Resonance fluorescence in a waveguide geometry

We show how to calculate the first- and second-order statistics of the scattered fields for an arbitrary intensity coherent-state light field interacting with a two-level system in a waveguide geometry. Specifically, we calculate the resonance fluorescence from the qubit, using input-output formalism. We derive the transmission and reflection coefficients, and illustrate the bunching and antibunching of light that is scattered in the forward and backward directions, respectively. Our results agree with previous calculations on one- and two-photon scattering as well as those that are based on the master equation approach.

Figure 1

Schematic of the system under investigation. A right-going coherent state (${\mathit{r}}_{\mathrm{in}}$) at frequency $k$ and an arbitrary intensity, propagating in a waveguide denoted by the long horizontal lines, is incident on a two-level system with energy separation $\Omega$ and a spontaneous emission rate ${\tau }^{-1}$. After interacting with the qubit, the transmitted (${\mathit{r}}_{\mathrm{out}}$) and the reflected (${\ell }_{\mathrm{out}}$) light has both a coherent ($\omega =k$) and an incoherent ($\omega \ne k$) component.

Figure 2

Coherent part of the transmitted (solid) and reflected (dashed) fluorescence spectrum for $R={\omega }_{\mathrm{R}}{\tau }^{\prime }=\{0.1,2,5\}$ corresponding to the blue, red, and green curves respectively. The reflected fluorescence for $R=5$ (dashed green curve) is plotted after being multiplied by 5. Inset shows the incoherent part ($\omega \ne k$ case as depicted in Fig. 1) of the spectrum with the Mollow triplet for zero detuning ($D=0$).

Figure 3

Plots of $g^{\left(2\right)}$ for $D=2$ and $R=\{0.2,4\}$ for the red and black curves, respectively. The dashed blue curve is the normalized two-photon wave function. (a) One-mode case; (b) two-mode, transmitted case; (c) two-mode, reflected case. As can be seen, the two-photon calculations are indistinguishable from resonance fluorescence ones for $R=0.2$ but not for $R=4$.