One-photon wave packet interacting with two separated atoms in a one-dimensional waveguide: Influence of virtual photons

We present a theoretical study of a one-photon wave packet scattered by two atoms in a one-dimensional waveguide. We investigate the role of terms beyond the rotating-wave approximation to correctly take into account the effects of the virtual photons that are exchanged between the atoms. These terms are shown to drastically influence the reflected and the transmitted fields, imposing strict constraints on their temporal envelopes.

Figure 1

(a) Configuration of the atoms and the initial photon wave packet in the waveguide. The dimension of the waveguide transverse section is $d$. The atoms are in the ground level and are separated by a distance $l$. The resonant wavelength is ${\lambda }_0$. (b) Atoms + field states involved in the interaction process with RWA or non-RWA nature of the coupling.

Figure 2

Quantum paths leading to the modification of ${\beta }_1$, the excited-state amplitude of atom 1. Paths are associated with (a) absorption of the initial photon, (b) relaxation of atom 1 with emission of a photon in the forward or backward direction, (c) relaxation of atom 2 with emission of a photon in the backward ($M_1$ amplitude) or forward ($M_3$ amplitude) direction and that interacts further with atom 1 (RWA terms), and (d) excitation of atom 1 with emission of a photon in the backward ($M_2$ amplitude) or forward ($M_4$ amplitude) direction and that interacts further with atom 2 (non-RWA terms). Similar photon diagrams exist for the modification of ${\beta }_2$.

Figure 3

Pulse-area theorem. Temporal behavior of the transmitted field with (a) real part and (b) imaginary part. We represent the curves for several ratios of $\Gamma /\Delta$. The interatomic distance is $k_{0}l=\frac{\pi }{4}$. The temporal area vanishes in all cases. In (c) the corresponding spectra are represented, and the incident spectrum is represented by the dashed-dotted line. The resonance frequency is not transmitted.