Influence of Impurity on the Rate of Single Photon Superradiance in Disordered N Qubit Chain

We investigate the rate of superradiant emission for a number of artificial atoms (qubits) embedded in a one-dimensional open waveguide. More specifically, we study the 1D (N+1)- qubit chain where N qubits are identical in respect to their excitation frequency $\Omega$ but have different rates of spontaneous emission $\Gamma_n$, and a single impurity qubit which is different from N qubits by its excitation frequency $\Omega_P$ and rate of spontaneous emission $\Gamma_P$. This system is shown to have two hybridized collective states which accumulates the widths of all qubits. The energy spectrum of these states and corresponding probabilities are investigated as the function of the frequency detuning between the impurity and other qubits in a chain. It is shown that the inclusion of impurity qubit alter the resonance widths of the system only in a narrow range of the frequency detuning between qubits and impurity, where the resonance widths experience a significant repulsion. The photon transmission through disordered N- qubit chain with impurity qubit is also considered. It is shown that a single photon transport through this system is described by a simple expression which predicts for specific photon frequency the existence of a complete transmission peak and transparency window between frequencies $\Omega$ and $\Omega_P$.