Single-photon scattering on a qubit. Space-time structure of the scattered field

We study the space-time structure of the scattered field induced by the scattering of a narrow single-photon Gaussian pulse on a qubit embedded in 1D open waveguide. For a weak excitation power we obtain explicit analytical expressions for space and time dependence of reflected and transmitted fields which are, in general, are different from plain travelling waves. The scattered field consists of two parts: a damping part which represent a spontaneous decay of the excited qubit and a coherent, lossless part. We show that for large distance $x$ from qubit and at times $t$ long after the scattering event our theory provides the result which is well known from the stationary photon transport. However, the approach to the stationary limit is very slow. The scattered field decreases as the inverse powers of $x$ and $t$ as both the distance from the qubit and the time after the interaction increase.