  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.
