Single photons versus coherent-state input in waveguide quantum electrodynamics: Light scattering, Kerr, and cross-Kerr effect

While the theoretical studies in waveguide quantum electrodynamics predominate with single-photon and two-photon Fock-state (photon number states) input, the experiments are primarily carried out using a faint coherent light. We create a theoretical toolbox to compare and contrast linear and nonlinear light scattering by a two-level or a three-level emitter embedded in an open waveguide carrying Fock-state or coherent-state inputs. We particularly investigate light transport properties and the Kerr and cross-Kerr nonlinearities of the medium for the two types of inputs. A generalized description of the Kerr and cross-Kerr effect for different types of inputs is formulated using the first-order correlation function to compare the Kerr and cross-Kerr nonlinearity between two photons in these models.

Figure 1

Schematics of (a) a 2LE and (b) a ladder-type 3LE side coupled to an open waveguide carrying (a) a probe or (b) a probe and a drive beam. The coupling strengths of the probe beam (red) and the drive beam (blue) with the emitter are $g_p$ and $g_d$, respectively.

Figure 2

Kerr vs cross-Kerr effect with Fock-state photons in waveguide QED. The linear, nonlinear (Kerr), and total phase shifts $({\varphi }_p^{\left(1\right)},{\varphi }_p^{\left(2\right)},{\varphi }_p)$ of two transmitted photons inside an open waveguide side coupled to a 2LE. The cross-Kerr phase shift $\delta {\varphi }_{pd}$ of a single-photon probe beam by a single-photon drive beam, where both beams interact with two allowed transitions of a ladder-type or a $V$-type 3LE. The parameters are $v_g=1,{\mathrm{\Gamma }}_p/{\omega }_{21}={\mathrm{\Gamma }}_d/{\omega }_{21}=0.1,{\mathrm{\Delta }}_d=0$, and $I_{1p}=I_{1d}=0.0125{\omega }_{21}/v_g$.