Few-photon transport in a waveguide coupled to a pair of colocated two-level atoms

We calculate the one- and two-photon scattering matrices of a pair of collocated nonidentical two-level atoms coupled to a waveguide. We show that by proper choice of a two-photon input, the background fluorescence by the atoms may be completely quenched, as a result of quantum interference, and that when the atoms’ detuning is smaller than their linewidths, extremely narrow fluorescence features emerge. Furthermore, the system emits a two-photon bound state which can display spatial oscillations or quantum beats, and can be tuned from bunched to antibunched statistics as the total photon energy is varied.

Figure 1

Schematic representation of a pair of nonidentical two-level atoms in a waveguide geometry. $k_{1,2}$ and $p_{1,2}$ denote the incoming and scattered photons, respectively. ${\Omega }_{1,2}$ and ${\tau }_{1,2}$ are the transition frequencies and decay times of the two atoms, respectively.

Figure 2

Plot of the normalized resonance fluorescence, $|B(k_{1,2},p_{1,2}){/\tau |}^2$, assuming ${\tau }_1={\tau }_2=\tau =1/\gamma$. ${\Omega }_c$, ${\Omega }_d$, and ${\Delta }_{i,o}$ are defined in the main text. Let $\overline{E}\equiv (E-2{\Omega }_c)\tau$ and ${\overline{\Delta }}_{i,o}\equiv {\Delta }_{i,o}\tau$. (a) $\overline{E}=3$, ${\Omega }_d=\gamma$. (b) $\overline{E}=3$, ${\Omega }_d=0.5\gamma$. (c) $\overline{E}=3$, ${\Omega }_d=0$. (d) $\overline{E}=0$.

Figure 3

$P_2^{\left(R\right)}\left(x\right)$, the joint detection probability density of the transmitted two-photon state. (a) $k_1={\Omega }_1$, ${\Omega }_d=2\gamma$, $(k_2-{\Omega }_2)=\{2,1,\frac{1}{4}\}\gamma$ in solid black, dotted blue, and dashed red curves, respectively. (b) $k_1={\Omega }_1$, ${\Omega }_d=6\gamma$, $(k_2-{\Omega }_2)=\{6,0.25\}\gamma$ in solid black and dashed red curves, respectively. In (a) and (b), insets show the frequency $k_2$ overlaid on the single-photon transmission $|{\overline{t}}_k{|}^2$. (c) $k_1={\Omega }_1$, $k_2={\Omega }_c=\{0.75,0.5,0.25\}\gamma$ for the black (solid), blue (dotted), and red (dashed) curves, respectively.