# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 2 Statement of the Problem and Calculation Technique
### Hamiltonian in Wannier-Fock Basis  
Let us denote with \(\left|a_{p}\right\rangle\) and \(\left|b_{p}\right\rangle\), the Wannier wave functions centered at \(p\)-th atom in the excited and ground states, respectively (Figure 1). The two neighboring atoms are coupled via the electron tunneling, such that only intra-band tunnel transitions are permitted. It means that the electron due to the tunneling can go from state \(\left|a_{p}\right\rangle\) to state \(\left|a_{p+1}\right\rangle\) and from state \(\left|b_{p}\right\rangle\) to state \(\left|b_{p+1}\right\rangle\) only. The transitions between ground and excited states of different atoms are forbidden: \(\big{\langle}\,a_{p\pm 1}\big{|}b_{p}\big{\rangle}\approx 0\). Let the chain be exposed to a single-mode quantum light, which electric field operator is \(\mathbf{\hat{E}}=\mathbf{\hat{E}}^{(+)}+\mathbf{\hat{E}}^{(-)}=\mathbf{e} \sqrt{\hbar\omega/2\varepsilon_{0}V}(\hat{a}+\hat{a}^{+})\), where \(V\) is the normalizing volume, \(\mathbf{e}\) is the unit polarization vector, \(\mathbf{\hat{E}}^{(+)}=\mathbf{e}(\hbar\omega/2\varepsilon_{0}V)^{1/2}\hat{a}\), \(\mathbf{\hat{E}}^{(-)}=\mathbf{e}(\hbar\omega/2\varepsilon_{0}V)^{1/2}\hat{a}^ {+}\) are positive and negative frequency components of field operator, \(\hat{a},\hat{a}^{+}\) are creation-annihilation operators, respectively (an \(\exp(-i\omega t)\) time dependence of the light is implicit). The field is assumed to be homogeneous over the chain axis. Such assumption corresponds to the chain excited by the normally incident wide laser beam, or specially-symmetric eigen-mode of microcavity, photonic crystal; etc. The chain is driven by an electrostatic (dc) field directed along the axis with responsibility for BO. We will consider the case of dipole interaction in the regime of strong coupling and assume the resonant condition \(\omega_{0}\approx\omega\) to be fulfilled. The system under consideration exhibits complex single-particle BO of the electrons entangled with photons, for which the theoretical framework will be introduced. The system is described by the total Hamiltonian (see Appendix A):  
\[\hat{H}=\hat{H}_{0}+\hat{H}_{ph}+\hat{H}_{T}+\hat{H}_{I,ph}+\hat{H}_{I,dc}. \tag{1}\]  
Here, the first term represents the free motion of the chain with absence of both light and dc field. It is given by \(\hat{H}_{0}=(\hbar\omega_{0}/2)\sum_{p}\hat{\sigma}_{zp}\), where \(\hat{\sigma}_{zp}=\big{|}a_{p}\big{\rangle}\big{\langle}a_{p}\big{|}-\big{|} b_{p}\big{\rangle}\big{\langle}b_{p}\big{|}\). The second term \(\hat{H}_{ph}=\hbar\omega\hat{a}^{+}\hat{a}\) is the Hamiltonian of free electromagnetic field. The term  
\[ \begin{array}{rl}\hat{H}_{T}=&-t_{a}\underset{p}{\sum}\big{(}\big{|}a_{p} \big{\rangle}\big{\langle}a_{p+1}\big{|}+\big{|}a_{p}\big{\rangle}\big{\langle} a_{p-1}\big{|}\big{)}-\\ &-t_{b}\underset{p}{\sum}\big{(}\big{|}b_{p}\big{\rangle}\big{\langle}b_{p+1} \big{|}+\big{|}b_{p}\big{\rangle}\big{\langle}b_{p-1}\big{|}\big{)}\end{array}  \tag{2}\]  
is Hamiltonian of electron tunneling, \(t_{a,b}\) are the penetration energies of potential barrier at the excited and ground states, respectively. The component  
\[\hat{H}_{I,ph}=\hbar\delta\underset{p}{\sum}\hat{\sigma}_{p}^{+}\hat{a}+\text{ H.c.} \tag{3}\]  
describes the atom-light interaction, where \(g=-\sqrt{\omega/2\hbar\varepsilon_{0}V}(\mathbf{e}\cdot\mathbf{d}_{ab})\) is the interaction constant, \(\mathbf{d}_{ab}\) is the dipole moment, H.c. means Hermitian conjugation. The transition dipole moments in the chain is assumed to be vectors of identical values and orientations. The operators \(\hat{\sigma}_{p}^{+}=\big{|}a_{p}\big{\rangle}\big{\langle}b_{p}\big{|}\), \(\hat{\sigma}_{p}^{-}=\big{|}b_{p}\big{\rangle}\big{\langle}a_{p}\big{|}\) are creation-annihilation operators of excited state in the \(p\)-th atom. The Hamiltonian (3) is written in the RWA form [26].