Generation of spin cat states in an engineered Dicke model

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.