Limits of atomic entanglement by cavity-feedback : from weak to strong coupling

We theoretically investigate the entangled states of an atomic ensemble that can be obtained via cavity-feedback, varying the atom-light coupling from weak to strong, and including a systematic treatment of decoherence. In the strong coupling regime for small atomic ensembles, the system is driven by cavity losses into a long-lived, highly-entangled many-body state that we characterize analytically. In the weak coupling regime for large ensembles, we find analytically the maximum spin squeezing that can be achieved by optimizing both the coupling and the atom number. This squeezing is fundamentally limited by spontaneous emission to a constant value, independent of the atom number. Harnessing entanglement in many-body systems is of fundamental interest [1] and is the key requirement for quantum enhanced technologies, in particular quantum metrology [2]. In this respect, many efforts have been devoted to prepare entangled states in atomic ensembles because of their high degree of coherence and their potential for precision measurement. Spin squeezed states as well as number states have been produced following methods based either on coherent evolution in the presence of a non-linearity in the atomic field [3--5], or on quantum non-demolition measurement [6--8]. Among methods of the first kind, cavity feedback [5, 9] is one of the most promising: it has already allowed for the creation of highly squeezed states [5] and the effective non-linearity introduced by the atom-cavity coupling can be easily switched off, making it very attractive for metrol-ogy applications. In this Letter, we analyze the entangled states that can be produced by cavity feedback in different coupling regimes from weak to strong, and derive the ultimate limits of the metrology gain, extending the optimization of squeezing to unexplored domains of parameters values. After optimization of both the coupling strength and the atom number, we find a maximum squeezing limit that depends only on the atomic structure.