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We study the dynamics of a single-photon pulse traveling through a linearqubit chain coupled to continuum modes in a one-dimensional (1D) photonicwaveguide. We derive a time-dependent dynamical theory for qubits' amplitudesand for transmitted and reflected spectra. We show that the requirement for thephoton-qubit coupling to exist only for positive frequencies can significantlychange the dynamics of the system. First, it leads to the additionalphoton-mediated dipole-dipole interaction between qubits which results in theviolation of the phase coherence between them. Second, the spectral lines oftransmitted and reflected spectra crucially depend on the shape of the incidentpulse. We apply our theory to one-qubit and two-qubit systems. For these twocases, we obtain the explicit expressions for the qubits' amplitudes and forthe photon radiation spectra as time tends to infinity. For the incidentGaussian wave packet we calculate the line shapes of transmitted and reflectedphotons.

Dynamical theory of single-photon transport through a qubit chain  coupled to a one-dimensional nanophotonic waveguide. Beyond the Markovian  approximation