Dissipative phase transition in the open quantum Rabi model

We demonstrate that the open quantum Rabi model (QRM) exhibits a second-order dissipative phase transition (DPT) and propose a method to observe this transition with trapped ions. The interplay between the ultrastrong qubit-oscillator coupling and the oscillator damping brings the system into a steady state with a diverging number of excitations, in which a DPT is allowed to occur even with a finite number of system components. The universality class of the open QRM, modified from the closed QRM by a Markovian bath, is identified by finding critical exponents and scaling functions using the Keldysh functional integral approach. We propose to realize the open QRM with two trapped ions where the coherent coupling and the rate of dissipation can be individually controlled and adjusted over a wide range. Thanks to this controllability, our work opens a possibility to investigate potentially rich dynamics associated with a dissipative phase transition.