# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## Appendix A Derivation of Hamiltonian in the Wannier-Fock Basis
The total Hamiltonian may be presented as \(\hat{H}=\hat{H}_{0}+\hat{H}_{ph}+\hat{H}_{I,ph}+\hat{H}_{I,dc}\), where the second component \(\hat{H}_{ph}=\hbar\omega\hat{a}^{+}\hat{a}\) corresponds to the free single-mode photons motion, and \(\hat{a},\hat{a}^{+}\) are creation and annihilation operators of photons. We describe the atom-photon interaction in the dipole approximation by the third term \(\hat{H}_{I,ph}=-e\hat{\mathbf{E}}(\mathbf{r},t)\cdot\hat{\mathbf{r}}\)[26], where  
\[\hat{\mathbf{E}}=\mathbf{e}\sqrt{\frac{\hbar\omega}{2\epsilon_{0}V}}\left( \hat{a}+\hat{a}^{+}\right) \tag{10}\]  
is the single-mode quantum light operator of electric field, \(V\) is the normalizing volume, and \(\mathbf{e}\) is the unit polarization vector. The quantum state of light will be described in terms of photons number states (Fock-states) \(|m\rangle\). The basis states should be used in our analysis correspond to the independent photons number states and Wannier electrons \(|\alpha_{j},m\rangle\) (Wannier-Fock basis). Let us proceed to the calculation of matrix elements  
\[H_{I,ph}^{mm,lj,aa^{\prime}}=\langle m,\alpha_{I}|\hat{H}_{I,ph}\big{|}\alpha^{ \prime}_{j},n\rangle=-e\sqrt{\frac{\hbar\omega}{2\epsilon_{0}V}}\langle\alpha _{I}|\mathbf{e}\cdot\hat{\mathbf{r}}_{j}\big{|}\alpha^{\prime}_{j}\rangle \langle m|\left(\hat{a}+\hat{a}^{+}\right)|n\rangle. \tag{11}\]  
Let us shift the origin for a given \(j\) to the center of the \(j\)-th atom via the relation \(\hat{\mathbf{r}}_{j}=\mathbf{R}_{j}\hat{I}+\hat{\mathbf{r}}^{\prime}\). Using the Wannier functions orthogonality for different electron bands, we have \(\big{\langle}\alpha_{j}\big{|}\hat{\mathbf{r}}_{j}\big{|}\alpha^{\prime}_{j} \big{\rangle}=\mathbf{R}_{j}\delta_{aa^{\prime}}+\big{\langle}\alpha_{j}|\hat {\mathbf{r}}^{\prime}|\alpha^{\prime}_{j}\big{\rangle}\). Together with orthogonality of the Fock states it makes (11)  
\[H_{I,ph}^{mn,jl,aa^{\prime}}=-e\sqrt{\frac{\hbar\omega}{2\epsilon_{0}V}}\Big{(} \sqrt{n}\delta_{m,n-1}+\sqrt{n+1}\delta_{m,n+1}\Big{)} \begin{cases}[(\mathbf{e }\cdot\mathbf{R}_{j})\delta_{aa^{\prime}}+(\mathbf{e}\cdot\mathbf{d}_{aa^{ \prime}})],&j=l\\ \delta_{aa^{\prime}}\widetilde{d}_{j,a},&l=j+1\\ 0,&\text{others}\end{cases} . \tag{12}\]  
Here \(\mathbf{d}_{aa^{\prime}}=\big{\langle}\alpha_{j}\big{|}\mathbf{r}_{j}\big{|} \alpha^{\prime}_{j}\big{\rangle}\) is the matrix of the dipole moments, \(\widetilde{d}_{j,a}=\big{\langle}\alpha_{j}\big{|}(\mathbf{e}\cdot\mathbf{r}_ {j})\big{|}\alpha_{j+1}\big{\rangle}\). The second term at the top line characterizes quantum transitions in the separate atom. For real atoms (excluding hydrogen) only off-diagonal elements, \(\mathbf{d}_{aa^{\prime}}\) are non-zero due to the inversion symmetry [70]. For some types of artificial atoms (such as quantum dots) the inversion symmetry is not obligatory. In this case on-diagonal elements of dipole matrix may be non-zero [37], which corresponds to the special type of intra-level motion [37]. The middle line in (12) corresponds to the interaction of neighboring atoms stimulated by photons (photon-assistant tunneling over the same level). The last term \(\hat{H}_{I,dc}=-e\mathbf{E}_{dc}(\mathbf{r},t)\cdot\hat{\mathbf{r}}\) describes the dipole interaction of the chain with the dc field. Its matrix elements are  
\[H_{I,dc}^{mn,lj,aa^{\prime}}=\langle m,\alpha_{I}|\hat{H}_{I,dc}\big{|}\alpha^{ \prime}_{j},n\rangle=-e\delta_{mn}\mathbf{E}_{dc}\cdot\langle\alpha_{I}|\hat {\mathbf{r}}|\alpha^{\prime}_{j}\rangle. \tag{13}\]  
Using the orthogonality of Wannier states, we obtain  
\[H_{I,dc}^{mn,jl,aa^{\prime}}=-e\delta_{mn}\mathbf{E}_{dc} \begin{cases}\mathbf{ R}_{j}\delta_{aa^{\prime}}+\mathbf{d}_{aa^{\prime}},&j=l\\ \delta_{aa^{\prime}}\widetilde{d}_{j,a}\mathbf{e},&l=j+1\\ 0,&\text{others}\end{cases} . \tag{14}\]  
The first term in the top line corresponds to BO, the second one describes the Stark-effect [70].