# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## Appendix A Derivation of Hamiltonian in the Wannier-Fock Basis  
The Hamiltonian of conductive electron movement in the chain associated with the interatomic tunneling, which is described by the periodic potential \(V(\mathbf{r})\), is  
\[\hat{H}_{0}=\frac{\hat{\mathbf{p}}^{2}}{2m}+V(\mathbf{r}). \tag{1}\]  
The eigenmodes of Hamiltonian (1) are two Bloch modes correspondent to valence (\(b\)) and conductive (\(a\)) zones and denoted as \(\psi_{\alpha,h}(\mathbf{r})=|\alpha,h\rangle\), \(\alpha=a\), \(b\), \(h\) being a scalar quasi-momentumdirected along the chain. Because the electrons are strongly confined inside the atoms, we will use as a basis, Wannier functions \(\phi_{\alpha,\mathbf{R}_{j}}(\mathbf{r})=\big{|}a_{j}\big{\rangle}\)[69], defined as a linear combinations of Bloch modes  
\[\phi_{\alpha,\mathbf{R}_{j}}(\mathbf{r})=\frac{1}{\sqrt{N}}{\sum_{h}e^{-ih( \mathbf{e}\cdot\mathbf{R}_{j})}\psi_{\alpha,h}(\mathbf{r})}. \tag{10}\]  
Let us mention some properties of Wannier functions, which are important for our analysis. Wannier function \(\phi_{\alpha,\mathbf{R}_{j}}(\mathbf{r})\) is strongly localized inside the \(j\)-th atom. Inversion of Equation (10) gives the expression of Bloch states in the terms of Wannier functions:  
\[\psi_{\alpha,h}(\mathbf{r})=\frac{1}{\sqrt{N}}{\sum_{\mathbf{R}_{j}}e^{ih( \mathbf{e}\cdot\mathbf{R}_{j})}\phi_{\alpha,\mathbf{R}_{j}}(\mathbf{r}- \mathbf{R}_{j})}. \tag{11}\]  
Wannier functions satisfy the orthogonality relation \(\big{\langle}\alpha_{j}\big{|}\alpha_{l}\big{\rangle}=\delta_{\alpha\alpha^{ \prime}}\delta_{jl}\). Projecting \(\hat{H}_{0}\) on the Bloch modes and using (11), we obtain the dispersion law for free-tunneling electron in the next form:  
\[\varepsilon_{\alpha}(h)=\big{\langle}\alpha,h\big{|}\hat{H}_{0}\big{|}\alpha, h\big{\rangle}=\frac{1}{N}{\sum_{j,l}e^{-ih\mathbf{e}\cdot(\mathbf{R}_{j}- \mathbf{R}_{l})}\big{\langle}\alpha_{j}\big{|}\hat{H}_{0}\big{|}\alpha_{l} \big{\rangle}}. \tag{12}\]  
Denoting  
\[\big{\langle}\alpha_{j}\big{|}\hat{H}_{0}\big{|}\alpha_{l}\big{\rangle}=\left\{ \begin{array}{cc}\varepsilon_{\alpha}(0),&j=l\\ t_{\alpha}(\mathbf{R}_{j}-\mathbf{R}_{l}),&j\
eq l\end{array} \right., \tag{13}\]  
we can rewrite (12) in the following form:  
\[\varepsilon_{\alpha}(h)=\varepsilon_{\alpha}(0)+\frac{1}{N}\sum_{ \begin{array} []{c}j,l\\ j\
eq l\end{array} }e^{-ih\mathbf{e}\cdot(\mathbf{R}_{j}-\mathbf{R}_{l})}t_{ \alpha}\big{(}\mathbf{R}_{j}-\mathbf{R}_{l}\big{)}. \tag{14}\]  
The Hamiltonian (12) may be presented with the Wannier basis in the form of a block diagonal matrix  
\[\hat{H}_{0}=\left( \begin{array}{cc}\hat{H}_{0a}&0\\ 0&\hat{H}_{0b}\end{array} \right), \tag{15}\]  
with diagonal-on elements  
\[\hat{H}_{0\alpha}=\varepsilon_{\alpha}(0){\sum_{j}}|\alpha_{j}\big{\rangle} \big{\langle}\alpha_{j}\big{|}+\sum_{ \begin{array}{c}j,l\\ j\
eq l\end{array} }t_{\alpha}(\mathbf{R}_{j}-\mathbf{R}_{l})\big{|}\alpha_{l} \big{\rangle}\big{\langle}\alpha_{j}\big{|}, \tag{16}\]  
where \(\alpha=a,b\). The tunneling coupling exist only between the neighboring atoms (tight-binding approximation). Thus, we omit for brevity the spatial arguments in tunneling matrix elements: \(t_{a,b}\big{(}\mathbf{R}_{j}-\mathbf{R}_{j+1}\big{)}=t_{a,b}\), \(t_{a,b}\big{(}\mathbf{R}_{j}-\mathbf{R}_{j-1}\big{)}=t_{a,b}^{*}\). The Hamiltonian (15) subject to (16) reads  
\[ \begin{array}{c}\hat{H}_{0}=\varepsilon_{b}(0){\sum_{j}}\big{|}b_{j}\big{\rangle} \big{\langle}b_{j}\big{|}+\varepsilon_{a}(0){\sum_{j}}\big{|}a_{j}\big{\rangle} \big{\langle}a_{j}\big{|}+\\ \sum_{j}\big{(}t_{b}\big{|}b_{j}\big{)}\ \ \big{\langle}b_{j+1}\big{|}+t_{b}^{*} \big{|}b_{j}\big{\rangle}\big{\langle}b_{j-1}\big{|}\big{\rangle}+\\ \sum_{j}\big{(}t_{a}\big{|}a_{j}\big{\rangle}\ \ \big{\langle}a_{j+1} \big{|}+t_{a}^{*}\big{|}a_{j}\big{\rangle}\big{\langle}a_{j-1}\big{|}\big{)} \end{array}  \tag{17}\]  
Hamiltonian (17) in the secondary quantization form is equivalent to the sum of intra-level term and tunneling Hamiltonian.