# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 5 Some Other Potential Experimental Implementations
### Josephson Junction
The tilt leads to BO of the particle in the quasi-charge space, which allows for the identification of the Bloch-frequency as dual plasma frequency \(\omega_{B}\Leftrightarrow\omega_{c}=\pi(\Delta_{0}L_{2})^{1/2}/e\). The inductive strong coupling of a Josephson junction with a microwave resonator leads to its dressing with microwave photons. It corresponds to the periodic emission-absorption of photons and inter-band transitions by the fictitious charged particle (RO). Thus, the qualitative picture of junction behavior is similar to the BO of dressed electrons considered above: The periodic inter-band transitions (RO of fluxonium) are accompanied by the charge oscillations through the bias-voltagedriving. For formal identification of the coupling parameters of the system, \"junction + microwave resonator\", it is necessary to analyze in more detail the total Hamiltonian. It reads  
\[\hat{H}=\hat{H}_{A}+\hat{H}_{ph}+\hat{H}_{I,ph}+\hat{H}_{I,dc}, \tag{39}\]  
where \(\hat{H}_{A}\) is a Hamiltonian of the Josephson junction in an inductive environment, \(\hat{H}_{ph}\) is Hamiltonian of microwave photons in the resonator, \(\hat{H}_{I,ph}\) is a Hamiltonian of junction-cavity interaction, and \(\hat{H}_{I,dc}\) is a support of biased voltage. For further analysis, it is convenient to use the secondary quantization technique. The Hamiltonian of photons reads  
\[\hat{H}_{ph}=\hat{\Phi}_{r}^{2}/2L_{r}+\hat{Q}_{r}^{2}/2C_{r}, \tag{40}\]  
where \(\hat{Q}_{r}=i\sqrt{\frac{\hbar}{2Z_{r}}}(\hat{a}-\hat{a}^{+})\), \(\hat{\Phi}_{r}=i\sqrt{\frac{\hbar Z_{r}}{2}}(\hat{a}+\hat{a}^{+})\), \(\hat{a},\hat{a}^{+}\) are creation-annihilation operators for microwave photons, which satisfy the conventional bosonic commutative relations. By following the concept of quasi-charge-phase duality, the Fourier-Bloch states \(\Psi_{\alpha,Q}(\varphi)\) exchanged by the Wannier functions (see Appendix A) and the Hamiltonian is written in the Wannier-basis. We denote the two first states with quasi-charge \(Q\) as \(\ket{e_{Q}}\) and \(\ket{g_{Q}}\) and introduce the raising operators \(\hat{\sigma}_{Q}=\ket{e_{Q}}\langle g_{Q}|\) and \(\hat{\sigma}_{Q}^{+}=\ket{g_{Q}}\langle e_{Q}|\). The atomic Hamiltonian may be rewritten as \(\hat{H}_{A}=\hbar\omega_{p}\underline{\omega}_{Q2}/2\), where \(\hat{\sigma}_{Q2}\) is Pauli inversion matrix. For junction-cavity interaction we have [54]  
\[\hat{H}_{I}=\frac{2e}{\hbar}G\hat{\Phi}_{r}\phi_{eg}\underline{\sum_{Q}}\left( \hat{\sigma}_{Q}^{+}+\hat{\sigma}_{Q}^{-}\right), \tag{41}\]  
where \(G\approx(\hbar/2e)^{2}L_{1}(L_{2}L_{r})^{-1}\), \(\phi_{eg}=\langle g_{Q}|\hat{\sigma}_{Q}^{+}|e_{Q}\rangle\) is the matrix element of effective dipole moment. The Hamiltonian is written in rotating-wave approximation. It is equal to the Hamiltonian (3) if the coupling coefficient identified with the value  
\[g\Leftrightarrow\frac{\left|\phi_{eg}\right|^{2}}{\sqrt{2}}\sqrt{\frac{R_{0} }{2\pi Z_{r}}}\frac{L_{1}}{L_{2}}\omega_{p}, \tag{42}\]  
where \(R_{0}\) is the quantum of resistance. The validity of rotating-wave approximation implies all relevant energies to be smaller than the plasmonic frequency of the Josephson junction and resonant frequency of resonator. In particular, we impose the characteristic energy, \(E_{L_{I}}\), associated with the inductance \(L_{2}\) to be smaller than \(\hbar\omega_{p}\). It corresponds to the values \(L_{2}\approx 200\) nH, which are quite reachable [56]. Second, we need also \(g<<\omega_{p}\). For standard value \(Z_{r}=50\) k\(\Omega\)[56] we have \(L_{1}/L_{2}\approx 10^{-3}\) for this condition been imposed. The coherence time and energy relaxation time both are of the order \(\cong\) us [56]. Thus, the number of BO flops is high enough for their experimental observation in the fluxonium atom in microwave frequency range. The real cases may be described by lossless model considered above.