Non-Hermitian Hamiltonian approach to the microwave transmission through one- dimensional qubit chain

We investigate the propagation of microwave photons in a one-dimensional open waveguide interacting with a number of artificial atoms (qubits). Within the formalism of projection operators and non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme for the calculation of the transmission and reflection factors of the microwave signal in a waveguide which contains an arbitrary number \emph{N} of non-interacting qubits. We considered in detail the resonances and photon mediated entanglement for two and three qubits in a chain. We showed that in non Markovian case the resonance widths, which define the decay rates of the entangled state, can be much smaller than the decay width of individual qubit. It is also shown that for identical qubits in the long wavelength limit a coherent superradiant state is formed with the width being equal to the sum of the widths of spontaneous transitions of \emph{N} individual qubits. The results obtained in the paper are of general nature and can be applied to any type of qubits. The specific properties of the qubit are only encoded in the two parameters: the qubit energy $\Omega$ and the rate of spontaneous emission $\Gamma$.