. First-order DPTs are characterized by a discontinuity of the steady state at the critical point, and have been observed in the paradigmatic example of the single-photon driven Kerr resonator [1,2,3]. Second-order DPTs, which exhibit a continuous but non-differentiable steady state at the critical point, have yet to be fully experimentally characterized, despite multiple theoretical investigations [4,5] . We present here a complete experimental and theoretical analysis of both first and second-order DPTs within a single parametrically-driven nonlinear superconducting resonator. We explore its non-equilibrium dynamics, showing the emergence of spontaneous symmetry breaking and bistability. Using the spectral theory of Liouvillian superoperators, we develop efficient procedures to characterize the timescales associated with these critical processes, even when simultaneously occurring. We perform a thorough study of quantum fluctuations when scaling towards the thermodynamic limit. Beyond confirming the theoretical predictions [5], our work shows a compelling example of engineering and control of criticality in superconducting circuits, paving the way for their use in critical-based sensors [6] and for quantum information technology [7]. References : [1] Chen, Q. M. (2023). Quantum behavior of the Duffing oscillator at the dissipative phase transition. Nature Communications, 14(1), 2896. [2] Rodriguez, S. R. K. (2017). Probing a dissipative phase transition via dynamical optical hysteresis. Physical review letters, 118(24), 247402. [3] Fink, T.. (2018). Signatures of a dissipative phase transition in photon correlation measurements. Nature Physics, 14(4), 365-369. [4] Minganti, F. (2021). Continuous dissipative phase transitions with or without symmetry breaking. New Journal of Physics, 23(12), 122001. [5] Minganti, F. (2018). Spectral theory of Liouvillians for dissipative phase transitions. Physical Review A, 98(4), 042118. [6] Di Candia, R. (2023). Critical parametric quantum sensing. npj Quantum Information, 9(1), 23. [7] Gravina, L. (2023). Critical Schrödinger Cat Qubit. PRX Quantum, 4(2), 020337.
