  We investigate two different kinds of quantum trajectories for a nonlinear
photon resonator subject to two-photon pumping, a configuration recently
studied for the generation of photonic Schroedinger cat states. In the absence
of feedback control and in the strong-driving limit, the steady-state density
matrix is a statistical mixture of two states with equal weight. While along a
single photon-counting trajectory the systems intermittently switches between
an odd and an even cat state, we show that upon homodyne detection the
situation is different. Indeed, homodyne quantum trajectories reveal switches
between coherent states of opposite phase.
