## 3 Strong Coupling Regime: Analytical Approximation
### Rabi-Bloch Oscillations: Quasiclassical Model  
BO of the charge particles is a pure quantum effect which can be explained in a simple quasiclassical model [21; 24; 25]. The similar concept may be developed for quasi-particles (Rabi-oscillated electrons in our case). The periodicity of the lattice leads to a band structure of the energy spectrum and the corresponding eigen energies \(\hbar\
u_{1,2}(\varphi)\) given by Equation (11). The eigen states \(\left|\Psi_{1,2}(h)\right\rangle\) given by Equations (12) and (13), are labeled by the continuous quasimomentum \(h_{t}\)\(\hbar\
u_{1,2}(h,n)\) and \(\left|\Psi_{1,2}(h)\right\rangle\) are periodic functions of \(h\) with period \(2\pi/a\), and \(h\) is conventionally restricted to the first Brillouin zone \([-\pi/a;\pi/a]\). Under the influence of dc field, weak enough not to induce interband transitions, a given state \(\left|\Psi_{1,2}(h_{0})\right\rangle\) evolves up to a phase factor into the state \(\left|\Psi_{1,2}(h(t))\right\rangle\), with \(h(t)\) variation according to  
\[\frac{dh}{dt}=-\frac{eE_{\text{DC}}}{\hbar}, \tag{23}\]  
or \(h(t)=-e\hbar^{-1}E_{\text{DC}}t+h_{0}\). Thus, this evolution is periodic with a Bloch frequency corresponding to the time required for the quasimomentum to scan a full Brillouin zone. It leads to the exchanges  
\[\varphi\to h(t)a=-\Omega_{B}t+\varphi_{0}, \tag{24}\]  
\[\Lambda(\varphi)\rightarrow\Lambda(t)=\frac{\Omega_{R}}{2(t_{a}-t_{b})\cos( \Omega_{B}t-\varphi_{0})+\sqrt{4(t_{a}-t_{b})^{2}\cos^{2}(\Omega_{B}t-\varphi_ {0})+\Omega_{R}^{2}}}, \tag{25}\]  
\[\Omega(\varphi)\rightarrow\Omega(t)=\sqrt{4(t_{a}-t_{b})^{2}\cos^{2}(\Omega_ {B}t-\varphi_{0})+\Omega_{R}^{2}}, \tag{26}\]  
where \(h_{0}=\varphi_{0}a\).  
Equations (24)-(26) allow the spectrum of the tunneling current in the BO regime to be analyzed. Following quasiclassical concepts, the tunneling current may be considered as RO, with modulated amplitude and frequency. As it leads from Equations (24)-(26), the spectral presentation of tunnel current is given by  
\[J_{T}(t)\Rightarrow\sum_{m=-\infty}^{\infty}a_{m}e^{i(\overline{\Omega}-m \Omega_{B})t}+c.c., \tag{27}\]  
where \(\overline{\Omega}=(2\pi)^{-1}\int_{0}^{2\pi}\sqrt{4(t_{a}-t_{b})^{2}\cos^{2} \varphi+\Omega_{R}^{2}}d\varphi\) and \(a_{m}\) are the spectral amplitudes. Below, we consider the amplitude relations in the current spectra based on the numerical solution of motion equations. Here, we note that the small detuning for which \(\overline{\Omega}-m\Omega_{B}\approx\delta\Omega_{B}\), with \(\delta<<1\), is associated with a special type of resonance motion. The above represents a superposition of harmonic oscillations, which evolves slowly in time, with a period \(2\pi/(\delta\Omega_{B})>>T_{B,R}\), where \(T_{B,R}\) are the Rabi and Bloch periods, respectively. Evaluating the particle position as \(x(t)\cong\int_{0}^{t}J_{T}(\tau)d\tau\), we obtain \(x(t)\cong(\delta\Omega_{B})^{-1}e^{i(\overline{\Omega}-m\Omega_{B})t}+c.c.\) It shows the strong enhancement of spatial oscillations, which is similar to the super-Bloch oscillations recently predicted and experimentally observed in optical lattices with cold atoms [40]. In contrast with super-Bloch oscillations, in our case, there exists two tunneling channels (with respect to the different zones). Thus, the influence of ac field does not add up to renormalization of tunneling factors, but leads to their mutual influence via Rabi-oscillations.