  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.
