  We investigate the driven quantum phase transition between the oscillating
motion and the classical nearly free rotations of the Josephson pendulum
coupled to a harmonic oscillator in the presence of dissipation. We refer to
this as the Josephson-Rabi model. This model describes the standard setup of
circuit quantum electrodynamics, where typically a transmon device is embedded
in a superconducting cavity. We find that by treating the system quantum
mechanically this transition occurs at higher drive powers than expected from
an all-classical treatment, which is a consequence of the quasiperiodicity
originating in the discrete energy spectrum of the bound states. We calculate
the photon number in the resonator and show that its dependence on the drive
power is nonlinear. In addition, the resulting multi-photon blockade phenomenon
is sensitive to the truncation of the number of states in the transmon, which
limits the applicability of the standard Jaynes-Cummings model as an
approximation for the pendulum-oscillator system. We calculate the nth order
correlation functions of the blockaded microwave photons and observe the
differences between the rotating-wave approximation and the full multilevel
Josephson-Rabi Hamiltonian with the counter-rotating terms included. Finally,
we compare two different approaches to dissipation, namely the
Floquet-Born-Markov and the Lindblad formalisms.
