content_image/424700.jpg content_image/424699.jpg content_image/424701.jpg content_image/424706.jpg content_image/424707.jpg content_image/424708.jpg content_image/424710.jpg content_image/424712.jpg content_image/424713.jpg content_image/424702.jpg content_image/424703.jpg content_image/424709.jpg content_image/424704.jpg content_image/424705.jpg content_image/424711.jpg
the ξ 2 1 and ξ 2 2 actually measure the spin squeezing of the atomic ensemble in the x − y plane. In Eq.(20c), λ$_{min}$ is the minimal eigenvalue of matrix 

where 

is the covariance matrix and 

is the global correlation matrix. Parameters ξ 2 $_{1}$, ξ 2 $_{2}$, and ξ 2 3 were defined by Kitagawa and Ueda, Wineland et al., and Toth et al., respectively. If ξ 2 2 < 1 ( ξ 2 3 < 1) , the spin squeezing occurs, and we can safely say that the multi- partite state is entangled [19, 20]. Although we cannot say that the squeezed state according to ξ 2 1 < 1 is entan- gled, it is indeed closely related to quantum entanglement [21]. 

For the quasi-spins in the dark state, since the averages 〈 J$_{x}$J$_{z}$ 〉 = 〈 J$_{y}$J$_{z}$ 〉 = 0 vanish as the same reason of the vanishing of 〈 J$_{x}$ and 〈 J$_{y}$ 〉 , the ξ 2 1 and ξ 2 3 are simplified as [14, 21] 

where 

characterizes the spin squeezing along the z − direction. Because 〈 J 2 − 〉 = 0 for the quasi-spins in the dark state, ξ 2 1 and ξ 2 3 are always greater than 1 and thus there is no spin squeezing of the atomic ensemble in the x − y plane. In the following discussions, we only use 

to characterize the z − component spin squeezing of the atomic ensemble in the dark states. 

We consider the special case that when the wave vector difference K between the quantum and the classical field is zero. Since U | K =0 is identity matrix, the quai-spin wave operator in Eq. (16) actually is proportional to the collective operator and the reduced density matrix in Eq. (14) is spanned by the symmetric Dicke states with total quasi-spin J = N/ 2 . Thus 

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<img src='content_image/424699.jpg'>

<img src='content_image/424701.jpg'>

<img src='content_image/424706.jpg'>

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an d the straightforward calculation gives 

<img src='content_image/424702.jpg'>

<img src='content_image/424703.jpg'>

where 

is mean z − component spin square of the Dicke state | n − N/ 2 〉 and 

<img src='content_image/424709.jpg'>

For convenience we define equivalent squeezing parame- ter [14] 

which characterizes the spin squeezing when 0 < ζ 2 3 ≤ 1 . 

## C. Numerical pictures of the atomic squeezing in dark states 

The squeezing parameters of the dark state | d$_{n}$ ( θ ) 〉 without wave vector difference ( K = 0 ) are illustrated in Fig. 2, 3 and 4. In all three figures, the total number of the atoms N are chosen as 20 . Since the reduced density matrix in Eq. (14) is symmetric when θ is replaced by θ + π , the range of the θ is chosen as [0 , π ) . Fig. 2 shows 

<img src='content_image/424704.jpg'>

<img src='content_image/424705.jpg'>

<img src='content_image/424711.jpg'>

