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Based on the map in Eq. (45) for DPC, one can find the following relations for single quasi-spin operators [14] 

Then the corresponding relations for the collective oper- ators are 〈 J$_{z}$ ( p ) 〉 = s 〈 J$_{z}$ 〉 0 and 

Substituting the relevant expectation values and the cor- relation function into Eqs. (25,26) leads to the explicit expression of the spin-squeezing parameters 

In the same sense to investigate the evolution of the con- currence, the two-qubit density matrix is present as 

with relations y D = s $^{2}$y, uD = s $^{2}$u and 

Thus we obtain the evolution of the concurrence as 

## D. Depolarizing channel 

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The subindex D in the above Eqs. (60,61,63,63) rep- resents that the evolutions of the spin-squeezing and the concurrence are taken under DPC. The sudden death of the spin-squeezing and the concurrence implied in Eq. (60,63) are shown in Fig. 12. In contrast of the ADC case, the atomic squeezing ζ 2 3 always disappear later than the concurrence C . 

Now, two competitive processes dominate generating the spin-squeezing. One is adiabatically manipulating the parameter θ from 0 to π/ 2 , which increases the spin- squeezing and the concurrence for storing the information of phonons into the atomic ensemble. However, the other process resulting from the various decoherence channels decreases the spin-squeezing and the concurrence, which means that the system loses information continuously. Therefore, the competition between this two processes leads to the existence of an optimal time for storing infor- mation, after which the information stored in the atomic ensemble is always losing. We can define two time scales: if only considering the adiabatic manipulation, one time scale is t$_{1}$ representing the time when the spin squeezing increases to the half maximum value; if only considering the decoherence processes, the other time scale is t$_{2}$ rep- resenting the time when the spin squeezing decreases to the half maximum value. Only when t$_{1}$ > t$_{2}$ the optimal time exists. 

We would like to demonstrate such competition in a typical quantum memory based on the cold $^{87}$Rb atomic ensemble, which is released from a magneto- optical trap at a temperature of about 100 µK [7]. In such quantum memory, the ground state | g 〉 , the metastable state | m 〉 and the excited state | e 〉 are chosen as ∣ 5 S$_{1}$$_{/}$$_{2}$, F = 1 , m$_{F}$ = 1 〉 , ∣ 5 S$_{1}$$_{/}$$_{2}$, F = 2 , m$_{F}$ = − 1 〉 and ∣ 5 S$_{1}$$_{/}$$_{2}$, F = 2 , m$_{F}$ = 0 〉 , respectively. An off-resonant σ $^{−}$-polarized write pulse contributes to the transition from | g 〉 to | e 〉 and the Stokes photon with σ − polar- ization associates with the transition from | e 〉 to | m 〉 . The dark state for storing the photonic information is described in Eq. (7). 

To realize the adiabatic creation of the non-classical correlation, we adopt the hyperbolic tangent type pulse sequence for both Rabi frequencies [26] as 

instead of the usual gaussian type ones [27] or solitonary type ones [28]. Here, Ω$_{m}$ is the maximum Rabi frequency, τ is the pulse length and a corresponds to the half width of the pulse. The pulse sequences of both Rabi frequen- cies are depicted in Fig. 13. Only at the time interval 

## V. OPTIMAL TIME FOR GENERATING ATOMIC SQUEEZING 

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