## 3 Strong Coupling Regime: Analytical Approximation
### Travelling Rabi-Waves  
Assuming the driven dc field to be absent (\(\Omega_{B}=0\)), we have from (2) and (3):  
\[i\frac{\partial a_{p}}{\partial t}=\frac{\omega}{2}a_{p}+t_{a}\bigl{(}a_{p+1} +a_{p-1}\bigr{)}-\Omega_{R}\cos\omega t\cdot b_{p}, \tag{4}\]  
\[i\frac{\partial b_{p}}{\partial t}=-\frac{\omega}{2}b_{p}+t_{b}\bigl{(}b_{p+1} +b_{p-1}\bigr{)}-\Omega_{R}\cos\omega t\cdot a_{p}. \tag{5}\]The idea of simplifying Equations (4) and (5) is based on the so-called rotating-wave approximation (RWA), which is frequently encountered in quantum optics [1]. It takes into account only the co-rotating field with the system, and neglects the counter-rotating part, which is proportional to \(\exp(2i\omega)\)[1]. This part oscillates very rapidly, and its average over the time larger than \(\omega^{-1}\) is zero, if \(\Omega_{R}<<\omega\) (strong coupling regime). It corresponds to the exchanges \(b_{p}\cos\omega t\rightarrow(1/2)b_{p}\exp(-i\omega t)\), \(a_{p}\cos\omega t\rightarrow(1/2)a_{p}\exp(i\omega t)\), and leads to the equations:  
\[i\frac{\partial a_{p}}{\partial t}=\frac{\omega}{2}a_{p}+t_{a}\big{(}a_{p+1}+ a_{p-1}\big{)}-\frac{1}{2}\Omega_{R}e^{-i\omega t}b_{p}, \tag{6}\]  
\[i\frac{\partial b_{p}}{\partial t}=-\frac{\omega}{2}a_{p}+t_{b}\big{(}b_{p+1}+ b_{p-1}\big{)}-\frac{1}{2}\Omega_{R}e^{i\omega t}a_{p}. \tag{7}\]  
For \(\Omega_{R}\) and \(\omega\) have been comparable, the average of the counter-rotating part does not vanish due to the high amplitude of such rapid oscillations (ultra-strong coupling). The equations of motion, in contrast with the system (6) and (7), become analytically unsolvable and should be integrated numerically. The system (6) and (7) is a generalization of Jaynes-Cummings model for the single atom to the case of atomic chain. The ansatz for solutions of Equations (6) and (7) in the form of a travelling wave is  
\[\left( \begin{array}{c}a_{p}(t)\\ b_{p}(t)\end{array} \right)=e^{i(p\varphi-vt)}\left( \begin{array}{c}\widetilde {a}e^{-i\frac{1}{2}\omega t}\\ \widetilde{b}e^{i\frac{1}{2}\omega t}\end{array} \right), \tag{8}\]  
where \(\widetilde{a},\widetilde{b}\) are unknown coefficients, \(\varphi\) is the phase shift per unit cell, \(\upsilon\) is the eigen frequency. Substituting (6)-(8), we obtain the system with respect to \(\widetilde{a},\widetilde{b}\):  
\[\left( \begin{array}{cc}\upsilon-2t_{a}\cos\varphi&\Omega_{R}/2\\ \Omega_{R}/2&\upsilon-2t_{b}\cos\varphi\end{array} \right)\cdot\left( \begin{array} []{c}\widetilde{a}\\ \widetilde{b}\end{array} \right)=0. \tag{9}\]  
Setting determinant equal to zero, we obtain the characteristic equation with respect to \(\upsilon\):  
\[(\upsilon-2t_{a}\cos\varphi)(\upsilon-2t_{b}\cos\varphi)-\frac{\Omega_{R}^{2 }}{4}=0, \tag{10}\]  
with two solutions:  
\[\upsilon_{1,2}(\varphi)=\left\{(t_{a}+t_{b})\cos\varphi\pm\sqrt{\left(t_{a}-t _{b}\right)^{2}\cos^{2}\varphi+\frac{\Omega_{R}^{2}}{4}}\right\} \tag{11}\]  
The linearly independent eigen modes correspondent to eigen frequencies (11) are:  
\[|\Psi_{1}(t)\rangle=\frac{1}{\sqrt{N\Big{(}1+\Lambda^{2}\Big{)}}}\underset{p} {\sum}e^{i(p\varphi-\upsilon_{1}t)}\Big{(}|a_{p}\rangle e^{-i\frac{1}{2} \omega t}-\Lambda|b_{p}\rangle e^{i\frac{1}{2}\omega t}\Big{)} \tag{12}\]  
\[|\Psi_{2}(t)\rangle=\frac{1}{\sqrt{N\Big{(}1+\Lambda^{2}\Big{)}}}\underset{p} {\sum}e^{i(p\varphi-\upsilon_{2}t)}\Big{(}\Lambda|a_{p}\rangle e^{-i\frac{1}{2} \omega t}+|b_{p}\rangle e^{i\frac{1}{2}\omega t}\Big{)} \tag{13}\]  
where \(N\) is a number of atoms in the chain and  
\[\Lambda=\frac{\Omega_{R}}{2\Big{[}\left(t_{a}-t_{b}\right)\cos\varphi+\sqrt{ \left(t_{a}-t_{b}\right)^{2}\cos^{2}\varphi+\Omega_{R}^{2}/4}\Big{]}} \tag{14}\]The general solution for Rabi-oscillations propagated along the chain in the form of travelling waves is given by the linear superposition of eigen modes (12), (13) \(\left|\Psi(t)\right\rangle=C_{1}\left|\Psi_{1}(t)\right\rangle+C_{2}\left|\Psi_{2 }(t)\right\rangle\), where \(C_{1,2}\) are arbitrary integration constants.