# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 3 Approximate Analytical Solution
### Details  
We consider the partial solutions of System (7) and (8) in the absence of a dc field (\(\mathbf{E}_{dc}=0\)) and the case of zero detuning (\(\omega=\omega_{0}\)). We use the ansatz \(a_{p+1,n}(t)=a_{p,n}(t)e^{i\varphi}\), \(b_{p+1,n}(t)=b_{p,n}(t)e^{i\varphi}\), \(a_{p,n}(t)=\widetilde{a}_{p}e^{-ivt}\), \(b_{p,n}(t)=\widetilde{b}_{pn}e^{-ivt}\), where \(\varphi\) is a given phase shift per unit cell, \(\
u\) is the unknown eigen frequency, \(\widetilde{a}_{pn},\widetilde{b}_{pn}\) are unknown constant coefficients. Making use of Equations (7) and (8) we obtain for them the matrix equation  
\[\left( \begin{array}{cc}\hbar\
u-2t_{a}\cos\varphi&-g\sqrt{n+1}\\ -g\sqrt{n+1}&\hbar\
u-2t_{b}\cos\varphi\end{array} \right)\cdot\left( \begin{array} []{c}\widetilde{a}_{pn}\\ \widetilde{b}_{p,n+1}\end{array} \right)=0. \tag{17}\]  
The eigenfrequencies are found from its characteristic equation  
\[(\hbar\
u-2t_{a}\cos\varphi)(\hbar\
u-2t_{b}\cos\varphi)-g^{2}(n+1)=0 \tag{18}\]  
and given by  
\[\hbar\
u_{1,2}(\varphi)=\bigg{\{}(t_{a}+t_{b})\cos\varphi\pm\sqrt{\left(t_{a}- t_{b}\right)^{2}\cos^{2}\varphi+g_{n}^{2}}\bigg{\}}. \tag{19}\]  
where \(g_{n}=g\sqrt{n+1}\). The correspondent eigenstates are  
\[|\Psi_{n,1}\rangle=\frac{e^{-i(n+1)\omega t}\cdot e^{-i\
u_{1}t}}{\sqrt{N(1+ \Delta_{n}^{2}(\varphi))}}\sum_{p}e^{ip\varphi}\big{(}|a_{p},n\rangle-\Delta_ {n}(\varphi)|b_{p},n+1\rangle\big{)}, \tag{20}\]  
\[|\Psi_{n,2}\rangle=\frac{e^{-i(n+1)\omega t}\cdot e^{-i\
u_{2}t}}{\sqrt{N(1+ \Delta_{n}^{2}(\varphi))}}\sum_{p}e^{ip\varphi}\big{(}\Delta_{n}(\varphi)|a_ {p},n\rangle+|b_{p},n+1\rangle\big{)}, \tag{21}\]  
where  
\[\Delta_{n}(\varphi)=\frac{g_{n}}{\left(t_{a}-t_{b}\right)\cos\varphi+\sqrt{ \left(t_{a}-t_{b}\right)^{2}\cos^{2}\varphi+g_{n}^{2}}}. \tag{22}\]  
The States (14) and (15) describe the travelling of transitions between the states \(\big{|}a_{p},n\rangle\) and \(\big{|}b_{p},n+1\rangle\) along the chain (so called Rabi-waves [41]). The periodicity of the lattice leads to a band structure of the energy spectrum of the dressed electrons in the entangled eigenstates \(|\Psi_{\alpha,n}(h)\rangle\) with the corresponding eigenenergies \(\hbar\
u_{1,2}(h,n)\). They are labeled by the discrete number of photons \(n\) and the continuous quasi-momentum \(h\). Under the influence of a dc field, weak enough not to induce inter-band transitions, the state \(|\Psi_{\alpha,n}(h_{0})\rangle\) evolves up to a phase factor into the state \(|\Psi_{\alpha,n}(h(t))\rangle\) with \(h(t)\) variation according to Equation (16). This evolution is periodic with a Bloch frequency \(\omega_{B}\) corresponding to the time required for the quasi-momentum to scan a full Brillouin zone. It leads to the exchange,  
\[\Delta_{n}(\varphi)\rightarrow\Delta_{n}(t)=\frac{g_{n}}{\left(t_{a}-t_{b} \right)\cos(\omega_{B}t-\varphi_{0})+\sqrt{\left(t_{a}-t_{b}\right)^{2}\cos^{ 2}(\omega_{B}t-\varphi_{0})+g_{n}^{2}}}. \tag{23}\]  
The distribution of photonic probabilities for the state \(|\Psi_{n,1}\rangle\) is given by  
\[p_{1}(m  
For photonic number variances we have  
\[\delta n_{1}=\delta n_{2}=\frac{\Delta_{n}^{2}(t)}{\left(1+\Delta_{n}^{2}(t) \right)^{2}}. \tag{27}\]  
Here, von Neumann entropy [44] accumulated at the given states of light is equal for both states and given by  
\[S_{n,1}(t)=S_{n,2}(t)=\frac{1}{1+\Delta_{n}^{2}(t)}\ln\biggl{(}\frac{1}{1+ \Delta_{n}^{2}(t)}\biggr{)}+\frac{\Delta_{n}^{2}(t)}{1+\Delta_{n}^{2}(t)}\ln \biggl{(}\frac{\Delta_{n}^{2}(t)}{1+\Delta_{n}^{2}(t)}\biggr{)}. \tag{28}\]  
Due to BO, characteristics of quantum light, such as the photonic probabilities, the average number of photons, and the Neiman entropy are oscillatory functions with Bloch frequency \(\omega_{B}\). The amplitude of these non-monochromatic oscillations is strongly dependent on the relation between penetrations of potential barrier in the ground and excited states.