## 5 Numerical Analysis
Figure 8 demonstrates the distribution function for the field model (57) and its change for the large time intervals \(T=406\),\(500\ t_{0}\ (1.0\times 10^{-10}\ \mathrm{s})\) and \(T=1\),\(219\),\(500\ t_{0}\ (3.0\times 10^{-10}\ \mathrm{s})\) at the field strength \(E_{a}=9.19\times 10^{-8}\ E_{0}(10\ \mathrm{V/cm})\). The top row shows images with a linear scale for the color code of the distribution function. This allows to demonstrate that the generated carriers are concentrated in momentum space  
Figure 2.1 The distribution function in the constant electric field (57) with \(E_{a}=0.01\ E_{0}\ (1.088\times 10^{6}\ \mathrm{V/cm})\) at \(t=10\ t_{0}\ (2.46\times 10^{-15}\ \mathrm{s})\) after switching on the field (**left panel**) and at \(t=50\ t_{0}\ (1.23\times 10^{-14}\ \mathrm{s})\) after switching on the field (**right panel**). Figure 2.2 The stages of evolution of the distribution function under the action of the field of type (59) with the number of periods \(m=1,2,4\) and \(10\), respectively. in a very narrow area close to zero values of the momentum in the direction perpendicular to the direction of the field. The bottom row shows the same distribution function with a logarithmic scale of the color coding. In this case the complicated structure of the distribution function outside the main area of the carrier generation becomes apparent. The main area, however, is absolutely dominant. Other parameters of the electric field can change this picture. This issue requires further research. Figure 2.3 The residual distribution function in momentum space after the action of the field of type (57) with \(E_{a}=10\) V/cm for a duration of \(1.0\times 10^{-10}\) s (**left column**) and \(3.0\times 10^{-10}\) s (**right column**). A logarithmic color scale used for the bottom panel. Figure 2.4 The residual distribution function in momentum space after the action of the electric field (60) with \(E_{a}=0.01\)\(E_{0}\) (\(1.088\times 10^{6}\) V/cm), \(\omega=2\pi 0.1\). **Left**: \(\sigma=5\). **Right**: \(\sigma=10\). The set of parameters used here is quite realizable in the experiment. We note that the behavior of the distribution function in momentum space has not undergone fundamental changes in comparison to the ones in Figure 2.5 Figure 8 demonstrates agreement with the results of the work [29] where the same parameters have been used for the numerical calculations in the framework of another formalism. The behavior of a quasiparticle plasma under the action of periodic fields (59) and (60) with increasing pulse duration is not trivial. Figure 9 shows the distribution function for the field model (60) with the parameters \(\omega=2\pi\times 2.46\times 10^{-4}\ t_{0}^{-1}\) (corresponding to \(2\pi\times 1.0\)  THz), \(E_{a}=9.19\times 10^{-6}\ E_{0}\) (corresponding to 1000 V/cm) for \(\sigma=3,10,25,\) and \(50\). Let us now discuss the analysis of some observable values. We have studied the behavior of the density of carriers (42) in dependence on the amplitude of the electric field for the three models (57), (58) and (60). A summary of the results is presented below. It should be noted that these results are determined solely by the filling of the conduction band due to the quasiparticle excitation by an external electric field. The presence of thermally excited carriers has not been taken into account as well as the relaxation processes since the considered times are much shorter than the relaxation time. Figure 10 demonstrates that the number of quasiparticles created during the action of the field increases quadratically with increasing electric field strength. The quadratic dependence can be traced quite rigorously in relatively weak fields \(E_{a}\leq 0.001\ E_{0}\ (\lesssim\)\(10^{5}\) V/cm). The increment of the pair number density is slowing down somewhat with further increase of the electric field. This can be explained by a saturation effect.