# Multiple Phase Stepping Generation in Alkali Metal Atoms: A Comparative Theoretical Study
## 3 Computational Results  
### Behaviour of the Amplitude and Energy for Probe and Driver Pulses Undergoing Propagation  
We have adopted relative units. Rabi frequencies are expressed as \(\Omega_{\rm p}=2\sqrt{2}\
u_{\rm p}\) and \(\Omega_{\rm r}=2\sqrt{2}\
u_{\rm r}\) for probe and drive, respectively. In Figure 2, we have shown three time intervals for the interaction time of the atom with the radiation fields. An extension of the probe pulse is formed beyond the duration \(\rm T_{1}=\{\tau\,|\,\tau\in[0,\,85]\}\). The domain \(\rm T_{2}=\{\tau\,|\,\tau\in[85,\,170]\}\) specifies the second probe pulse in the train and is related to the first created pulse in the drive branch. The domain \(\rm T_{2}\) presents split pulses in the branch probes for \({}^{87}\)Rb and \({}^{39}\)K vapours. The last domain, \(\rm T_{3}=\{\tau\,|\,\tau\in[170,\,250]\}\), presents the domain for the second generated pulse.  
We compute the relative energy of the probe pulses as  
\[E_{p;\rm T_{\alpha}}(\zeta)=\frac{\int_{\rm T_{\alpha}}\Omega_{p;\rm T\alpha} (\zeta,\tau)d\tau}{\int_{\rm T\alpha}\Omega_{p;\rm T\alpha}(0,\tau)d\tau},\ \alpha=1,\ 2,\ 3. \tag{5}\]We present the gain in the drive channel according to  
\[E_{r,\mathrm{T}\alpha}(\xi)=\frac{\int_{\mathrm{T}\alpha}\Omega_{r,\mathrm{T} \alpha}(\zeta,\tau)d\tau}{\int_{\mathrm{T}_{1}}\Omega_{r,\mathrm{T}_{1}}(0,\tau )d\tau},\ \alpha=1,\ 2,\ 3. \tag{6}\]  
Figures 3 and 4 expose the space dependence of the relative energies for both probe and drive pulses, where one is subtracted from the first drive pulse's energy to indicate the gain. In the representation \(E_{1j}\), j = 1-3, the first suffix 1 indicates the lower (first) \(\Lambda\) subsystem is concerned, as shown in Figure 1. Figure 3 exhibits the attenuation of probe pulses for different alkali vapours. Propagation in the probe channel is less attenuated in sodium vapours. Initially, the probe pulses have more energy than others. The attenuation in rubidium and potassium is similar, whereas depletion did not depend on whether or not pulses matched. Figure 4 displays amplification in the first drive pulse and gain in time sections T\({}_{2}\) and T\({}_{3}\), which is due to the generated Stokes field in the Raman transition. In alkali vapours, different gain mechanisms concerning hf structure can occur [38; 39]. In the numerical results, we have focused our attention on the amplification of the generated Stokes field and the time dependence of its phase sequences, as shown in Figures 3 and 4. It is noted that, in our treatment, we have an incomplete LS experiment. This is because of the absence of the read pulse in the drive channel. Thus, the Stokes field generation is viewed as energy retrieval without invoking the traditional read pulse. Moreover, the phase sequence is associated with the restored Stokes field. The energy restored in the drive channel is written by the probe train. Our input pulses are not equivalent pulses, in the sense that it has not had the same energy and the same mean duration, initially. Therefore, it is difficult to compare the restoring efficiency of different alkali gases that can appear in alkali atoms.  
Figure 3: Attenuation of the relative energies for probe pulses in different alkali vapours. Energies E\({}_{11}\), E\({}_{12}\), and E\({}_{13}\) for probe and drive fields to correspond to time sections \(\tau\) = [0, 85], \(\tau\) = [85; 170], and \(\tau\) = [170; 250], respectively.