Single-photon scattering on a two-qubit system. Spatio-temporal structure of the scattered field

In this paper, we study the spatiotemporal distribution of the photon electric field produced by the scattering of a single photon narrow pulse from a system of two identical qubits coupled to continuum modes in a one-dimensional (1D) open waveguide. We derive the time-dependent dynamical equations for qubits' and photon amplitudes which allow the calculation of the photon backward and forward scattering fields in the whole space: before qubits, between qubits, and behind the qubits. The scattered field consists of several contributions that describe a free field of incoming photon, a spontaneous exponential decay of excited qubits, a slowly decaying part that dies out as the inverse powers of $t$, and a lossless part that represents a steady state solution as $t\rightarrow\infty$. For our system, we find the transmittance and reflectance fields as both time and distance from the qubits tend to infinity. We show that as the time after the event of scattering tends to infinity, the steady state photon the field is being formed in the whole one-dimensional space. If the distance $d$ between qubits is equal to the integer of the wavelength $\lambda$, the field energy exhibits temporal beatings between the qubit frequency $\Omega$ and the photon frequency $\omega_S$ with the period $T=2\pi/(\omega_S-\Omega)$.