# Multiple Phase Stepping Generation in Alkali Metal Atoms: A Comparative Theoretical Study
## 2 Excitation and Atomic System Scheme  
The atomic energy level scheme for alkali metal atoms such as \({}^{23}\)Na, \({}^{87}\)Rb, and \({}^{39}\)K is depicted in Figure 1. The fine structure allowed transitions specified as \(3\)\({}^{2}S_{1/2}\)--\(3\)\({}^{2}P_{1/2}\), \(5\)\({}^{2}S_{1/2}\)--\(5\)\({}^{2}P_{1/2}\), and \(4\)\({}^{2}S_{1/2}\)--\(4\)\({}^{2}P_{1/2}\) for \({}^{23}\)Na, \({}^{87}\)Rb, and \({}^{39}\)K vapours, respectively. The kets \(\left|\,1\right\rangle\) to \(\left|\,4\right\rangle\) specifies the state of the hf levels, as depicted in Figure 1. \(\Delta_{p}\) and \(\Delta_{r}\) denote the detuning of fields from the transition \(\left|\,1\right\rangle\leftrightarrow\left|\,3\right\rangle\) and \(\left|\,2\right\rangle\leftrightarrow\left|\,13\right\rangle\). The experimental data were taken from [22, 23, 24, 25]. In detail, the experiment data for potassium were taken from [25] and those sodium from [23]. Furthermore, the experimental data for rubidium were found in [24], whereas the updated data were taken from [22]. The Liouville-von Neumann (LvN) equation [26, 27],  
\[-i\frac{\partial\rho_{s}(t)}{\partial t}=\hat{\mathcal{L}}_{t}\rho_{s}(t);\ \hbar=1, \tag{1}\]  
describes temporal development of the reduced density matrix (DM), \(\rho_{s}(t)\), where \(\hat{\mathcal{L}}_{t}\) stands for the Liouvillian super operator [28] in the Liouville space [29, 30]. In the RDM description, we have adopted the formalism and notations in [28] and its extensions to the multilevel atoms with hf structures [31, 32, 33]. Our model variables are defined as follows: (\(z\), \(t\)) is the space and time coordinates in a laboratory frame, \(\tau=\gamma(t-z/c)\) is dimensionless retarded time, \(\gamma\) is the spontaneous decay rate of the atomic state P\({}_{1/2}\), \(c\) is the speed of light, \(\zeta=\alpha\) (\(z+ct\)) is the dimensionless space  
Figure 1: Optical transitions in the D\({}_{1}\) line hf-structure for \({}^{23}\)Na, \({}^{87}\)Rb, and \({}^{39}\)K vapours. variable, \(\alpha^{{}^{\prime}}\) is the absorption coefficient, \(v\) is the atom-field, \(\Omega=\sqrt{8}v\) is the Rbi frequency, and \(v=v/\gamma\) is the relative coupling of the atom-field. The shapes for the probe and drive pulses for various vapours are shown in Figure 2. Along the boundary \(\zeta=0\), the pulses are formed as truncated-Gaussian pulses in time, which are characterized by different widths. We have used the driving pulse of one pulse, whereas the probing pulses formed a Gaussian train. Through \({}^{23}\)Na vapour, we used pulses with envelopes that were not exactly matched in time, as the driving is narrower than the first probing pulse in the train. Let the pulse-width be denoted as \(w\) and \(\gamma\) is the spontaneous decay rate of the fine-structure transition. Consequently, the triples {\(\{\gamma_{\text{Na}},w_{\text{Na}}\}\), \(\{\gamma_{\text{Rb}},w_{\text{Rb}}\}\), and \(\{\gamma_{\text{K}},w_{\text{K}}\}\)} are obtained for the specific gases addressed in this study. The width compression in \({}^{87}\)Rb and \({}^{39}\)K gives rise to determined pulses through long times and near initial times. However, this is not true for the \({}^{23}\)Na case, in particular for the first and second pulses. In the pulses' far-wings, the overlap is significant. During computations, we have kept the amplitude \(v_{0}\) the same, in agreement with \(\frac{v_{0}^{2\alpha}}{\gamma_{\text{Na}}}=\frac{v_{0}^{2\beta}}{\gamma_{ \text{Rb}}}=\frac{v_{0}^{2}}{\gamma_{\text{K}}}\). Thus, the pulses degenerate with respect to the amplitudes of the field. The degeneracy is removed as the intensity \(I\) is expressed as \(\left(\frac{\Omega}{\gamma}\right)^{2}=\frac{1}{2I_{\text{s}}}\), where \(I_{\text{s}}\) is the saturation intensity.