  We study trajectories of collective spin states of an ensemble of spinors.
The spinors considered here are either trapped ions in free space or atoms
confined in a cavity, both systems of which are engineered through their
interactions with light fields to obey an effective Dicke model. In an
appropriate limit of the Dicke model, one obtains one-axis twisting dynamics of
the collective spin and evolution after a finite time to a spin cat state, or,
in the long-time limit, the Dicke state $|S,0\rangle_x$, conditioned upon there
being no photon emissions from the system (i.e., no quantum jumps). If there is
a jump, however, the system evolves probabilistically into one of a finite
number of entangled-state cycles, where the system then undergoes a persistent
sequence of jumps between two Dicke state superpositions in a rotated basis.
The different cycles can be distinguished by the frequency at which jumps
occur.
