## 6 Graphene as Active Medium  
The nonlinear properties of graphene allow to activate it by some basic electric field for driving by the current of another weak signal. Below we will consider an example when a rather strong basic field is aligned with the \(x_{1}\)--axis while the probe field aligned with the \(x_{2}\)--axis allows the application of perturbation theory.  
The corresponding perturbation theory can be constructed both on the system of Equation (27) and in the framework of the Markovian approximation (33). The latter variant allows to proceed with analytical calculations.  
In the framework of such an approximation we can limit ourselves in the weak field approximation to the Markovian solution (33) and (34). The distribution function in this approximation is \(f_{M}\approx f^{(0)}+f^{(1)}\),  
\[f^{(0)}=\frac{1}{2}(1-e^{-f^{(0)}}),\ \ \ f^{(1)}=J^{(1)}e^{-f^{(0)}}, \tag{61}\]  
where labels \((0)\) and \((1)\) correspond to the \"basic\" field \(A_{1}(t)\) (nonperturbative solutions) and to the perturbing field \(A_{2}(t)\), respectively. The function \(J(\vec{p},t)\) is defined by Equation (33), so that  
\[f^{(1)}(\vec{p},t)=8\int\limits_{t_{0}}^{t}dt^{\prime}\lambda^{(1)}(\vec{p},t^ {\prime})\int\limits_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\lambda^{(0)}(\vec{p },t^{\prime\prime})\cos\theta^{(0)}(t^{\prime},t^{\prime\prime}), \tag{62}\]  
Figure 12: From left to right: density of charge carriers (42), density of conduction current and energy density the carriers for range of the time 25,000 \(t_{0}-\) 500,000 \(t_{0}\) (\(6.15\times 10^{-12}s-1.23\times 10^{-10}s\)). The constant electric field strength \(E_{a}=1\times 10^{-8}\ E_{0}\ (1.088\ \mathrm{V/cm}\)).  
The Hamiltonian functions (4) and (67) have the same pseudospin structure. Therefore one can follow the way of derivation of KE (24) in the theory with the Hamiltonian function (67). The quasienergy (8) and the amplitude (10) are changed only by the following formal substitutions  
\[v_{F}P_{1}\to h^{\prime}_{\vec{p}}(t),\ v_{F}P_{2}\rightarrow-h^{\prime\prime}_ {\vec{p}}(t). \tag{70}\]  
This results in  
\[\varepsilon(\vec{p},t)=\sqrt{h^{*}_{\vec{p}}(t)h_{\vec{p}}(t)}=|h_{\vec{p}}(t)|, \tag{71}\]  
\[\lambda(\vec{p},t)=\frac{1}{|h_{\vec{p}}(t)|^{2}}\left\{h^{\prime\prime}_{ \vec{p}}(t)h^{\prime}_{\vec{p}}(t)-\dot{h}^{\prime}_{\vec{p}}(t)h^{\prime\prime }_{\vec{p}}(t)\right\}. \tag{72}\]  
An analogous KE can be obtained also for the case of the multilayer graphene model [30].  
The conduction and polarization currents have the following form (the flavor number in the given model is equal to 2 [3])  
\[j^{\text{cond}}_{k}(t)=-4e\gamma\int[dp]f(\vec{p},t)[F^{(1)}_{k} (\vec{p},t)\cos\chi+F^{(2)}_{k}(\vec{p},t)\sin\chi], \tag{73}\] \[j^{\text{pol}}_{k}(t)=-4e\gamma\int[dp]f(\vec{p},t)[-F^{(1)}_{k} (\vec{p},t)\sin\chi+F^{(2)}_{k}(\vec{p},t)\cos\chi]\, \tag{74}\]  
with the vector formfactors  
\[F^{(1)}_{k}(\vec{p},t)=\sum_{\alpha}\delta^{(k)}_{\alpha}\sin \left(\frac{1}{\hbar}\vec{p}\vec{\delta}_{\alpha}\right), \tag{75}\] \[F^{(2)}_{k}(\vec{p},t)=\sum_{\alpha}\delta^{(k)}_{\alpha}\cos \left(\frac{1}{\hbar}\vec{p}\vec{\delta}_{\alpha}\right) \tag{76}\]  
and \(\chi\) being the angle of the unitary rotation in the matrix of the type (7),  
\[\chi=-h^{\prime\prime}_{\vec{p}}(t)/h^{\prime}_{\vec{p}}(t). \tag{77}\]  
Let us rewrite Equations (73) and (74) for the currents to obtain  
\[j^{\text{cond}}_{k}(t) = -4e\gamma\int[dp]f(\vec{p},t)\sum_{\alpha}\delta^{(k)}_{\alpha} \sin\left(\chi+\frac{1}{\hbar}\vec{p}\vec{\delta}_{\alpha}\right), \tag{78}\] \[j^{\text{pol}}_{k}(t) = -4e\gamma\int[dp]u(\vec{p},t)\sum_{\alpha}\delta^{(k)}_{\alpha} \cos\left(\chi+\frac{1}{\hbar}\vec{p}\vec{\delta}_{\alpha}\right) \tag{79}\]  
as the final result of this section. The numerical evaluation of the currents for particular external field models is delegated to future work.