Graduate Quantum synchronization description used in a paper

[Danny Boy]

In the paper "Steady-state spin synchronization through the collective motion of trapped ions" it states the following:

"Steady-state synchronization of atomic dipoles forms the foundation for ultra-stable optical lasers utilizing
narrow-linewidth atoms coupled to a lossy cavity mode. The cavity mode acts as a channel for synchronization of the atomic dipoles (spins) resulting in a macroscopic collective dipole in steady-state composed of correlated atoms. Synchronization here refers to the development of a preferred relative phase (correlations) between every pair of spins. "

Question:
Does anyone know how the statement in bold could be expressed mathematically (to add some clarity)?

Thanks for any assistance.

 

[azntoon]

Answer: The statement in bold can be expressed mathematically as: Let ϕ_i and ϕ_j be the relative phases between spins i and j, then the steady-state synchronization of atomic dipoles is given by the correlationC_{ij} = E[cos(ϕ_i - φ_j)]