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group velocity v$_{g}$ of the forward (backward)-propagating light pulse is shown in Fig.2. One finds that: 1), the ap- proximate solution with cutoff at finite ℓ yields a non-zero v$_{g}$ , and 2), v$_{g}$ ( ℓ ) reaches the maximum value at ℓ = 1, and then vanishes with √$_{2}$$_{ℓ}$$_{+ 1}$$_{/ℓ}$$_{.}$ Since the real sys- tem corresponds to the infinite-order limit, one “seems” to conclude that the recreated light forms a stationary pulse. 

For cold atomic systems where temperature is low but nonzero, one has γ$_{0}$ = 0 and γ$_{n}$ ̸ = 0 for n ̸ = 0. In this case, we set γ$_{n}$ = | n | a Γ, with a the decay constant. This decay model can well describe the decays of the coef- ficients of the ground-spin/optical coherence in various cold atomic systems. For instance, in the laser-cooled cold atomic ensembles, the higher-order coefficients have a phase grating of e ink$_{c}$z across the atomic gases, and thus will decay due to atomic random motion. The de- cay rate can be estimated by the time needed for the atoms moving across one wavelength of the phase grat- ing γ$_{n}$ ∼ | v$_{s}$ λ | = n | k$_{c}$v$_{s}$ 2 π | [17]. In the Bose condensation, the higher-order coefficients can be regarded as a particle excitation with momentum | n | k$_{c}$ ℏ , At average, they move out of the atomic gases after a time of L $_{(}$| n | k$_{c}$ ℏ m ) − 1 [18], and the decay rate can be approximated by γ$_{n}$ ∼ | n | k$_{c}$ ℏ mL . 

## III. NUMERICAL SOLUTION 

To check the validity of the approximate solution in Fig. 2 and further find the dynamics of the recreated light 

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pulse in cold atomic systems ( γ$_{n}$ = | n | a Γ) for which the approximate treatment is unavailable, we directly simu- late Eqs.(6)–(9). Naturally, a cutoff takes place at finite ℓ and accordingly 5+4 ℓ equations are involved in each sim- ulation. The result for real systems ( ℓ → ∞ ) is obtained from the extrapolation of simulations for finite ℓ . 

We first consider the zero-decay limit ( γ$_{n}$ = 0). The initial condition is taken such that: 1), only the zeroth component S$_{0}$ ( z, t = 0) of the ground-spin coherence is nonzero while all other components S$_{2}$$_{n}$ ( z, 0) are zero; S$_{0}$ assumes a Gaussian shape S$_{0}$ ( z, 0) = e − ( z/L$_{0}$ ) 2 with L$_{0}$ the length of the wave packet; 2), all components of the optical coherence are zero P$_{2}$$_{n}$$_{+1}$ ( z, 0) = 0, and 3), no probe light exists at the beginning E + p ( z, 0) = E − p ( z, 0) = 0. Further, the wave-packet length is set at L$_{0}$ = 5 l$_{abs}$ , with l$_{abs}$ = Γ c g $^{2}$N the absorption length. To be in the slow- light regime, we chose the parameters to be Ω$_{c}$ = 0 . 69Γ, g $^{2}$N = 138Γ $^{2}$. The equations are directly solved by Lax- Friedrichs method with sufficiently small step. The sim- ulation is up to ℓ = 100, and the group velocity v$_{g}$ of the recreated forward (backward)-propagating light pulse is measured. The results are shown in Fig. 2. As the ap- proximate analysis, v$_{g}$ ( ℓ ) reaches its maximum at ℓ = 1 and then starts to decrease. Nevertheless, the decrease of v$_{g}$ becomes slower and slower after ℓ ≈ 14 and eventu- ally stays unchanged at v$_{g}$ = 0 . 47 ± 0 . 006. From Fig. 2, 

