We present a full multimode treatment of the interaction of the quantized radiation field with a single two-level atom in a one-dimensional waveguide configuration. Starting with an incident pulse consisting of an arbitrary (finite) number of photons in a general initial state, we derive the equations of motion and a closed-form expression for the shape of the pulse after its interaction with the atom. We then specialize our results to the two-photon case where a number of analytical results can be derived, for both unidirectional and bidirectional systems. We study the effects of different pulse shapes, the manifestations of the entangled, so-called bound state of the two photons, single- and two-photon detection probabilities, and provide simple approximate results for the strong-coupling (or adiabatic, or long-pulse) regime. We also discuss the requirements for a true unidirectional setup, and the application of such a setup to photon sorting proposed by Witthaut, Lukin, and S\o{}rensen [Europhys. Lett. 97, 50007 (2012)].
We consider an infinite waveguide supporting a single propagating mode for which the excitation (say the electric or the magnetic field) is assigned in a given section assumed as the origin of the coordinate z along its axis. In steady state the state evolution along z is the solution of the spatial Cauchy problem along such coordinate. As soon as the radiation involves a low number of photons, or even reduces to a single one, the classical treatment must be replaced by a quantum one. Moreover the effect of dissipation must be taken into account by either a microscopic treatment of the properties of the dielectric material and of the metallic boundaries, or, as long as only the mathematical form of the equations be of interest, by using a spatial version of Lindblad equations. We choose here the second alternative. From the obtained equations it is possible to extend the treatment to multimode finite terminated waveguides.
Waveguide Quantum Electrodynamics (WQED) offers a suitable stage for controlling the interaction of light with atoms, allowing for collective phenomena such as super- and subradiance. In a chiral waveguide setup, the quantum state evolves through all the Hilbert space, rendering an exact theoretical treatment exponentially hard and unobtained to date for more than
∼20 atoms. In this work, we use a computationally efficient higher order mean-field approximation to model the radiation dynamics in a chirally coupled array of atoms, showing good agreement with recent experimental results. Further, based on a perturbative approximation of the full dynamics, we develop an analytical solution that captures photon statistics for a moderate atom number,
N, and a homogeneous atom-waveguide coupling,
β. Finally, we show that capturing the onset of second-order coherence from a fully inverted state requires a fourth-order mean-field approximation, as lower-order treatments fail to account for the necessary four-body correlations. These results illustrate the complex behavior of a symmetry-lacking system, and the methods discussed here provide systematic analytical solutions to which semi-classical methods such as the cumulant expansion or the truncated Wigner approximation can be benchmarked.