## 4 Ultrastrong Coupling: Results and Discussion
### The Case of Small Bloch-Frequency  
In this subsection, we consider the case of Rabi-frequency comparable with the quantum transition frequency, but strongly exceeding Bloch-frequency (\(\Omega_{R}>>\Omega_{B}\), while \(\Omega_{R}\cong\omega_{0}\)). The spatial-temporal behavior of the tunnel current, for different types of relative values of tunneling factors at ground and excited states, is presented in Figures 3 and 4. The current oscillates with Bloch frequency. The amplitude for the case \(t_{a}=t_{b}\) exceeds the case \(t_{a}>>t_{b}\). The current amplitude is high frequency modulated. In general, the scenario looks like as in the case of strong coupling. Figure 4: Space-time distribution of the tunnel current density with RO and BO in the separated frequency ranges. \(t_{a}=t_{b}=0.04\). All other parameters are identical to Figure 3.  
Figure 3: Space–time distribution of the tunnel current density with Rabi-oscillations (RO) and Bloch-oscillations (BO) in the separated frequency ranges. Here, the quantum transition frequency is taken as the frequency unit, detuning is equal to zero (\(\omega=\omega_{0}\)), \(\Omega_{B}=0.04\), \(\Omega_{R}=0.8\), \(t_{a}=0.4\), \(t_{b}=0.04\), interatomic distance \(a=20\,\)nm. The initial state of the chain is an excited single Gaussian wave packet \(a_{j}(0)=g\exp\left[-(j-j^{\prime})^{2}a^{2}/\sigma^{2}\right]\), \(b_{j}(0)=0\). Gaussian initial position and width are \(j^{\prime}=80\), \(g=20\), respectively. Number of atoms \(N=128\). In Figures 5 and 6, the corresponding spectra are presented. Their qualitative behavior dramatically changes with respect to the strong coupling regime (see Equation (27)). The main reason leads from the strong analysis of the system in Equations (2) and (3), which is free from RWA, and based on Floquet theorem. Following it, the probability amplitudes of the \(j\)-th mode are presented in the form  
\[\left( \begin{array}{c}a_{p,j}(t)\\ b_{p,j}(t)\end{array} \right)=\sum_{m=-\infty}^{\infty}\left( \begin{array}{c}a _{pm,j}(\varphi)\\ b_{pm,j}(\varphi)\end{array} \right)e^{-im\omega_{0}t}e^{-iv_{j}(\varphi)t}, \tag{28}\]  
where \(a_{pm,j}(\varphi),b_{pm,j}(\varphi)\) are the periodical functions of the phase shift per unit cell, and \(\
u_{j}(\varphi)\) is the eigen value. Therefore, as it leads from Equation (21), the lines \(\omega=m\omega_{0}\) appear at the frequency spectrum, and the first of them are really observed at Figures 5 and 6, in contrast with relation (27). Following the quasiclassical approach, we exchange \(\varphi\rightarrow-\Omega_{B}t+\varphi_{0}\), thus the values \(a_{pm,j}(\varphi),b_{pm,j}(\varphi)\) become time dependent periodic functions with a Bloch-period. Such amplitude modulation means the appearance of lines \(\omega=m\omega_{0}+n\Omega_{B}\), where \(m,n=0,\pm 1,\pm 2,...,\pm\infty\). The frequency-modulated factors \(\exp[iv_{j}(\varphi)]\), for qualitative analysis, may be averaged over the Bloch-period [40]. The dominant support to observable values is defined by two first modes, with \(j=1\),\(2\). Thus, the discrete part of the spectra in the regime of ultrastrong coupling corresponds to the narrow lines with the frequencies  
\[\omega=m\omega_{0}+n\Omega_{B}\pm p\overline{\Omega} \tag{29}\]  
where \(p=0,\pm 1\), and  
\[\overline{\Omega}=\frac{1}{2\pi}\int_{0}^{2\pi}[\
u_{1}(\varphi)-\
u_{2}( \varphi)]d\varphi \tag{30}\]  
Figure 5: Frequency spectrum of the tunneling current density with RO and BO in the separated frequency ranges at the atomic position with number \(j=90\). The quantum transition frequency, Rabi-frequency and Bloch-frequency are denoted by a red-colored dashed line, blue-colored dashed–dotted line and green-colored dotted line, respectively. The quantum transition frequency is taken as the unit. \(\Omega_{B}=0.04\), \(\Omega_{R}=0.8\). All other parameters are identical to Figure 3.  
This frequency may be identified with the frequency of RO.