## 4 Ultrastrong Coupling: Results and Discussion
### The Role of the Losses  
One of distinctive features of ultrastrong coupling is a comparable value of inter-band transition with radiative and dissipative damping value. Thus, the dissipative and radiative broadening leads to the confluence of neighboring spectral lines. As a result, the discrete lines will interchange with areas of continuous spectra. These discrete lines will have non-Lorentz asymmetric form. This means that the simple lossless model is not applicable for detailed analysis of the spectra thin structure--it allows only prediction of the spectral line positions and their relative amplitudes. For estimating the role of losses, we will use the simple phenomenological model used by [PERSON] et al., for the analysis of population trapping, lasing without inversion, and electromagnetically induced transparency [1]. This model assumes that two levels decay at a rate \(\gamma\)[1; 66]. This means that the ground state is not  
Figure 14: The frequency spectrum of the dipole moment density with RBO in the ultrastrong coupling regime at the atomic position \(j=60\). \(\Omega_{R}=\Omega_{B}=0.5\), \(t_{a}=t_{b}=5.0\). Figure 13: The frequency spectrum of the tunnel current density with RBO in the ultrastrong coupling regime at the atomic position \(j=60\). \(\Omega_{R}=\Omega_{B}=0.5\), \(t_{a}=t_{b}=5.0\). All other parameters are identical to Figure 11.  
really ground, and corresponds to the excited state at the lowest energy level. The wavefunction for such model is given by  
\[\ket{\Psi(t)}=\frac{1}{\sqrt{N\Big{(}1+\Lambda^{2}\Big{)}}}\underset{p}{\sum} \big{(}A_{p}(t)\big{|}a_{p}\big{\rangle}+B_{p}(t)\big{|}b_{p}\big{\rangle}\big{)} e^{-i\mu t}e^{-\gamma t/2}, \tag{31}\]  
where \(A_{p}(t),B_{p}(t)\) are the same coefficients as in the lossless system. The tunneling current and its frequency spectrum for \(Q=15\) are shown in Figures 15 and 16. One can see that the oscillation process quickly reduces (Figure 15). The number of the resolved spectral lines decreases with decay appearance (Figure 16), because of the large number of lines run into, due to the broadening. The main lines are kept, however, their positions are shifted and amplitudes are decreased with respect to the lossless case. The frequency spectrum of the tunneling current for \(\Omega_{B}=\Omega_{R}=0.5\) is shown in Figure 17. The mechanism of the spectral lines confluence due to the losses, is especially well-defined in this partial case via the transformation of duplets to their corresponding broadened singlets. The non-Lorentz origin of these spectral lines manifests itself in their asymmetry. Such qualitative behavior agrees with general principles of the oscillations' theory [67]. The detailed description of losses must be based on the concept of open quantum systems [68], and master equation technique especially adopted to the case of ultrastrong coupling [69]. It goes beyond the subject of this article, and may be one of the topics for future research activity. Figure 15: The space–time distribution of tunneling current density with RBO in the ultrastrong coupling regime with dissipation. Insert: the individual energy spectrum of the single atom in the chain. The model assumes that two levels decay at a rate \(\gamma\), due to spontaneous emission. Thus, the ground state is not really ground; it corresponds to the excited state at the lowest energy level. The dissipation property is characterized by a Q-factor, defined as \(Q=\omega_{0}/2\gamma\). Realistic values for ultrastrong coupling cases are Q = 15–30 [43]. Here, the quantum transition frequency is taken as the frequency unit, detuning is zero \((\omega=\omega_{0})\), \(\Omega_{B}=0.7\), \(\Omega_{R}=0.8\), \(t_{a}=t_{b}=7.0\), interatomic distance \(a=20\,\mathrm{nm}\), and \(Q=15\). The initial state of the chain is an excited single Gaussian wave packet \(a_{j}(0)=g  
Another topic for future activity is the implementation of the scenario considered before. The subject of this article is limited by the guidance of some candidates for such potential experiments.