  We follow the passage from complex amplitude bistability to phase bistability
in the driven dissipative Jaynes-Cummings oscillator. Quasidistribution
functions in the steady state are employed, for varying qubit-cavity detuning
and drive parameters, in order to track a first-order dissipative quantum phase
transition up to the critical point marking a second-order transition and
spontaneous symmetry breaking. We demonstrate the photon blockade breakdown in
the dispersive regime, and find that the coexistence of cavity states in the
regime of quantum bistability is accompanied by pronounced qubit-cavity
entanglement. Focusing on the role of quantum fluctuations in the response of
both coupled quantum degrees of freedom (cavity and qubit), we move from a
region of minimal entanglement in the dispersive regime, where we derive
analytical perturbative results, to the threshold behaviour of spontaneous
dressed-state polarization at resonance.
