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Entangled-State Cycles of Atomic Collective-Spin States

We study quantum trajectories of collective atomic spin states of $N$ effective two-level atoms driven with laser and cavity fields. We show that interesting ``entangled-state cycles'' arise probabilistically when the (Raman) transition rates between the two atomic levels are set equal. For odd (even) $N$, there are $(N+1)/2$ ($N/2$) possible cycles. During each cycle the $N$-qubit state switches, with each cavity photon emission, between the states $(|N/2,m>\pm |N/2,-m>)/\sqrt{2}$, where $|N/2,m>$ is a Dicke state in a rotated collective basis. The quantum number $m$ ($>0$), which distinguishes the particular cycle, is determined by the photon counting record and varies randomly from one trajectory to the next. For even $N$ it is also possible, under the same conditions, to prepare probabilistically (but in steady state) the Dicke state $|N/2,0>$, i.e., an $N$-qubit state with $N/2$ excitations, which is of particular interest in the context of multipartite entanglement.