  In open quantum systems, first- and second-order dissipative phase
transitions (DPTs) can emerge in the thermodynamic limit from the competition
between unitary evolution, driving terms, and dissipation. The order of a DPT
is defined by the continuity properties of the steady state. Until now,
second-order DPTs have predominantly been investigated theoretically, while
first-order DPTs have been observed in key experiments based on the theory of
the single-photon driven Kerr resonator. We present here the first
comprehensive experimental and theoretical analysis of both first and
second-order DPTs in a two-photon (i.e., parametrically) driven Kerr
superconducting resonator. Firstly, we characterize the steady state and its
main features at the second- and first-order critical points: squeezing below
vacuum and coexistence of two phases with different photon numbers,
respectively. Then, by continuously monitoring the system along quantum
trajectories, we study the non-equilibrium dynamics across the critical points.
We witness the hysteresis cycles associated with the first-order DPT and the
spontaneous symmetry breaking due to the second-order DPT. Applying the
spectral theory of the Liouvillian superoperator, we develop efficient
procedures to quantify the critical slowing down associated with the timescales
of these processes. When scaling towards the thermodynamic limit, these
timescales span five orders of magnitude. Our results corroborate the
predictions derived using the Liouvillian theory of DPTs. This work stands as a
compelling example of engineering and controlling of criticality in
superconducting circuits. It marks a significant advancement in the use of
two-photon driven Kerr resonators for criticality-enhanced quantum information
applications.
