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We report a theoretical study of the electromagnetic waves (EWs) propagationthrough an array of superconducting qubits, i.e. coherent two-level systems,embedded in a low-dissipative transmission line. We focus on the near-resonantcase as the frequency of EWs $\omega \simeq \omega_q$, where $\omega_q$ is thequbit frequency. In this limit we derive the effective dynamic nonlinear waveequation allowing one to obtain the frequency dependent transmissioncoefficient of EWs, $D(\omega)$. In the linear regime and a relatively widefrequency region we obtain a strong resonant suppression of $D(\omega)$ in bothcases of a single qubit and chains composed of a large number of denselyarranged qubits. However, in narrow frequency regions a chain of qubits allowsthe resonant transmission of EWs with greatly enhanced $D(\omega)$. In thenonlinear regime realized for a moderate power of applied microwave radiation,we predict and analyze various transitions between states characterized by highand low values of $D(\omega)$. These transitions are manifestations ofnonequilibrium steady states of an array of qubits achieved in this regime.

Electromagnetic waves propagation through an array of superconducting qubits: manifestations of non-equilibrium steady states of qubits