## 4 Ultrastrong Coupling: Results and Discussion
### Stark Effect in One-Dimensional Atomic Chains  
In this section, we consider the 1D chain of two-level atoms driven by dc field only. For a single atom, the interaction with a dc field leads to the well-known Stark effect, which consists of the shifting of energy levels [48]. For reachable field values, the shift is small, and considered as a perturbation of the energy spectrum (weak coupling regime) [48]. The dc field couples all eigen states of the atom, thus the multi-level model with summation over the complete set of atomic terms is applicable to real atoms. Here, we show that the Stark effect manifests itself in 1D chains too, but its qualitative picture differs from the single-atomic case. The main reason is the performance ability of ultrastrong coupling, on which we focus below. The system under consideration is shown in Figure 18. The dipole moments of the atoms is assumed to be oriented orthogonally to the chain axis, and the dc field is directe d under the angle \(\alpha\), with respect to the axis. The analytical model, in general, is similar to the RBO-model developed and used before. The wave function is given by  
\[|\Psi(t)\rangle=\sum_{p}\bigl{(}a_{p}(t)|a_{p}\rangle+b_{p}(t)|b_{p}\rangle \bigr{)} \tag{32}\]  
where the probability amplitudes satisfy the system  
\[i\frac{\partial a_{p}}{\partial t}=\left(\frac{1}{2}\omega_{0}-p\Omega_{B} \right)a_{p}+t_{a}\bigl{(}a_{p+1}+a_{p-1}\bigr{)}-\Omega_{R}b_{p}, \tag{33}\]  
\[i\frac{\partial b_{p}}{\partial t}=-\biggl{(}\frac{1}{2}\omega_{0}+p\Omega_{B }\biggr{)}b_{p}+t_{b}\bigl{(}b_{p+1}+b_{p-1}\bigr{)}-\Omega_{R}a_{p}, \tag{34}\]  
Figure 18: General illustration of the periodic two-level atomic chain used as a model, indicating Stark effect in the ultrastrong coupling regime. The chain is placed obliquely between the armatures of plane capacitor with applied dc voltage. The dipole moments (blue arrowheads) are oriented orthogonally to the chain axis. Dc field (red arrowheads) are oriented angularly with respect to the chain axis. The choice of the angle allows for obtaining the arbitrary relative value between Rabi- and Bloch frequencies. where \(\Omega_{R}=e\hbar^{-1}d_{ab}E_{Dc}\sin\alpha\), \(\Omega_{B}=e\hbar^{-1}aE_{Dc}\cos\alpha\), are the Rabi- and Bloch frequencies, respectively. In contrast with RBO, these two frequencies are defined by the same field (dc), but their relative values may be arbitrary, via the corresponding choice of the field direction (angle\(\alpha\)). Let us assume that  
\[\Omega_{R}\approx\omega_{0} \tag{35}\]  
and  
\[\Omega_{R}>>\Omega_{B}. \tag{36}\]  
Equation (35) corresponds to the resonant condition. It is a counterintuitive resonance, which does not couple with any oscillating forces, and represents the attribute of a ultrastrong coupling regime. This resonance legitimates the using of a two-level model, in contrast with the single-atomic case. This inequality (36) means high amplitude of BO. It allows us to separate the inter-atomic and intra-atomic motions, and describe the last one in terms of quasiclassical approximation. The wave function of inter-atomic motion is given as a superposition of eigen states (12) and (13). The account of intra-atomic motion is made via exchange (24). As a result, the total motion is described by the wave function (15) with  
\[\Omega(t)=\sqrt{\left(t_{a}-t_{b}\right)^{2}\cos^{2}(\Omega_{B}t-\varphi_{0})+ \Omega_{R}^{2}}, \tag{37}\]  
\[\Lambda(t)=\frac{\Omega_{R}}{\left(t_{a}-t_{b}\right)\cos(\Omega_{B}t-\varphi_ {0})+\sqrt{\left(t_{a}-t_{b}\right)^{2}\cos^{2}(\Omega_{B}t-\varphi_{0})+ \Omega_{R}^{2}}}. \tag{38}\]  
The application of this solution in relation to tunneling current (21) leads to the spectrum (29), in the case of Stark effect. Another approach consists in the numerical solution of Equations (33) and (34), which is simplified by the absence of oscillation coefficients. The typical behavior of space-time distribution of inversions and tunneling currents, as well as the frequency spectrum of tunneling currents, are shown in Figures 19-21.