Input-output formalism for few-photon transport in one-dimensional nanophotonic waveguides coupled to a qubit

We extend the input-output formalism of quantum optics to analyze few-photon transport in waveguides with an embedded qubit. We provide explicit analytical derivations for one- and two-photon scattering matrix elements based on operator equations in the Heisenberg picture.

Figure 1

Schematic representation of two photons in a waveguide at frequencies $k$ and $p$ moving to the right toward a two-level atom with energy levels $|g\rangle$ and $|e\rangle$, respectively. $\mathrm{\Omega ̃}$ is the separation between the energy levels. Coupling of the two-level atom to the modes in the waveguide is proportional to ${\tau }^{-1}$. Long horizontal lines denote the waveguide geometry.

Figure 2

Linearization of a surface-plasmon-like waveguide dispersion relation $\mathrm{\omega ̃}(\beta )$ around a wave vector ${\beta }_0$. The slope of the line is equal to the group velocity $v_g$. The photon states in the text are assumed to have frequencies in the vicinity of ${\omega }_0$ so that the linearization is justified.