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

Abstract

We report a theoretical study of the electromagnetic waves (EWs) propagation through an array of superconducting qubits, i.e. coherent two-level systems, embedded in a low-dissipative transmission line. We focus on the near-resonant case as the frequency of EWs $\omega \simeq {\omega }_q$, where ${\omega }_q$ is the qubit frequency. In this limit we derive the effective dynamic nonlinear wave equation allowing one to obtain the frequency dependent transmission coefficient of EWs, $D(\omega )$. In the linear regime and a relatively wide frequency region we obtain a strong resonant suppression of $D(\omega )$ in both cases of a single qubit and chains composed of a large number of densely arranged qubits. However, in narrow frequency regions a chain of qubits allows the resonant transmission of EWs with greatly enhanced $D(\omega )$. In the nonlinear regime realized for a moderate power of applied microwave radiation, we predict and analyze various transitions between states characterized by high and low values of $D(\omega )$. These transitions are manifestations of nonequilibrium steady states of an array of qubits achieved in this regime.