Dynamical theory of single-photon transport through a qubit chain coupled to a one-dimensional nanophotonic waveguide

We study the dynamics of a single-photon pulse traveling through a linear qubit chain coupled to continuum modes in a one-dimensional (1D) photonic waveguide. We derive a time-dependent dynamical theory for qubits’ amplitudes and for transmitted and reflected spectra. We show that the requirement for the photon-qubit coupling to exist only for positive frequencies can significantly change the dynamics of the system. First, it leads to the additional photon-mediated dipole-dipole interaction between qubits which results in the violation of the phase coherence between them. Second, the spectral lines of transmitted and reflected spectra crucially depend on the shape of the incident pulse and on the initial distance between the pulse center and the first qubit in the chain. We apply our theory to one-qubit and two-qubit systems. For these cases we obtain the explicit expressions for the qubits’ amplitudes and for the photon radiation spectra as time tends to infinity. Specific calculations are performed for superconducting qubits operating in GHz frequency range. For the incident Gaussian wave packet we calculate the line shapes of transmitted and reflected photons.