It has become common practice to model large spin ensembles as an effective pseudospin with total angular momentum J = N x j, where j is the spin per particle. Such approaches (at least implicitly) restrict the quantum state of the ensemble to the so
-called symmetric Hilbert space. Here, we argue that symmetric states are not generally well-preserved under the type of decoherence typical of experiments involving large clouds of atoms or ions. In particular, symmetric states are rapidly degraded under models of decoherence that act identically but locally on the different members of the ensemble. Using an approach [Phys. Rev. A 78, 052101 (2008)] that is not limited to the symmetric Hilbert space, we explore potential pitfalls in the design and interpretation of experiments on spin-squeezing and collective atomic phenomena when the properties of the symmetric states are extended to systems where they do not apply.
It is generally believed that dispersive polarimetric detection of collective angular momentum in large atomic spin systems gives rise to: squeezing in the measured observable, anti-squeezing in a conjugate observable, and collective spin eigenstates
in the long-time limit (provided that decoherence is suitably controlled). We show that such behavior only holds when the particles in the ensemble cannot be spatially distinguished-- even in principle-- regardless of whether the measurement is only sensitive to collective observables. While measuring a cloud of spatially-distinguishable spin-1/2 particles does reduce the uncertainty in the measured spin component, it generates neither squeezing nor anti-squeezing. The steady state of the measurement is highly mixed, albeit with a well-defined value of the measured collective angular momentum observable.
