  We study a dissipative Kerr-resonator subject to both single- and two-photon
detuned drives. Beyond a critical detuning threshold, the Kerr resonator
exhibits a semiclassical first-order dissipative phase transition between two
different steady-states, that are characterized by a $\pi$ phase switch of the
cavity field. This transition is shown to persist deep into the quantum limit
of low photon numbers. Remarkably, the detuning frequency at which this
transition occurs depends almost-linearly on the amplitude of the single-photon
drive. Based on this phase switching feature, we devise a sensitive quantum
transducer that translates the observed frequency of the parametric quantum
phase transition to the detected single-photon amplitude signal. The effects of
noise and temperature on the corresponding sensing protocol are addressed and a
realistic circuit-QED implementation is discussed.
