content_image/388270.jpg content_image/388269.jpg
<img src='content_image/388270.jpg'>

it looks rather secure to conclude that the group velocity v$_{g}$ takes a finite value for ℓ → ∞ , suggesting that the recreated light splits into two counter-propagating light pulses. This defies the earlier approximate solution that the light forms a stationary light pulse. 

We then consider the effect of the wave-packet length L$_{0}$ for the γ$_{n}$ = 0 case and set L$_{0}$/l$_{abs}$ = 5 , 10 , 20 , 30 , 40, and 50. We define the sum mode E$_{s}$ = E + p + E − p and difference mode E$_{d}$ = E + p − E − p , and measure the inten- sity of the light pulses | E$_{s}$ | $^{2}$+ | E$_{d}$ | 2 as a function of time and position z . The results for ℓ = 30, where the simula- tion already reaches the steady state, are shown in Fig. 3. One observes that, for all the cases, the generated light always splits into two counter-propagating light pulses. Nevertheless, as L$_{0}$ grows, the time for the occurrence of splitting becomes longer and longer. In experiments that are finished within short times, one may not be able to observe such a splitting. 

Next, we study the decay model for cold atomic sys- tems ( γ$_{n}$ = | n | a Γ). Shown in Figs. 4 (a)-(f) are the dy- namics for L$_{0}$ = 5 l$_{abs}$ with a = 0 . 2 , 0 . 02 , 0 . 01 , 0 . 005 , 0 . 001 and 0. When the decay rate is large–i.e., a is large, the generated light forms a stationary light pulse with dissi- pating; see Fig.4(a)-(c). In contrast, for sufficient small a (Fig.4(d)-(f)), two counter-propagating light pulses ap- 

<img src='content_image/388269.jpg'>

pear. 

Figures 5 (a)-(f) display the dynamics of the generated light for a = 0 . 001 with L$_{0}$/l$_{abs}$ = 50 , 40 , 30 , 20 , 10, and 5. It can be seen that the generated light is stationary for long wave length L$_{0}$ while splits into two light pulses for small L$_{0}$ . We should mention that, however, the dy- namics shown in Fig.5 is up to time 100 / Γ, much shorter than 500 / Γ in Fig.3(c)-(f). To see the behavior of the light pulse for longer time, we performed the calculation for L$_{0}$/l$_{abs}$ = 50 up to the time 500 / Γ and did not ob- serve any splitting. 

We also calculated the total light strength I = ∫ ( | E$_{s}$ | 2 + | E$_{d}$ | $^{2}$) dz remaining in region {− 3 L$_{0}$, 3 L$_{0}$ } as a function of time. The results for L$_{0}$ = 5 l$_{abs}$ and different a are shown in Fig.6. Indeed, as a decreases, the remain- ing light strength I becomes weaker and weaker, reflect- ing that the loss of photons becomes more and more se- rious due to the propagating of the splitting light pulses. 

The earlier simulation yields directly observable phe- nomena. In this section, we aim to provide a qualitative understanding by exploring the associated dispersion re- lation. 

## IV. DISPERSION RELATION 

