Dicke states and Bloch states

In this paper, we show that if the density matrix of a pure Bloch state is averaged over the common phase, which connects the excited state of every atom with its ground state, the result is identical to a mixture of Dicke states with r = N/2 and a narrowly peaked distribution of m. It follows that Dicke states are essentially indistinguishable from Bloch states, except by experiments sensitive to phi, or by precise measurements of m. We also show that a thermally prepared quasi-Bloch state can be expanded in two ways: into a mixture of sub-maximal Bloch states, and alternately after averaging over phi into a mixture of Dicke states.