5 FIG. 5: (Color online) The polar plot of the squeezing pa - ra m eter ξ 2 3 versus the vector difference K . The total num ber of the atoms N , the excitation number n and the parame ter θ are chosen as 20 , 4 and π/ 2 . Here, ξ 2 3 = 1 is the b a se- li ne to measure whether the dark state is squeezed or no t. O bviously, the dark state is squeezed only when the vecto r difference K ∈ [ − K$_{c}$ , K$_{c}$ ] . FIG. 3: (Color online) The polar plot of the squeezing pa- ram e ter ξ 2 3 versus the parameter θ for the excitation nu mber n = 0 (blue dotted line), n = 10 (orange dashed line) and n = 2 0 (red solid line). The total number of the atoms N are chos e n as 20 . The greatest spin-squeezing is obtained for the symmetric Dicke state when the half of the quasi-spins are excited ( n = N/ 2 ) and θ = π/ 2 . FI G. 4: (Color online) The contour plot of the squeezing pa- ra m eter ξ 2 3 versus the excitation number n and parameter θ . Th e total number of the atoms N are chosen as 20 . The gre atest spin-squeezing ( ζ 2 3 = 1 ) is obtained for the symmet- ric Dicke state when the half of the quasi-spins are excited ( n = N/ 2 ) and θ = π/ 2 . and the photon Fock state, this non-vanishing squeez- ing mainly results from the sub-Poisson distribution of the photons, which will be explicitly demonstrated in the last subsection of the Sec. II. Fig. 3 is the polar plot of the squeezing parameter ρ = ξ 2 $_{3}$( θ ) versus θ for different excitation numbers n . For the state without excitation ( n = 0 ), the atomic ensemble always has no spin-squeezing because all of the quasi-spins are in their ground states. The greatest spin-squeezing is obtained for the symmetric Dicke state when the half of the quasi- spins are excited ( n = N/ 2 ) and θ = π/ 2 . Fig. 4 is the contour plot of the squeezing parameter ζ 2 3 versus the excitation number n and parameter θ . Obviously, the greatest spin-squeezing ( ζ 2 3 = 1 drawn by red color) can be obtained by adiabatically manipulating the pa- rameter θ to π/ 2 at the half excitation of the quasi-spins ( n = N/ 2 ). If the wave vectors between the quantum and the clas- sical field are different, the nonzero K results in the non- symmetrical Dicke bases in Eq. (14) and the spin squeez- ing decreases rapidly as the K deviates from 0 . Fig. 5 is the polar plot of the squeezing parameter ρ = ξ 2 $_{3}$( θ ) ver- sus K . The parameters are chosen as N = 20 , n = 4 , and θ = π/ 2 . Since ξ 2 3 = 1 is the baseline to measure the atomic squeezing, the dark state is squeezed only when the vector difference K ∈ [ − K$_{c}$ , K$_{c}$ ] , where K$_{c}$ is shown in Fig. 5. Fig. 6 is the contour plot of the squeezing pa- rameter ζ 2 3 versus the vector difference K and parameter θ . The total number of the atoms N and the excitation number n are chosen as 20 , 4 . In Fig. 6, [ − K$_{c}$ , K$_{c}$ ] is still th at the squeezing parameter ζ 2 3 varies with the excita- tio n number n for different θ . The dark state is always sq ueezed except without excitation ( n = 0 ) or fully ex- cited ( n = N ). It is noticed that when θ = 0 , π/ 2 , the dark state is squeezed even when the excitation number n is equal to the total number of the atoms N . Because the dark state actually mix the quasi-spins wave state